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# Number of groups of order n

Given a positive integer
 n
, it is not a simple matter to determine how many isomorphism types of groups of order
 n
there are. If
 n
is the square of a prime, then there are exactly two possible isomorphism types of groups of order
 n
, both of which are Abelian. If
 n
is a higher power of a prime, then results of Graham Higman and Charles Sims give asymptotically correct estimates for the number of isomorphism types of groups of order
 n
, and the number grows very rapidly as the power increases.

## Cyclic groups

Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group. Depending on the prime factorization of
 n
, some restrictions may be placed on the structure of groups of order
 n
, as a consequence, for example, of results such as the Sylow theorems. For example, every group of order
 p q
is cyclic when
 p < q
are primes with
 q  −  1
not divisible by
 p
. The cyclic numbers (
 n
such that there is just one group of order
 n
) are the numbers
 n
such that
 n
and
 φ (n)
are relatively prime.

## Solvable groups

If
 n
is squarefree, then any group of order
 n
is solvable. A theorem of William Burnside, proved using group characters, states that every group of order
 n
is solvable when
 n
is divisible by fewer than three distinct primes. By the Feit–Thompson theorem, which has a long and complicated proof, every group of order
 n
is solvable when
 n
is odd. For every positive integer
 n
, most groups of order
 n
are solvable. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and
 12
solvable groups of order
 60
) but the proof of this for all orders uses the classification of finite simple groups.

## Simple groups

For any integer there are at most
 2
simple groups of that order, and there are infinitely many pairs of non-isomorphic simple groups of the same order.

## Number of Abelian groups of order n

The number of Abelian groups of order
 n
is multiplicative, i.e.
 a (m n) = a (m) a (n), (m, n) = 1
. The number of Abelian groups of order
 p k
(prime powers), is the number of partitions of
 k
(A000041). Thus
${\displaystyle a(n)=a{\bigg (}\prod _{i=1}^{\omega (n)}{p_{i}}^{\alpha _{i}}{\bigg )}=\prod _{i=1}^{\omega (n)}p(\alpha _{i}),\,}$
where
 p (k)
is the number of partitions of
 k
.

(...)

## Table of number of distinct groups of order n

Number of distinct groups of order
 n

Order
 n
Prime
factorization
of
 n
 ω (n)
Number of
groups[1]
Number of
simple
groups
Number of
Abelian
groups

 ω (n)

 i  = 1
p (αi)
Number of
non-Abelian
groups
Number of
solvable
groups
Number of
non-solvable
groups
Comment
1 0 1   1 0   0
2
 2 1
1 1   1 0   0 Order
 p
3
 3 1
1 1   1 0   0 Order
 p
4
 2 2
1 2   2 0     Order
 p 2
5
 5 1
1 1   1 0   0 Order
 p
6
 2 1  ⋅   3 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
7
 7 1
1 1   1 0   0 Order
 p
8
 2 3
1 5   3 2     Order
 p 3
9
 3 2
1 2   2 0   0 Order
 p 2
10
 2 1  ⋅   5 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
11
 11 1
1 1   1 0   0 Order
 p
12
 2 2  ⋅   3 1
2 5   2 3     NOT squarefree
13
 13 1
1 1   1 0   0 Order
 p
14
 2 1  ⋅   7 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
15
 3 1  ⋅   5 1
2 1   1 0   0 Order
 p q
with
 p < q
primes and
 q  −  1
NOT divisible by
 p
16
 2 4
1 14   5 9     Order
 p 4
17
 17 1
1 1   1 0   0 Order
 p
18
 2 1  ⋅   3 2
2 5   2 3     NOT squarefree
19
 19 1
1 1   1 0   0 Order
 p
20
 2 2  ⋅   5 1
2 5   2 3     NOT squarefree
21
 3 1  ⋅   7 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
22
 2 1  ⋅   11 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
23
 23 1
1 1   1 0   0 Order
 p
24
 2 3  ⋅   3 1
2 15   3 12     NOT squarefree
25
 5 2
1 2   2 0   0 Order
 p 2
26
 2 1  ⋅   13 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
27
 3 3
1 5   3 2   0 Order
 p 3
28
 2 2  ⋅   7 1
2 4   2 2     NOT squarefree
29
 29 1
1 1   1 0   0 Order
 p
30
 2 1  ⋅   3 1  ⋅   5 1
3 4   1 3     Order
 p q r
31
 31 1
1 1   1 0   0 Order
 p
32
 2 5
1 51   7 44     Order
 p 5
33
 3 1  ⋅   11 1
2 1   1 0   0 Order
 p q
with
 p < q
primes and
 q  −  1
NOT divisible by
 p
34
 2 1  ⋅   17 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
35
 5 1  ⋅   7 1
2 1   1 0   0 Order
 p q
with
 p < q
primes and
 q  −  1
NOT divisible by
 p
36
 2 2  ⋅   3 2
2 14   4 10     NOT squarefree
37
 37 1
1 1   1 0   0 Order
 p
38
 2 1  ⋅   19 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
39
 3 1  ⋅   13 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
40
 2 3  ⋅   5 1
2 14   3 11     NOT squarefree
41
 41 1
1 1   1 0   0 Order
 p
42
 2 1  ⋅   3 1  ⋅   7 1
3 6   1 5     Order
 p q r
43
 43 1
1 1   1 0   0 Order
 p
44
 2 2  ⋅   11 1
2 4   2 2     NOT squarefree
45
 3 2  ⋅   5 1
2 2   2 0   0 NOT squarefree
46
 2 1  ⋅   23 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
47
 47 1
1 1   1 0   0 Order
 p
48
 2 4  ⋅   3 1
2 52   5 47     NOT squarefree
49
 7 2
1 2   2 0   0 Order
 p 2
50
 2 1  ⋅   5 2
2 5   2 3     NOT squarefree
51
 3 1  ⋅   17 1
2 1   1 0   0 Order
 p q
with
 p < q
primes and
 q  −  1
NOT divisible by
 p
52
 2 2  ⋅   13 1
2 5   2 3     NOT squarefree
53
 53 1
1 1   1 0   0 Order
 p
54
 2 1  ⋅   3 3
2 15   3 12     NOT squarefree
55
 5 1  ⋅   11 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
56
 2 3  ⋅   7 1
2 13   3 10     NOT squarefree
57
 3 1  ⋅   19 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
58
 2 1  ⋅   29 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
59
 59 1
1 1   1 0   0 Order
 p
60
 2 2  ⋅   3 1  ⋅   5 1
3 13   2 11 12 1 NOT squarefree
61
 61 1
1 1   1 0   0 Order
 p
62
 2 1  ⋅   31 1
2 2   1 1   0 Order
 p q
with
 p < q
primes and
 q  −  1
divisible by
 p
63
 3 2  ⋅   7 1
2 4   2 2   0 NOT squarefree
64
 2 6
1 267   11 256     Order
 p 6

## Sequences

A000001 Number of groups of order
 n, n   ≥   1
.
 {1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, ...}
A003277 Cyclic numbers, i.e.
 n
such that
 n
and
 φ (n)
are relatively prime; also
 n
such that there is just one group of order
 n
, i.e. A000001
 (n) = 1
.
 {1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, ...}
A000679 Number of groups of order
 2 n, n   ≥   0
.
 {1, 1, 2, 5, 14, 51, 267, 2328, 56092, 10494213, 49487365422, ...}
A000688 Number of Abelian groups of order
 n, n   ≥   1
. (Number of factorizations of
 n
into prime powers greater than
 1
.)
 {1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, ...}
A060689 Number of non-Abelian groups of order
 n, n   ≥   1
.
 {0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 9, 0, 3, 0, 3, 1, 1, 0, 12, 0, 1, 2, 2, 0, 3, 0, 44, 0, 1, 0, 10, 0, 1, 1, 11, 0, 5, 0, 2, 0, 1, 0, 47, 0, 3, 0, 3, 0, 12, 1, 10, 1, 1, 0, 11, 0, 1, 2, 256, 0, 3, 0, 3, 0, 3, 0, 44, ...}
A066295 Number of Abelian groups of order
 n, n   ≥   1,
minus the number of non-Abelian groups of order
 n, n   ≥   1
.
 {1, 1, 1, 2, 1, 0, 1, 1, 2, 0, 1, −1, 1, 0, 1, −4, 1, −1, 1, −1, 0, 0, 1, −9, 2, 0, 1, 0, 1, −2, 1, −37, 1, 0, 1, −6, 1, 0, 0, −8, 1, −4, 1, 0, 2, 0, 1, −42, 2, −1, 1, −1, 1, −9, 0, −7, 0, 0, 1, −9, 1, 0, 0, −245, 1, −2, 1, −1, 1, −2, ...}
A051532 The Abelian orders (or Abelian numbers):
 n
such that every group of order
 n
is Abelian.
 {1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, ...}
A060652 The non-Abelian orders (or non-Abelian numbers):
 n
such that some group of order
 n
is non-Abelian.
 {6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, ...}
A034382 Number of labeled Abelian groups of order
 n
.
 {?, ...}
A000113 Number of transformation groups of order
 n
.
 {?, ...}
A000041 Number of partitions of
 n
(the partition numbers).
 {?, ...}