

A018216


Maximal number of subgroups in a group with n elements.


8



1, 2, 2, 5, 2, 6, 2, 16, 6, 8, 2, 16, 2, 10, 4, 67, 2, 28, 2, 22, 10, 14, 2, 54, 8, 16, 28, 28, 2, 28, 2, 374, 4, 20, 4, 78, 2, 22, 16, 76, 2, 36, 2, 40, 12, 26, 2, 236, 10, 64, 4, 46, 2, 212, 14, 98, 22, 32, 2, 80, 2, 34, 36, 2825, 4, 52, 2, 58, 4, 52, 2, 272
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OFFSET

1,2


COMMENTS

For n >= 2 a(n)>=2 with equality iff n is prime.
The minimal number of subgroups is A000005, the number of divisors of n, attained by the cyclic group of order n.  Charles R Greathouse IV, Dec 27 2016


LINKS



FORMULA

(C_2)^m has A006116(m) subgroups, so this is a lower bound if n is a power of 2 (e.g., a(16) >= 67).  N. J. A. Sloane, Dec 01 2007


EXAMPLE

a(6) = 6 because there are two groups with 6 elements: C_6 with 4 subgroups and S_3 with 6 subgroups.


PROG

(GAP) a:=function(n)
local gr, mx, t, g;
mx := 0;
gr := AllSmallGroups(n);
for g in gr do
t := Sum(ConjugacyClassesSubgroups(g), Size);
mx := Maximum(mx, t);
od;
return mx;


CROSSREFS



KEYWORD

nonn,nice


AUTHOR

Ola Veshta (olaveshta(AT)mydeja.com), May 23 2001


EXTENSIONS

More terms from Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Jun 13 2003


STATUS

approved



