A174511 The number of isomorphism classes of subgroups of the symmetric group S_n.

1, 2, 4, 9, 16, 29, 55, 137, 241, 453, 894, 2065, 3845, 7766, 16438, 43626, 84246

Offset

1,2

Comments

Two subgroups are considered to be isomorphic here if they are isomorphic as abstract groups, not as permutation groups. - N. J. A. Sloane, Nov 28 2010

Links

Table of n, a(n) for n=1..17.

Andreas Distler and Tom Kelsey, The semigroups of order 9 and their automorphism groups, arXiv preprint arXiv:1301.6023 [math.CO], 2013. - From N. J. A. Sloane, Feb 19 2013

Jeffrey Ketchersid, Verification scripts, isomorphisms, and invariants for a(14)-a(15), GitHub, 2026.

Jeffrey Ketchersid, Verification scripts, isomorphisms, and invariants for a(16), GitHub, 2026.

Jeffrey Ketchersid, Verification scripts, isomorphisms, and invariants for a(17), GitHub, 2026.

Jack Schmidt, Enumerating all subgroups of the symmetric group.

Example

a(3) = 4 since S_3 contains up to isomorphism exactly one subgroup of each of the orders 1,2,3,6.

Prog

(GAP)
a:=[];
for n in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] do
  G:=SymmetricGroup(n);
  R:=ConjugacyClassesSubgroups(G);
  RR:=ListX(R, Representative);
  T:=[RR[1]];
  for g in RR do
    flag:=false;
    for h in T do
      if IsomorphismGroups(g, h)<>fail then
        flag:=true;
        break;
      fi;
    od;
    if flag=false then Add(T, g); fi;
  od;
  Add(a, Size(T));
od;
Print(a, "\n");

Crossrefs

Cf. A000638, A005432.

Sequence in context: A000291 A081055 A034446 * A034452 A034449 A082894

Adjacent sequences: A174508 A174509 A174510 * A174512 A174513 A174514

Keyword

nonn,more

Author

W. Edwin Clark, Nov 28 2010

Extensions

a(11) and a(12) from Stephen A. Silver, Feb 24 2013

a(13) (as calculated by Jack Schmidt) from L. Edson Jeffery, May 26 2013

a(14)-a(15) from Jeffrey Ketchersid, Feb 13 2026

a(16) added by Jeffrey Ketchersid, Mar 05 2026

a(17) added by Jeffrey Ketchersid, Apr 08 2026

Status

approved