A243748 -id:A243748 - cp's OEIS frontend

A277566 Irregular table by rows: Orders of subgroups of the symmetric group S_n.

1, 1, 2, 1, 2, 3, 6, 1, 2, 3, 4, 6, 8, 12, 24, 1, 2, 3, 4, 5, 6, 8, 10, 12, 20, 24, 60, 120, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 36, 48, 60, 72, 120, 360, 720, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 21, 24, 36, 40, 42, 48, 60, 72, 120, 144, 168, 240, 360, 720, 2520, 5040, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 30, 32, 36, 40, 42, 48, 56, 60, 64, 72, 96, 120, 128, 144, 168, 180, 192, 240, 288, 336, 360, 384, 576, 720, 1152, 1344, 1440, 2520, 5040, 20160, 40320

Offset: 1

Charles R Greathouse IV, Nov 28 2016

Each row begins with 1 (the order of the trivial group) and ends with n! (the order of S_n itself).

			1;
1, 2;
1, 2, 3, 6;
1, 2, 3, 4, 6, 8, 12, 24;
1, 2, 3, 4, 5, 6, 8, 10, 12, 20, 24, 60, 120;
1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 36, 48, 60, 72, 120, 360, 720;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 21, 24, 36, 40, 42, 48, 60, 72, 120, 144, 168, 240, 360, 720, 2520, 5040;
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 30, 32, 36, 40, 42, 48, 56, 60, 64, 72, 96, 120, 128, 144, 168, 180, 192, 240, 288, 336, 360, 384, 576, 720, 1152, 1344, 1440, 2520, 5040, 20160, 40320;

Charles R Greathouse IV, Rows n = 1..12 of table, flattened
Götz Pfeiffer, Numbers of subgroups of various families of groups

Row lengths are given by A218913.

See A243748 for a closely related sequence.

LoadPackage("sonata");
a := function(n)
  return Length(Unique(Apply(Subgroups(SymmetricGroup(n)), Size)));
end;; # Charles R Greathouse IV, Nov 28 2016

A284210 Number of subgroups of order n of the symmetric group Sym(n) on n symbols.

Original entry on oeis.org

1, 1, 1, 7, 6, 280, 120, 25335, 11200, 276696, 362880, 374838255, 39916800, 2414617920, 11721790080

Offset: 1

Jens Voß, Mar 23 2017

The diagonal of A243748. - R. J. Mathar, Mar 30 2017 [edited by Peter Munn, Mar 06 2025]

			The group Sym(4) contains 3 cyclic groups of order 4, 3 non-normal elementary abelian groups of order 4 and one normal group of order 4, so A284210(4) = 3 + 3 + 1 = 7.

List([1..14], n -> Sum(List(Filtered(ConjugacyClassesSubgroups(SymmetricGroup(n)), c -> Size(Representative(c)) = n)), c -> Size(c)));

If n is prime, A284210(n) = (n-2)!.

cp's OEIS Frontend

A277566 Irregular table by rows: Orders of subgroups of the symmetric group S_n.

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