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A005432 Number of permutation groups of degree n (or, number of distinct subgroups of symmetric group S_n, counting conjugates as distinct).
(Formerly M1690)
1, 1, 2, 6, 30, 156, 1455, 11300, 151221, 1694723, 29594446, 404126228, 10594925360, 175238308453, 5651774693595, 117053117995400, 5320744503742316, 125889331236297288, 7598016157515302757 (list; graph; refs; listen; history; text; internal format)



Labeled version of A000638.


J. Labelle and Y. N. Yeh, The relation between Burnside rings and combinatorial species, J. Combin. Theory, A 50 (1989), 269-284.

L. Pyber, Ann. Math. 137 (1993), 203-220 shows c^{n^2(1+o(1))} <= a(n) <= d^{n^2(1+o(1)}, c=2^{1/16}, d=24^{1/6}; conjectures lower bound is accurate.

C. C. Sims, Computational methods in the study of permutation groups, pp. 169-183 of J. Leech, editor, Computational Problems in Abstract Algebra. Pergamon, Oxford, 1970.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


Table of n, a(n) for n=0..18.

Piotr Graczyk, Hideyuki Ishi, Kołodziejek Bartosz, Hélène Massam, Model selection in the space of Gaussian models invariant by symmetry, arXiv:2004.03503 [math.ST], 2020.

D. F. Holt, Enumerating subgroups of the symmetric group.

D. Holt, Enumerating subgroups of the symmetric group, in Computational Group Theory and the Theory of Groups, II, edited by L.-C. Kappe, A. Magidin and R. Morse. AMS Contemporary Mathematics book series, vol. 511, pp. 33-37. [Annotated copy]

L. Naughton, G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8.

Götz Pfeiffer, Numbers of subgroups of various families of groups

N. J. A. Sloane, Transforms

Index entries for sequences related to groups


Exponential transform of A116655. Binomial transform of A116693. - Christian G. Bower, Feb 23 2006


(MAGMA) n := 5; &+[ Length(s):s in SubgroupLattice(Sym(n)) ];

(GAP) List([2..5], n->Sum( List( ConjugacyClassesSubgroups( SymmetricGroup(n)), Size))); [Alexander Hulpke]


Cf. A000001, A000019, A000638.

Sequence in context: A327927 A192446 A218940 * A009422 A057221 A180892

Adjacent sequences:  A005429 A005430 A005431 * A005433 A005434 A005435




N. J. A. Sloane, Simon Plouffe


a(9) and a(10) from Alexander Hulpke, Dec 03 2004

More terms from a(11) and a(12) added by Christian G. Bower, Feb 23 2006 based on Goetz Pfeiffer's web page.

a(13) added by Liam Naughton, Jun 09 2011

a(14)-a(18) from Holt reference, Wouter Meeussen, Jul 02 2013



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Last modified August 3 16:28 EDT 2021. Contains 346439 sequences. (Running on oeis4.)