A261717 -id:A261717 - cp's OEIS frontend

A263447 Arrange the 26 sporadic simple groups in increasing order; a(n) = number of sporadic simple groups which are subquotients of the n-th largest sporadic simple group.

1, 2, 1, 1, 1, 3, 3, 1, 5, 3, 1, 1, 4, 3, 7, 6, 4, 5, 4, 1, 6, 12, 6, 9, 12, 20

Offset: 1

Sven Simon and Charles R Greathouse IV, Oct 18 2015

A group is a subquotient of itself, so a(n) >= 1.
It is well-known that a(26) = 20, the so-called "happy family". Trivially a(1) = 1 and a(2) = 2 since M_11 is a subquotient of M_12.
The sequence was generated from the diagram of subquotient relations on page 238 of the ATLAS, together with the update that J_1 is not involved in M (which replaces the single question mark in the table with a plus sign).

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites]. See page 238.

Wikipedia, Sporadic group

Cf. A001228, A261717 (another version).

Terms confirmed by N. J. A. Sloane, Oct 19 2015

A321224 Sporadic numbers: n is defined to be sporadic if the set of groups G not in {A_n, S_n} and having a core-free maximal subgroup of index n is nonempty and contains only sporadic simple groups.

Original entry on oeis.org

266, 506, 759, 1045, 1288, 1463, 3795

Offset: 1

Sébastien Palcoux, Aug 27 2019

A finite group G has a core-free maximal subgroup H of index n if and only if it is a primitive permutation group of degree n (acting on the set G/H of cosets).
There are no other sporadic numbers less than 4096 (see computation below).
According to Derek Holt, the next sporadic number is 4180, and the last one should be 492693551703971265784426771318116315247411200000000 (coming from the maximal subgroup 41:40 of the Monster, and assuming that L_2(13) is not maximal).
Derek Holt suggested another sequence where we also allow the extensions of the sporadic simple groups.

The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.

S. Palcoux, The sporadic numbers (version: 2019-07-22), MathOverflow.

Cf. A102842, A001228, A174601, A174848, A261717, A263447, A121236.

IsSporadic:=function(G)
   if not IsSimple(G) then
      return false;
   else
      return IsomorphismTypeInfoFiniteSimpleGroup(G).series="Spor";
   fi;
end;;
SporadicNumbers:=function(b1,b2)
   local L,i,n,a,j,G;
   L:=[];
   for i in [b1..b2] do
      n:=NrPrimitiveGroups(i);
      if n>2 then
         a:=0;
         for j in [1..n] do
            G:=PrimitiveGroup(i,j);
            if not G=SymmetricGroup(i) and not G=AlternatingGroup(i) and not IsSporadic(G) then
               a:=1;
               break;
            fi;
         od;
         if a=0 then
            Add(L,i);
         fi;
      fi;
   od;
   return L;
end;;
SporadicNumbers(1,4095);
# gives: [ 266, 506, 759, 1045, 1288, 1463, 3795 ]

cp's OEIS Frontend

A263447 Arrange the 26 sporadic simple groups in increasing order; a(n) = number of sporadic simple groups which are subquotients of the n-th largest sporadic simple group.

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