TITLE: Minimal maximal subgroup of the symmetric group
QUESTION [15 upvotes]: The question is pretty much in the title: What is the maximal subgroup of $S_d$ of maximal index (so minimal size)? A slight variant (I am not sure if it leads to a different answer) is: what if we restrict to transitive subgroups? (Of course, in both cases, there might be massive ties, but at least the index is well-defined).
Clarification: I want both asymptotic results, and, if possible, a pointer to a table for small $d.$

REPLY [5 votes]: A table of the largest indices of maximal subgroups of ${\rm S}_d$ for small $d$
can be computed with GAP:
    d | largest index of a maximal subgroup of S_d 
------+----------------------------------------------------------------
    2 : 2
    3 : 3
    4 : 2^2
    5 : 2*5
    6 : 3*5
    7 : 2^3*3*5
    8 : 2^3*3*5
    9 : 2^3*3*5*7
   10 : 2^3*3^2*5*7
   11 : 2^7*3^4*5*7
   12 : 2^7*3^4*5*7
   13 : 2^8*3^4*5^2*7*11
   14 : 2^8*3^4*5^2*7*11
   15 : 2^3*5^2*7^2*11*13
   16 : 3*5^3*7^2*11*13
   17 : 2^11*3^6*5^3*7^2*11*13
   18 : 2^11*3^6*5^3*7^2*11*13
   19 : 2^15*3^6*5^3*7^2*11*13*17
   20 : 2^15*3^6*5^3*7^2*11*13*17
   21 : 2^14*3^8*5^4*7^2*11*13*17*19
   22 : 2^11*3^7*5^3*7^2*11*13*17*19
   23 : 2^18*3^9*5^4*7^3*11*13*17*19
   24 : 2^18*3^9*5^4*7^3*11*13*17*19
   25 : 2^17*3^9*5^3*7^3*11^2*13*17*19*23
   26 : 2^18*3^9*5^4*7^3*11^2*13*17*19*23
   27 : 2^18*3^7*5^6*7^3*11^2*13*17*19*23
   28 : 2^22*3^9*5^6*7^3*11^2*13*17*19*23
   29 : 2^23*3^13*5^6*7^3*11^2*13^2*17*19*23
   30 : 2^23*3^13*5^6*7^3*11^2*13^2*17*19*23
   31 : 2^25*3^13*5^6*7^4*11^2*13^2*17*19*23*29
   32 : 2^25*3^13*5^6*7^4*11^2*13^2*17*19*23*29
   33 : 2^12*5^5*7^3*11^2*13^2*17*19*23*29*31
   34 : 3^9*5^4*7^2*11^2*13*17*19*23*29*31
   35 : 2^7*3^6*7^4*11^3*13^2*17^2*19*23*29*31
   36 : 2^27*3^13*5^7*7^4*11^3*13^2*17^2*19*23*29*31
   37 : 2^32*3^15*5^8*7^5*11^3*13^2*17^2*19*23*29*31
   38 : 2^32*3^15*5^8*7^5*11^3*13^2*17^2*19*23*29*31
   39 : 2^12*5^6*7^4*11^2*13^2*17^2*19^2*23*29*31*37
   40 : 2^31*3^14*5^8*7^5*11^3*13^3*17^2*19^2*23*29*31*37
   41 : 2^35*3^18*5^8*7^5*11^3*13^3*17^2*19^2*23*29*31*37
   42 : 2^35*3^18*5^8*7^5*11^3*13^3*17^2*19^2*23*29*31*37
   43 : 2^38*3^18*5^9*7^5*11^3*13^3*17^2*19^2*23*29*31*37*41
   44 : 2^38*3^18*5^9*7^5*11^3*13^3*17^2*19^2*23*29*31*37*41
   45 : 2^34*3^17*5^9*7^6*11^4*13^3*17^2*19^2*23*29*31*37*41*43
   46 : 3^12*5^6*7^3*11^2*13^2*17*19*23*29*31*37*41*43
   47 : 2^41*3^21*5^10*7^6*11^4*13^3*17^2*19^2*23*29*31*37*41*43
   48 : 2^41*3^21*5^10*7^6*11^4*13^3*17^2*19^2*23*29*31*37*41*43
   49 : 2^41*3^20*5^10*7^5*11^4*13^3*17^2*19^2*23^2*29*31*37*41*43*47
   50 : 2^41*3^21*5^10*7^6*11^4*13^3*17^2*19^2*23^2*29*31*37*41*43*47
   51 : 2^15*5^9*7^6*11^3*13^2*17^2*19^2*23^2*29*31*37*41*43*47
   52 : 3^5*5^10*7^7*11^3*13^3*17^3*19^2*23^2*29*31*37*41*43*47
   53 : 2^47*3^23*5^12*7^8*11^4*13^3*17^3*19^2*23^2*29*31*37*41*43*47
   54 : 2^47*3^23*5^12*7^8*11^4*13^3*17^3*19^2*23^2*29*31*37*41*43*47
------+----------------------------------------------------------------

The GAP function used to compute these values is
LargestIndexOfMaximalSubgroupOfSn := function ( n )

  local  Sn, maxes, indices;

  Sn := SymmetricGroup(n);
  maxes := MaximalSubgroupClassReps(Sn);
  indices := List(maxes,G->Factorial(n)/Size(G));
  return Maximum(indices);
end;

and the body of the table has been produced with
gap> for n in [2..54] do
>      Print("       ",String(n,2)," : ");
>      PrintFactorsInt(LargestIndexOfMaximalSubgroupOfSn(n));
>      Print("\n");
>    od;

This loop takes about $4$ seconds on my laptop; for $n > 54$ there appears
to be a sharp increase in runtime.