I am asking if you know unsolved or recently solved conjectures around numbers of subgroups in symmetric or alternating groups. In fact, is there a formula depending of $n$ to count subgroups of order $k$ in the symmetric group $S_n$ ? Particularly, how subgroups $S_n$ contains ?
I know it is possible to use GAP to find this on the cases for $n=1,\dots,15$, but i don't know if formulas around these questions have already been discovered. If you have references on the topic, don't hesitate.