How many subgroups of order 6 are in $S_6$?
I have been thinking on first counting the number of elements of order $6$ in $S_6$. Also, I have in mind using the fact the a group of order 6 is either isomoporhic to $S_3$ if its non abelian or to $Z_6$ if abelian. Do these facts help? Does someone have a better approach?
I think that's a good approach. You get an $S_3$ whenever you have an element $s$ of order 2 and an element $t$ of order 3 with $sts=t^2$.