Because of my interest in this question, I listed the subgroups of ${\frak S}_n$ for $1\le n\le4$. I found that the number of subgroups are, respectively, $1,2,6,24$. It might be a coincidence, or it could reveal a deep connection.

Is it always the case that ${\frak S}_n$ has exactly $n!$ (non-proper) subgroups ?

Edit. Because of my erroneous claim about ${\frak S}_4$, I tried to delete the question. But this cannot be done once an answer has been posted. So I asked the help of the front desk. Decision is pending.