TITLE: What About The Converse of Lagrange's Theorem?
QUESTION [9 upvotes]: Given a positive integer $n$, a group $G$ of order $n$, and a divisor $d$ of $n$, in what cases can we be assured of the existence of a subgroup $H$ of $G$ of order $d$? 
What's the situation in the case of the symmetric group on $m$ letters or for the alternating group? 
What's the most general statement in each case?

REPLY [5 votes]: See these two links to have additional information:
The inverse of Lagrange's Theorem is true for finite supersolvable group. 
and 
A kind of converse of Lagrange's Theorem. In the second one, there is a great classification for the problem done by missed @Arturo Magidin.