This is a 'Harder' problem 40 from Abstract Algebra(1996) by Herstein. I'm just not able to figure out how to do this. even though I found a very similar post. Following is a verbatim statement of the question.
If $G$ is a finite group, $H$ a subgroup of $G$ such that $n \nmid i_G(H)!$, where $n=|G|$, prove that there is a normal subgroup $N \neq (e)$ of $G$ contained in $H$.
P.S. I've been stuck on this for about a week, and now I'm throwing in the towel, so I'd really appreciate a solution, but I humbly implore you to give me hints instead so that I can kill this problem (sort of) on my own, although frankly, I've given up hope.