I saw an exercise that goes:
Let $G$ be a group of order 120, and $H$ be a subgroup of order 24. If at least one left coset of $H$ in $G$ is a right coset apart from $H$ itself, show that $H$ is normal.
Somehow I need to show that the remaining 3 left cosets are also right cosets. But I can't yet find a way to argue about that.