Let G be any non-trivial finite group.
Has G always a subgroup, whose index is prime ?
If G is solvable and |G| has a prime divisor $p$, such that $p^2$ does not divide $|G|$, this is the case because of Hall's theorem.
If $G$ is a $p$-group, the answer is also positive.
The group $A_5$, for example, is not solvable, but has subgroups with index $5$.
So, I wonder whether we always can find a subgroup with prime index.