Finite group with unique normal subgroup

Question

We all know that symmetric group Sn has a unique normal subgroup An for n>4. Is there any classification of finite groups with unique normal subgroup ? By unique I mean non trivial and improper normal subgroups.

Answer 1

There is a classification of finite solvable groups having a unique non-trivial normal subgroup.

Theorem: Let $G$ a finite solvable group. If $G$ has a unique nontrivial normal subgroup, then either $G$ is a cyclic $p$-group of order $p^2$, or $G$ is a semidirect product $G = P \rtimes Q$, where $P$ is an elementary abelian $p$-group and $Q$ is a cyclic group of order $q$, with $p$ and $q$ being distinct primes. Moreover, the action of $Q$ on $P$ is irreducible.