# The classification all finite groups which possess a single proper non-trivial normal subgroup

We know that

For $n≥5$, $A_n$ is the only proper nontrivial normal subgroup of $S_n$.

I am kindly asking to know the possible presented references including the following point, if anybody is aware of them.:

The classification all finite groups G whose possess a single proper non-trivial normal subgroup.

You will get similar examples by taking a finite, nonabelian simple group $S$, and extending it by an outer automorphism of prime order to a group $G$.

Somewhat dually, you can take a quotient $P$ of prime order of the Schur multiplier of $S$, and extend $P$ by $S$ to a group $G$.

Another class of soluble examples can be obtained by starting with two distinct primes $p, q$. Consider the period $n$ of $p$ modulo $q$. Then in the multiplicative finite field $\mathbf{F}_{p^{n}}$ there is a subgroup $Q$ of order $q$ that acts irreducibly on the additive group $P$ of $\mathbf{F}_{p^{n}}$. The semidirect product $G = PQ$ will have the property, with $P$ the only nontrivial, proper normal subgroup.

Coming back to insoluble examples, one can take a direct power $S^{p}$ of a nonabelian, finite simple group $S$, with $p$ prime. If you let a cyclic group $C_p$ of order $p$ permute cyclically the factors in $S^{p}$, you should get another example, with $S^{p}$ as the distinguished normal subgroup.