Infinite group with finitely many normal subgroups

Question

In theorem 2 of this, a finite number of non normal group is classified. I want to know the opposite of this question. Is there any classification of groups with finitely many normal subgroups?

Answer 1

A few remarks:

1 normal subgroup = trivial group

2 normal subgroups = simple groups

3 normal subgroups: these are groups $G$ with a minimal non-trivial subgroup $N$ such that $G/N$ is simple and such that $N$ is not a direct factor. For finite groups, $N$ is either $p$-elementary abelian for some $p$ with $G$ acting irreducibly on $N$, or $N$ is a power of a non-abelian simple group and $G/N$ acts transitively on the set of simple factors in $N$.

4 normal subgroups: either $G$ is the direct product of two simple groups (excluding the case of two isomorphic abelian simple groups), or the 4 normal subgroups form a chain.

In general, having finitely many normal subgroups has some consequences (like having minimal nontrivial normal subgroups) but however it seems hard to make a classification (beware that the minimal nontrivial normal subgroups can have infinitely many normal subgroups).

Also note that it implies that there is a minimal subgroup of finite index (which itself might have infinitely many normal subgroups).