# Does there exist some sort of classification of finite marginally simple groups?

Let’s call a group marginally simple if it does not have any non-trivial marginal subgroup (strict definition of marginal subgroups and brief overview of their properties can be found here: https://groupprops.subwiki.org/wiki/Marginal_subgroup). Does there exist some sort of classification of finite marginally simple groups?

$$G^n$$, with $$G$$ being a finite simple group, is always marginally simple as it has no nontrivial characteristic subgroups and all marginal subgroups are characteristic. Moreover, all groups that have no non-trivial characteristic subgroup (or even no non-trivial verbal subgroup, as shown here: Does there exist some sort of classification of finite verbally simple groups?) are of this form. However, maybe marginally simple groups are more abundant…

If $$G$$ is marginally simple it is either abelian or centerless, as center is a marginal subgroup.

If $$G$$ is abelian, then any marginal subgroup in it can be defined using a word $$x^n$$. That results in marginal subgroups of abelian groups being exactly the sets of all elements of order dividing $$n$$. And the only abelian groups that have no subgroups of such type are exactly $$C_p^n$$ for prime $$p$$, which makes them the only abelian marginally simple groups.

However, I do not know, how to deal with the case, when $$G$$ is centerless.

EDIT: Actually, there do exist finite marginally simple groups, which are not characteristically simple. One of the possible example series is $$S_n$$ for $$n \geq 5$$ (For what $n$ is $A_n$ a marginal subgroup of $S_n$?)