$H$ is called a nearly normal subgroup of a group $G$ if it is of finite index in its normal closure, $H^G:=\cup\_{g\in G} gHg^{-1}$, in $G$. Clearly, every normal subgroup or subgroup of finite index of $G$ is nearly normal. I am not interested in finite subgroups either. I am looking for infinite nearly normal subgroups of infinite index in well-known groups. By a well-known group, I mean a group that some of its basic properties like amenability, growth or its geometric nature (the spaces it acts) have been studied in the literature. 



Non-elementary examples of nearly normal subgroups
Take any infinite group with an infinite-index normal subgroup. Say, the free group on two generators with the commutator subgroup. Then take a finite index subgroup of the normal subgroup. This particular normal subgroup is the free group on countably infinite generators, and so has many finite-index subgroups. Most are not normal, but all are nearly normal.


This feels somewhat unsatisfying, though. Presumably the only way to get something satisfying is to find a nearly normal subgroup whose definition is more natural than the corresponding normal subgroup. I do not know any subgroups like that.


