Groups, in which almost all subgroups are near to normal

A subgroup $H$ of a group $G$ is said to be nearly normal, if $H$ has a finite index in its normal closure. These subgroups have been introduced by B. H. Neumann. In a present paper is studied the groups whose non polycyclic by finite subgroups are nearly normal. It is not hard to show that under some natural restrictions these groups either have a finite derived subgroup or belong to the class $S_{1}F$ (the class of soluble by finite minimax groups). More precisely, this paper is dedicated of the study of $S_{1}F$ groups whose non polycyclic by finite subgroups are nearly normal.