On the finite solvable $PNC$-groups

A subgroup $H$ of a finite group $G$ is said to be an NC-subgroup of $G$, if $H^G N_G (H) =G$, where $H^G$ denotes the normal closure of $H$ in $G$. A finite group $G$ is called a PNC-group, if any subgroup of $G$ is an NC-subgroup of $G$, and $G$ is said to be an ON-group, if for any subgroup $H$ of $G$, either $N_G (H)=H,\,H^G=G$, or $H \unlhd G$. In this paper, we firstly investigate the basic properties of solvable PNC-groups, and then give several sufficient conditions for $G$ to be a solvable PNC-group. In the end of this paper, we discover some characterizations for minimal non-ON-groups, ON-groups, non-abelian simple groups whose second maximal subgroups are solvable PNC-groups and groups whose proper (maximal) subgroups are solvable PNC-groups.