  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.
