Structure of finite groups with trait of non-normal subgroups II

A finite non-Dedekind group G is called an N AC-group if all non-normal abelian subgroups are cyclic. In this paper, all finite N AC-groups will be characterized. Also, it will be shown that the center of non-nilpotent N AC- groups is cyclic. If N AC-group G has a non-abelian non-normal Sylow subgroup of odd order, then other Sylow subgroups of G are cyclic or of quaternion type.