Let H be a subgroup of a finite group G. We say that H is a c#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H boolean AND K is a CAP-subgroup of G. In the present paper, we use c#-normality of subgroups to characterize the structure of finite groups, and establish some necessary and sufficient conditions for a finite group to be p-supersolvable, p-nilpotent and solvable. Our results extend and improve some recent ones.