Let G be a finite group and H a subgroup of G. We say that H is an \(\mathcal {H}\) -subgroup of G if \(N_{G}(H)\cap H^{g}\le H\) for all \(g\in G\) . The subgroup H is called an \(\mathcal {H}N\) -subgroup of G if there exists a normal subgroup T with \((|H|,|G:HT|)=1\) such that \(N_{T}(H)\cap H^{g}\le H\) for all \(g\in G\) . In this paper, some new criteria for a group G to be p-nilpotent and supersolvable are given when certain subgroups of prime power orders are \(\mathcal {H}N\) -subgroups of G. Our results improve and generalize some recent results in the literature.
