On the partial L-Π-property of subgroups of finite groups

Let H be a subgroup of a finite group G. We say that H satisfies the partial L-Π-property in G if H G, or if | G / K : N G / K (HK/K)| is a π (HK/K)-number for any G-chief factor of type HG/K with HG≤ K. In this paper, we investigate the structure of finite groups under the assumption that some subgroups of prime power order satisfy the partial L-Π-property.