On the partial Π-property of second minimal or second maximal subgroups of Sylow subgroups of finite groups

Let H be a subgroup of a finite group G. We say that H satisfies the partial Π-property in G if there exists a chief series ΓG:1=G0<G1<···<Gn=G of G such that for every G-chief factor Gi/Gi−1 (1≤i≤n) of ΓG, |G/Gi−1:NG/Gi−1(HGi−1/Gi−1∩Gi/Gi−1)| is a π(HGi−1/Gi−1∩Gi/Gi−1)-number. In this paper, we study the influence of some second minimal or second maximal subgroups of a Sylow subgroup satisfying the partial Π-property on the structure of a finite group.