Finite groups with some subgroups satisfying the partial $ \Pi $-property

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies the partial $\Pi$-property in $G$ if there exists a chief series $\varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G$ of $G$ such that for every $G$-chief factor $G_{i}/G_{i-1}$ $(1\leq i\leq n)$ of $\varGamma_{G}$, $| G / G_{i-1} : N _{G/G_{i-1}} (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})|$ is a $\pi (HG_{i-1}/G_{i-1}\cap G_{i}/G_{i-1})$-number. In this paper, we investigate how some subgroups satisfying the partial $\Pi$-property influence the structure of finite groups.