finite groups with $x$-quasipermutable subgroups of prime power order

let $h$, $l$ and $x$ be subgroups of a finite group$g$. then $h$ is said to be $x$-permutable with $l$ if for some$xin x$ we have $al^{x}=l^{x}a$. we say that $h$ is emph{$x$-quasipermutable } (emph{$x_{s}$-quasipermutable}, respectively) in $g$ provided $g$ has a subgroup$b$ such that $g=n_{g}(h)b$ and $h$ $x$-permutes with $b$ and with all subgroups (with all sylowsubgroups, respectively) $v$ of $b$ such that $(|h|, |v|)=1$. inthis paper, we analyze the influence of $x$-quasipermutable and$x_{s}$-quasipermutable subgroups on the structure of $g$. some known results are generalized.