Finite groups with $s$-normal maximal subgroups

Subgroup $H$ of $G$ is called $s$-normal in $G$, if $G$ has a subnormal subgroup $T$ such that $G=HT$ and $H\cap T\subseteq H_{\dots G}$. New criteria for the $p$-solvability and solvability of finite groups are obtained on the basis of this definition.