Some criteria for the nonsimplicity of finite groups

Let $|G|=\prod_{i=1}^n p_i^{\alpha_i}$, where $p_i$ are prime numbers, $p_i\ne p_j$ for $i\ne j$. Let $\pi(G)=\{p_1,\dots,p_n\}$, $s\in\pi(G)$ and let $\mathfrak{T}$ is the set of some Sylow subgroups of the group $G$, that are taken one at a time for every $p_i\in\pi(G)\setminus\{s\}$, $i=\overline{1,n-1}$. It is proved that if every subgroup from the set $\mathfrak{T}$ normalises some non-identity $s$-subgroup from $G$, $s>3$, then $G$ has solvable normal subgroup $R$ and $s$ divide $|R|$.