Chains in finite groups

Let $\mathbb{N}$ and $\mathbb{P}$ be the set of all positive integers and all primes, respectively. A subgroup $H$ of $G$ is called $\mathbb{P}^\infty$-subnormal in $G$ ($H$ $\mathbb{P}^\infty$-$sn$ $G$) if there is a chain $H=H_0\subset H_1\subset\dots\subset H_{n-1}\subset H_n=G$ such that $|H_i:H_{i-1}|\in\mathbb{P}^\infty$ for every $i=1,\dots,n$, where $\mathbb{P}^\infty=\{p^k\mid p\in\mathbb{P}, k\in\{0\}\subset\mathbb{N}\}$. We obtained finite simple non-abelian groups $G$ with $1$ $\mathbb{P}^\infty$-$sn$ $G$.