On nonabelian composition factors of a finite group that is prime spectrum minimal

Suppose that $L$ is a finite group, $\pi(L)$ is the set of prime divisors of the order $|L|$, and $\mathfrak{Y}$ is the class of finite groups $G$ such that $\pi(G) \not = \pi(H)$ for any proper subgroup $H$ of $G$. Groups from the class $\mathfrak{Y}$ will be called prime spectrum minimal. Many but not all finite simple groups are prime spectrum minimal. For finite simple groups not from the class $\mathfrak{Y}$, the question whether they are isomorphic to nonabelian composition factors of groups from the class $\mathfrak{Y}$ is interesting. We describe some finite simple groups that are not isomorphic to nonabelian composition factors of groups from the class $\mathfrak{Y}$ and construct an example of a finite group from $\mathfrak{Y}$ that has as its composition factor a finite simple sporadic McLaughlin group $McL$ not from the class $\mathfrak{Y}$.