Let $G$ be a finite simple group of type ${\rm L}_{n}^{\epsilon}(q)$ with the following conditions:

$n$ and $\dfrac{q^{n}-\epsilon}{(q-\epsilon)(n,q-\epsilon)}$ both prime, $n\geqslant3$ and $(n,q,\epsilon)\neq(3,4,+),(3,3,-),(3,5,-),(5,2,-)$

Show that:

1- Every minimal subgroup $L$ of order $s=\dfrac{q^{n}-\epsilon}{(q-\epsilon)(n,q-\epsilon)}$ is contained properly in a maximal subgroup $M$ isomorphic to $\mathbb{Z}_{s}\rtimes\mathbb{Z}_{n}$

2- The maximal subgroup $M$ is the only proper subgroup of $G$ which contains properly $L$, i.e. every minimal subgroup of order $s$ has a unique overgroup.

As far as I checked the "ATLAS" the above propositions both are true.