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Does there exist a finite group $G$ with the following properties?

$G$ has a minimal subgroup say $L$ isomorphic to $\mathbb{Z}_{p}$ and a maximal subgroup say $M$ isomorphic to $\mathbb{Z}_{pq}$ containing properly $L$ ($p$ and $q$ are some distinct primes), such that $L\ntrianglelefteq G$, $M\ntrianglelefteq G$ and $M$ is the only proper subgroup of $G$ containing properly $L$.