Let $G$ be a finite group with trivial Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$ is trivial) such that $G$ has more than two maximal subgroups and at least one of its maximal subgroup isn't of prime order. Do there exist two distinct non-trivial subgroups $H_1$ and $H_2$ of $G$ such that

• $H_1$ and $H_2$ are contained in a common maximal subgroup $M$
• For each maximal subgroup $M'$ of $G$, either $H_1 \leq M'$ or $H_2 \leq M'$ (or both)?

If it is hard to answer the question in general, can we answer it for certain classes of finite groups (say finite simple groups, symmetric groups,...)?

Let $G$ be a finite group with trivial Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$ is trivial) such that $G$ has more than two maximal subgroups and at least one of its maximal subgroup isn't of prime order. Do there exist two distinct non-trivial subgroups $H_1$ and $H_2$ of $G$ such that

• $H_1$ and $H_2$ are contained in a common maximal subgroup $M$
• For each maximal subgroup $M'$ of $G$, either $H_1 \leq M'$ or $H_2 \leq M'$ (or both)?

If it is hard to answer the question in general, can we answer it for certain classes of finite groups (finite simple groups, symmetric groups,…)?

Let $G$ be a finite group with trivial Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$ is trivial) such that $G$ has more than two maximal subgroups and atleast one of its maximal subgroup isnt of prime order. Do there exist two distinct non-trivial subgroups $H_1$ and $H_2$ of $G$ such that

• $H_1$ and $H_2$ are contained in a common maximal subgroup $M$
• For each maximal subgroup $M^'$ of $G$, either $H_1 \leq M'$ or $H_2 \leq M'$ (or both)?

If it is hard to answer the question in general, can we answer it for certain classes of finite groups (say finite simple groups, symmetric groups,...)?

Let $G$ be a finite group with trivial Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$ is trivial) such that $G$ has more than two maximal subgroups and at least one of its maximal subgroup isn't of prime order. Do there exist two distinct non-trivial subgroups $H_1$ and $H_2$ of $G$ such that

• $H_1$ and $H_2$ are contained in a common maximal subgroup $M$
• For each maximal subgroup $M'$ of $G$, either $H_1 \leq M'$ or $H_2 \leq M'$ (or both)?

If it is hard to answer the question in general, can we answer it for certain classes of finite groups (say finite simple groups, symmetric groups,...)?

Let $G$ be a finite group with trivial Frattini subgroup, i.e. the intersection of all maximal subgroups of $G$ is trivial.

Do there exist two distinct non-trivial subgroups $H_1$ and $H_2$ of $G$ such that

• $H_1$ and $H_2$ are contained in a common maximal subgroup $M$
• For each maximal subgroup $M^'$ of $G$, either $H_1 \leq M'$ or $H_2 \leq M'$ (or both)?

If it is hard to answer the question in general, can we answer it for certain classes of finite groups (say finite simple groups, symmetric groups,...)?

Let $G$ be a finite group with trivial Frattini subgroup (i.e. the intersection of all maximal subgroups of $G$ is trivial) such that $G$ has more than two maximal subgroups and atleast one of its maximal subgroup isnt of prime order. Do there exist two distinct non-trivial subgroups $H_1$ and $H_2$ of $G$ such that

• $H_1$ and $H_2$ are contained in a common maximal subgroup $M$
• For each maximal subgroup $M^'$ of $G$, either $H_1 \leq M'$ or $H_2 \leq M'$ (or both)?

If it is hard to answer the question in general, can we answer it for certain classes of finite groups (say finite simple groups, symmetric groups,...)?