TITLE: A question about finite simple groups
QUESTION [5 upvotes]: Let ‎$G‎‎$ ‎be a‎ ‎finite ‎simple ‎group. ‎Is ‎it ‎possible ‎to ‎find ‎two ‎‎
distinct ‎proper non-trivial ‎subgroups ‎‎‎$H_1‎‎$ and ‎‎‎‎$‎‎H_2$ ‎of ‎‎$G‎‎$ ‎‎
such ‎that ‎‎$\langle H_1 , H_2\rangle‎\lneq G‎‎‎$ ‎and ‎for ‎each ‎maximal ‎subgroup ‎‎$M‎‎$ ‎of ‎‎$G‎‎$‎, ‎
if ‎$‎‎H_i\nleq M$, ‎then ‎‎$‎‎H_j\leq M‎$ ‎‎‎(‎‎$i , j\in \{1,2\}‎‎$‎)?‎
If‎ it is ‎dificult ‎to ‎answer ‎the ‎question in the class of finite simple groups, ‎
‎could ‎we ‎answer ‎it ‎in ‎certain ‎classes ‎of ‎finite ‎simple ‎groups‎?‎
(such as alternating groups, minimal simple groups, ...). ‎
‎
Many thanks.
‎

REPLY [3 votes]: As mentioned in the comments, this is not possible in $A_5$. Suppose $H_1$ and $H_2$ are subgroups of $A_5$ with the desired property. The point stabilizers in $A_5$ are maximal, so each of them contains either $H_1$ or $H_2$. Thus one of the $H_i$ is contained in at least three point stabilizers, which implies that $H_i$ is trivial.