Intersection of maximal subgroups isomorphic to $PGL_2(p)$ is of form $PGL_2(p_0)$

Question

I want prove (or disprove) that given a finite group G, any two maximal subgroups that are isomorphic to $PGL(2,p)$ where p is prime, then their intersection is isomorphic $PGL(2,q_0)$ (for some $q_0$). I am aware of the fact that intersection of subfields is also a subfield. However, the first statement is unclear to me.

Any comments or suggestion is greatly appreciated !

Answer 1

There are many counterexamples.

Take say $G=Sym(4)$. It has four maximal subgroups with are isomorphic to $Sym(3)\cong PGL(2,2)$. Any two of these intersect in a group of order $2$, which is not isomorphic to any $PGL(2,q)$.

If you want an example with $PGL(2,p)$ insoluble, take $G=Alt(7)$. It has a conjugacy class of maximal subgroups isomorphic to $ Sym(5)\cong PGL(2,5)$. The intersection of two such subgroup varies, but can be $A_4$ or dihedral of order $12$, which are not isomorphic to any $PGL(2,q)$.