It seems that there are some examples. Let $G=D_{10}\times S_2$. Then the frattini subgroup of $G$ is trivial and the maximal subgroups are (that can be check by GAP): $M_1=\langle(6,7),(1,2,3,4,5)\rangle$, $M_2=\langle (2,5)(3,4),(1,2,3,4,5)\rangle$, $M_3=\langle(2,5)(3,4)(6,7),(1,2,3,4,5)\rangle$, $M_4=\langle(6,7),(2,5)(3,4)\rangle$, $M_5=\langle(6,7),(1,4)(2,3)\rangle$, $M_6=\langle(6,7),(1,2)(3,5)\rangle$, $M_7=\langle(6,7),(1,5)(2,4)\rangle$, and $M_8=\langle(6,7),(1,3)(4,5)\rangle$. Now consider $H_1=\langle(6,7)\rangle$ and $H_2=\langle(1,2,3,4,5)\rangle$.
2 of 3
use `\langle` and `\rangle` instead of `<` and `>`; better spacing
Arturo Magidin
- 7.2k
- 1
- 42
- 60
majid arezoomand
- 792
- 3
- 11