I know this question is discussed several times on this site, but I am trying to use a slightly different method than the ones already provided to solve it. Please would you check whether my solution is correct?
I have considered an action from a group $G$ to the set $X$ of all the subgroups of $G$ defined by $(g, H) \to gHg^{-1}$, where $H$ is a subgroup of $G$, and proven that $|Orb_G(H)| \le |G/H|$. Now I considered the case where $H \ne G$, and tried to argue that $|Orb_G(H)| \le |G/H| \lt |G|$. But I worry that the $|Orb_G(H)|$ counts a number of subgroups of $G$, whereas $|G|$ counts elements, so I don't think it's valid. In that case, please could I have some help?
Please note, I need to find a way to follow on from the weak inequality already proven.
EDIT: $G/H$ denotes the set of left cosets.