Orbit and stabilisers of subgroups of S4

Question

We are given that for a group $G$ and set $S$ denoting the set of subgroups of $G$:

$\rho(g,H) =gHg^{-1}$ where $g \in G, H \leqslant G$ defines a left action of G on S.

We are then asked to determine $Orb(H)$ and $Stab(H)$ for the following subgroups of $G=S_4$:

$H=V_4$, $H=Sym\{1,2,3\}$ and $H=\langle (1234) \rangle$.

Any help would be appreciated. For $H=V_4$, I thought $Orb(V_4)= V_4$ and $Stab(V_4)=S_4$ but this clearly doesn't agree with the Orbit-Stabiliser theorem so not too sure where I've gone wrong.

Answer 1

For $H=V_4$, I thought $\mathrm{Orb}(V_4)=\{V_4\}$ and $\mathrm{Stab}(V_4)=S_4$ but this clearly doesn't agree with the Orbit-Stabiliser theorem so not too sure where I've gone wrong.

You're wrong about being wrong. There is no disagreement with the orbit-stabizer theorem. Since the stabilizer is the whole group, the size of the quotient $G/\mathrm{Stab}$ is $1$, which is the size of the orbit $\{V_4\}$ (notice we must put $V_4$ in parentheses, since it is a single element of the orbit).

Any ideas for $H=S_3$ and $H=\langle (1234)\rangle$?