The conjugate of a sylow $p$-subgroup is a sylow $p$-subgroup Second Sylow theorem states that all Sylow $p$-subgroups are conjugate. But reviewing my proof it seems to me that we also prove that all the conjugates of a Sylow $p$-subgroup are Sylow $p$-subgroups.
I can include the proof if needed but can anybody confirm me this idea? Or give a counterexample?

In general if $H$ is a subgroup of $G$ and $g\in G$, then we have $|gHg^{-1}|=|H|$, so it follows that the conjugates of a Sylow subgroup are also Sylow subgroups.
