Can a Sylow $p$-subgroup be conjugate to anything other than another Sylow $p$-subgroup?

Question

The Sylow theorems say that all $p$-Sylow subgroups are conjugate.

Is that the only thing they are conjugate to? In other words could a $p$-Sylow subgroup also be conjugate to something that is not a $p$-Sylow group?

Or does the fact that every element with order dividing $p$ is in a $p$-Sylow group mean that that is never possible?

Answer 1

To get this question an answer, here is Jyrki Lahtonen's answer:

Conjugate subgroups are always isomorphic. Among other things this implies that they have the same cardinality.

So p-Sylow subgroups are always isomorphic to other p-Sylow subgroups.