I *think* this works, but it seems to easy to work. And in my experience, if it seems to simple to be true in group theory, it usually is.
I am supposing that x in G exists in a way that no element in G conjugates x into a subgroup H.
Now I am considering x's action on the set of H's cosets, to say then if x fixes a coset, then for some a in G we have . This is where I am not sure if I can do what I am doing. I then move the a over and get .
Since H is a subgroup, 1 is in it. So or equivalently . So conjugates x into H, contradicting my hypothesis, so x cannot fix a coset.