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 $\displaystyle xaH=aH$. This is where I am not sure if I can do what I am doing. I then move the a over and get $\displaystyle a^{-1}xaH=H$.

Since H is a subgroup, 1 is in it. So $\displaystyle a^{-1}xa1 \in H$ or equivalently $\displaystyle a^{-1}xa \in H$. So $\displaystyle a^{-1}$ conjugates x into H, contradicting my hypothesis, so x cannot fix a coset.