Finding the # of elements of a group that are not in any conjugate to its subgroup

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Also, just to be clear does G\H denote the right cosets where H<G and Hg = (hg: such that h is an element of H)?

Here's a solution with just a couple of details left for you:

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Thanks a lot for your help, but there is something I'm having trouble with. I'm not used to the H^x notation you were using so I changed one line so it makes more sense to me, but what I don't understand is the implication that I denoted with a red *.

Given a function f from X to Y, if f is 1:1, then $\displaystyle f(A\cap B)=f(A)\cap f(B)$ for any subsets A, B of X. So in this situation, let f be the function from G to G where $\displaystyle f\,:\,g\mapsto x_j^{-1}gx_j$. So switching to exponential notation (you can change if you like):
$\displaystyle <e>=H\cap H^{x_ix_j^{-1}}$ implies
$\displaystyle <e>=<e>^{x_j}=(H\cap H^{x_ix_j^{-1}})^{x_j}=H^{x_j}\cap H^{x_ix_j^{-1}x_j}=H^{x_j}\cap H^{x_i}$