I figured out the first part of this years ago, but completely forget how I did it. I looked at the second, but don't think I figured it out.

This I am sure is true, but don't remember why. Suppose that G is a finite group of size $n$, and H is a normal subgroup with |G/H| = $k$. Then at least $\frac{1}{k}$ of the conjugacy classes of G are within H.

This I don't know the answer to. If H is allowed to be an arbitrary subgroup, must H intersect at least $\frac{1}{k}$ of the conjugacy classes?

An example of a simple consequence of the first statement is that if we look at $S_n$ and $A_n$ is that at least half the partitions of $n$ have an even number of even parts.