Suppose $K$ is a number field which is Galois over $\mathbb{Q}$ and let $H(K)$ be the Hilbert class field of $K$. Also assume that $H(K)$ is Galois over $\mathbb{Q}$ with $Gal(H(K)/\mathbb{Q})$ abelian.
I want to prove that the conductors of $K$ as well as $H(K)$ are same. If $f(K)$ and $f(H(K))$ stand for the conductors of $K$ and $H(K)$, respectively then all I could establish is that $f(K) \mid f(H(K))$. How can I proceed to prove the other way?