Definition: (conductor of a number field) Let $K$ be a number field with abelian Galois group over $\Bbb Q$. The conductor $n$ is the smallest even number such that $K\subset \Bbb Q(\zeta_n).$
Assumption
The Hilbert class field $H(K)$ of $K$ is abelian over $\Bbb Q.$
My attempt:
Suppose the conductor of Hilbert class field $H(K)$ different from the conductor of $K.$ Say the conductor of $H(K)$ to be $m$ and the conductor of $K$ to be $n$. clearly $n$ divides $m.$ I think $m$ should have at least one different prime factor $p$ than $n$, thinking to conclude the prime above $p $ in $K$ ramifies in $H(K)$. Any help here very much appreciated.