Let $K$ be an imaginary quadratic field of discriminant $D_K$, let $n$ be a positive integer. Call $H_n$ the ring class field of $K$ of conductor $n$. Then, all primes of $K$ that do not divide $n$ are unramified in $H_n/K$.
Is it true that all primes of $\mathbb{Q}$ that do not divide $nD_K$ are unramified in the extension $H_n/\mathbb{Q}$?
I was hoping to say yes, but non-ramification is in general not transitive in field extensions...