Conductor of an extension and ray class field

Question

Let $K$ be a number field and let $L$ be a finite abelian extension of $K$. Denote with $\mathfrak{f}:=\mathfrak{f}(L/K)$ the conductor of $L/K$. I know that $L\subset K_{\mathfrak{f}}$ and if $L\subset K_{\mathfrak{m}}$ for some modulus $\mathfrak{m}$ then $\mathfrak{f}\mid \mathfrak{m}$ (where $K_{\mathfrak{m}}$ is the ray class field modulo $\mathfrak{m}$). Is it true that if $\mathfrak{f}\mid \mathfrak{m}$ then $L\subset K_{\mathfrak{m}}$?