Let $k$ be a finite abelian extension of $\mathbb{Q}$. Class field theory states that $k$ corresponds to some open subgroup of finite index $U_k \subset \mathbb{A}_{\mathbb{Q}}^*/ \mathbb{Q}^*$ where $\mathbb{A}_{\mathbb{Q}}$ denotes the adeles of $\mathbb{Q}$. Specifically the subgroup $U_k$ is the image of the idelic norm map. Moreover every open subgroup of finite index arises this way for some unique abelian extension.

The subgroup $U_k$ should know everything about the abelian extension $k$. So I was wondering, does it know the class group?

Let $U \subset \mathbb{A}_{\mathbb{Q}}^*/ \mathbb{Q}^*$ be an open subgroup of finite index with corresponding abelian extension $k$. Is there some way to calculate the class group of $k$, just knowing $U$?