In Cox's book "Primes of the form $x^2 + ny^2$", he proves that in a quadratic imaginary field $K$, if $\mathcal O$ is an order of conductor $f \in \mathbb Z$, we have that the class group $\mathrm{Cl}(\mathcal O)$ of $\mathcal O$ is isomorphic to the quotient $I\_K(f) / P\_{K,\mathbb Z}(f)$, where $I\_K(f)$ is the free abelian group generated by the prime ideals of $\mathcal O\_K$ coprime to $f\mathcal O\_K$, and $P\_{K,\mathbb Z}(f)$ is the subgroup of principal ideals $\alpha \mathcal O\_K$ where $\alpha$ is congruent mod $f\mathcal O\_K$ to an integer coprime to $f$ (proposition 7.22). **So $\mathrm{Cl}(\mathcal O)$ is a quotient of the ray class group relative to $f\mathcal O\_K$.**


I suspect a more general result, but I cannot find any reference. Let $K$ be a number field, and $\mathcal O$ an order of conductor $\mathfrak f \subset \mathcal O\_K$,



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> **Can $\mathrm{Cl}(\mathcal O)$ be expressed as a quotient of $\mathrm{Cl}\_\mathfrak f(K)$, the ray class group of $K$ relative to $\mathfrak f$ ?**
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The problem actually boils down to proving that the map $$\varphi : I\_K(\mathfrak f) \longrightarrow \mathrm{Cl}(\mathcal O) : \mathfrak a \longmapsto \mathfrak [\mathfrak a \cap \mathcal O]$$
is a surjection. Indeed, if it is the case, then $\mathrm{Cl}(\mathcal O) \cong I\_K(\mathfrak f) / \ker(\varphi)$, but it is easy to see that $P\_K(\mathfrak f) \subset \ker(\varphi)$ (where $P\_K(\mathfrak f)$ is from the definition of the ray class group, i.e. $\mathrm{Cl}\_\mathfrak f(K) \cong I\_K(\mathfrak f) / P\_K(\mathfrak f)$), so $\mathrm{Cl}(\mathcal O)$ would be a quotient of $\mathrm{Cl}\_\mathfrak f(K)$.



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> **Is $\varphi$ surjective ?**
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Class groups of orders
The answer is yes, essentially by weak approximation.


Let $\mathbb{I}\_K=\prod'\_{v}K\_v$ denote the finite Idelles over $K$. Now let $\prod\_v\mathcal{O}\_v^{\times}=\widehat{\mathcal{O}}^{\times}$ denote the multiplicative group of the completion of $\mathcal{O}$. Then (as Cox explains) you can write 


$$Cl(\mathcal{O}) = K^{\times}\backslash\mathbb{I}\_K/\widehat{\mathcal{O}}^{\times}.$$


Also letting $\mathbb{I}\_{K,f}:=\prod'\_{v\not\mid f}K\_v$ we have
$$I\_K(f) = \mathbb{I}\_{K,f}/\prod\_{v\not\mid f}\mathcal{O}\_{K,v}^{\times}.$$


Then the map $I\_K(f)\rightarrow Cl(\mathcal{O})$ is induced by the natural inclusion $\mathbb{I}\_{K,f}\hookrightarrow \mathbb{I}\_K$ so your question amounts 
to establishing the equality 
$$\prod\_{v\mid f}K^{\times}\_v = K^{\times}\cdot\prod\_{v\mid f}\mathcal{O}\_v^{\times}.$$


Weak approximation says that $K^{\times}$ is dense in $\prod\_{v\mid f}K^{\times}\_v$, which implies your claim.


