There is a classical formula stating:

Let $K$ be a number field with ring of integers $\mathcal{O}_K\subseteq K$ and 
let $\mathcal{O}\subseteq \mathcal{O}_K$ be any non-maximal order with conductor $\mathfrak{n}$. Then
$$\frac{\#(\mathcal{O}_K/\mathfrak{n}\mathcal{O}_K)^\times}{\#(\mathcal{O}/\mathfrak{n}\mathcal{O})^\times} = [\mathcal{O}_K^\times:\mathcal{O}^\times]\frac{\# \text{Pic}(\mathcal{O})}{\#\text{Pic}(\mathcal{O}_K)}.$$

My question is: do we know anything about the growth on the RHS, especially in the case K is a CM field? In particular, can we bound from below the LHS by the conductor $\mathfrak{n}$? I am in particular interested when the CM field corresponds to an isotypic CM abelian variety of dimension greater than $1$.