The lattice of primary ideals of orders in quadratic number fields

Let $O$ be an order in a quadratic number field $K$ with ring of integers $D$, such that the conductor $\mathfrak{F}=fD$ is a prime ideal of $O$, where $f\in \mathbb{Z}$ is a prime. We give a complete description of the $\mathfrak F$-primaryidealsof$O$. They form a lattice with a particular structure by layers; the first layer, which is the core of the lattice, consists of those $\mathfrak{F}$-primary ideals not contained in ${\mathfrak{F}}^2$. We get three different cases, according to whether the prime number $f$ is split, inert or ramified in $D$.