For $\mathcal{O}_{K}$, the integer ring of a global field, we denote $S$ to be any set of primes of a global field $K.$ Let $$\mathcal{O}_{K,S}:=\{x\in K\mid v_{\mathfrak{p}}\geq 0\text{ for }\mathfrak{p}\notin S\}$$ be the ring of $S$-integers of $K$ (see Neukirch, Schmidt, Wingberg Cohomology of Number Fields, Ch. VIII, § 3).
Ideal class group of $\mathcal{O}_{K,S}$ is called $S$-ideal class group, denoted by $Cl_{K,S}$.
Neukirch, Schmidt, Wingberg Cohomology of Number Fields, Ch. VIII, § 3, p. 452 states that
The $S$-ideal class group is the quotient of the usual ideal class group $\mathrm{Cl}_K$ of $K$ by the subgroup generated by the classes of all prime ideals in $S$.
I want to know concrete examples.
Let $K=\Bbb{Q}(\sqrt{-29})$. Let $S=\{2,17\}$. $K$ has class number $2$, thus $\mathrm{Cl}_{K,S}$ has order $1$ or $2$.
$\mathrm{Cl}_K$ is generated by $(3,2+\sqrt{-29})$. Prime ideals in $S$ is generated by $2$ and $17$. What is the $\mathrm{Cl}_{K,S}$ in this situation? Thank you for your help.