Let $K$ be a quadratic number field, $h(K)$ its class number. Is it true that: for every prime factor $r$ of $h(K)$, there is a prime $p$ that decomposes in $K$, and, if $(p) = IJ$, then $[I],[J]$ have order $r$ in the ideal class group?
Examples: $K=\mathbb{Q}(\sqrt{-87})$, $h(K) = 6$. For the prime factor $r=3$, pick $p=7$, then we have $(7) = (7, 16+\sqrt{-87})(7, 16-\sqrt{-87})$, and $[(7, 16\pm\sqrt{-87})]$ have order $3$ in the ideal class group.
$K=\mathbb{Q}(\sqrt{-159})$, $h(K) = 10$. For the prime factor $r=5$, pick $p=7$, then we have $(7) = (7, 124+3\sqrt{-159})(7, 124-3\sqrt{-159})$, and $[(7, 124\pm3\sqrt{-159})]$ have order $5$ in the ideal class group.