https://arxiv.org/abs/2609.37562 For every integer n≥2, we prove that every countable group embeds in a group of type FPn. We also construct a group of type Fn containing a copy of every recursively presented group. Consequently, a finitely generated group embeds in a group of type Fn if and only if it is recursively presented. This answers questions of Fournier-Facio and Zaremsky, and confirms a suggestion of Gromov.