  We define the Grothendieck group of an $n$-exangulated category. For $n$ odd,
we show that this group shares many properties with the Grothendieck group of
an exact or a triangulated category. In particular, we classify dense complete
subcategories of an $n$-exangulated category with an $n$-(co)generator in terms
of subgroups of the Grothendieck group. This unifies and extends results of
Thomason, Bergh--Thaule, Matsui and Zhu--Zhuang for triangulated,
$(n+2)$-angulated, exact and extriangulated categories, respectively. We also
introduce the notion of an $n$-exangulated subcategory and prove that the
subcategories in our classification theorem carry this structure.
