Grothendieck rings of ordered subgroups of $\mathbb{Q}$

Let $G$ be a proper subgroup of $\mathbb{Q}$ and $S_G$ be the set of primes $p$ for which $G$ is $p$-divisible. We show that the model-theoretic Grothendieck ring of the ordered abelian group $(G;+,<)$ is a quotient of $(\mathbb{Z}\mathrm{/}q\mathbb{Z})[T]\mathrm{/}(T+T^2)$, where $q$ is the largest odd integer that divides $p-1$ for all $p\mathrm{\notin }{S}_G$. This implies that the Grothendieck ring of $(G;+,<)$ is trivial in various salient cases, for example when $S_G$ is finite, or when $S_G$ does not contain some prime of the form $2^n+1$, $n\in \mathbb{N}$.