Some notes:

(1) Any measurable  proper subgroup of $\mathbb {R}^{1}$ is of measure $0$.

(2) Any non-measurable subgroup $G$ of $\mathbb {R}^{1}$ charges fully everywhere, i.e., for any interval $I$, $m^{\ast}(G \cap I)=|I|$, where $m^{\ast}(\cdot)$ denotes the outer Lebesgue measure.

(3) Non-measurable subgroup of $\mathbb {R}^{1}$ exists.