Let $C$ be a fixed elliptic curve over $ℚ$. The group $C(ℚ)$ is a finitely generated Abelian group and we have $$C(ℚ)≃ℤ^{r}⊕C(ℚ)^\mathrm{tors}$$ where $C(ℚ)^\mathrm{tors}$ is a finite abelian group (is the subgroup of elements of finite order in $C(ℚ)$).
My question is: How to find the subgroups of the group $C(ℚ)$? I think they have the forms: $ℤ^{m}⊕C(ℚ)^\mathrm{tors}$ where $0≤m≤r$. However, I have no idea to start