subgroup of direct product of two groups

Its seems true but I did not see in any group theory book. Let $G\times_{c} H$ be product of two abelian groups (both are infinite). Then $I$ is subgroup of product iff $I = A\times_{c} B$ where $A$ is a subgroup of $G$ and $B$ is a subgroup of $H$. Thanks

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This is not true: One has the diagonal embedding $G \to G \times G$ by $g \mapsto (g, g)$.