# Isomorphism of a quotient group and a subgroup [closed]

Let $$G_1$$ be a finite group with a subgroup $$G_2$$ and a normal subgroup $$H$$. Suppose that $$G_1/H$$ is isomorphic to a subgroup of $$G_2$$. Do we have $$G_1/H$$ is isomorphic to $$G_2$$?

## closed as off-topic by Shaun, user1729, Ernie060, Adrian Keister, José Carlos SantosMay 16 at 18:34

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Take $$G_1=\mathbb{Z}_2\oplus \mathbb{Z}_4$$, let $$G_2$$ be the $$\mathbb{Z}_4$$ summand, and let $$H:=\mathbb{Z}_2\oplus 2\mathbb{Z}_4=\{(0,0),(0,2),(1,0),(1,2)\}.$$ Then $$G_1/H$$ is a group of order $$2$$, so it's isomorphic to a subgroup of $$\mathbb{Z}_4\simeq G_2$$. But of course it's not isomorphic to $$\mathbb{Z}_4$$.