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In any group $G$, we can also consider the group isomorphism problem for its subgroups, so we might ask the following problems:

I, If $H_1\cong H_2$ for some two subgroup $H_i$ of $G$, if $H_1$ is normal, is $H_2$ also normal? I believe it is not true in general, the motivation is that we can expect to extend the isomorphism between $H_i$ to an isomorphism on $G$.

II, Define the number of different subgroups in $G$ up to isomorphism to be $\phi(G)$, are there any result on relating $\phi(G)$ to the propertity of $G$, such as its order when $G$ is finite or anything else if $G$ has any other further structures such as $G$ is a algebraic group. etc?

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In the group of symmetries of a square, the subgroup consisting of the identity and the 180-degree rotation about the center of the square is normal, but the subgroup consisting of the identity and the reflection about a diagonal is not normal. Nevertheless, these two 2-element subgroups are isomorphic.

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