Which of the following groups $Q$ is normal in $G$?

Question

Which of the following groups can be a normal $2$-Sylow subgroup of a non-abelian group $G$ of order $56$?

1) $\mathbb{Z}_8$

2) $\mathbb{Z_4}\times \mathbb{Z_2}$

3) $\mathbb{Z_2}\times \mathbb{Z_2}\times \mathbb{Z_2}$

I know that if a $2$-Sylow subgroup is the only such Sylow subgroup, then it will be normal in $G$, however, I am not sure how to apply it to this .

Thanks for any help.

[Editor's note: I rephrased the first sentence to match with my interpretation. IMHO it is the version that makes sense as a question, and the OP seems to like my answer, so I guess it is correct. Cheers, JL.]

Answer 1

Hint: Those candidate subgroups are all abelian. If a $2$-Sylow subgroup $P$ is normal, then conjugation by an element of order seven is an automorphism of $P$. Because the big group is non-abelian, that automorphism must be non-trivial, and thus of order seven. Which of those groups have automorphisms of order seven?