# Subgroup structure of quaternion group

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2. The center, given by $\pm 1$. This is isomorphic to the cyclic group of order two. (1)
3. The three four-element cyclic subgroups, generated by the elements $i,j,k$ respectively. They are all related by outer automorphisms, though no two of them are conjugate. (In fact, they're all normal). Each is isomorphic to the cyclic group of order four. (3)