Let $Z(G)$ be the center of a non-commutative group $G$. Show that the factor group $G/Z(G)$ has at least 4 distinct subgroups.
So I have a guess at this however I feel that the answer is a bit too easy to be correct. So we have at least the two trivial subgroups, as we have to maintain our identity element under the inner automorphism, and the entire group. However since we in a sense 'kill' all the commuting elements by quotient by the centre we also have the two distinct subgroups of the left coset and the right coset.
So the four distinct subgroups are the identity, the entire group, and the left and right cosets. Is this correct or am I missing something?