Let $G$ be a crystallographic group of the Euclidean plane. Define the lattice and the point group of $G$. Suppose that the lattice for $G$ is $\{(k, 0): k \in \mathbb{Z}\}$. Show that there are five different possibilities for the point group. Show that at least one of these point groups can arise from two groups $G$ that are not conjugate in the group of all isometries of the Euclidean plane.