Find a nonabelian subgroup of order $10$ in $D_{15}$ Find a nonabelian subgroup of order $10$ in $D_{15}$.
I know that I have to show that it has a reflection. How could I prove that if the subgroup contains all rotations, it wouldn't be a subgroup?

$D_{15}$ is the symmetries of a regular 15-gon. "Extend" a regular pentagon in Euclidean plane to a regular 15-gon so that any symmetry of the pentagon is in $D_{15}$. So, $D_{15}$ contains a copy of $D_5$.
