 Solution

Since $|D_5|=10$, Lagrange's theorem permits proper subgroup orders only $1,2,5$. The complete list isTo show completeness, an order-two subgroup must be generated by its unique nonidentity element, necessarily one of the five reflections. An order-five subgroup is cyclic, since its order is prime; its generator cannot be a reflection, and every nonidentity rotation generates the same rotation subgroup. There is only one order-one subgroup.

Thus there are seven proper subgroups when the trivial subgroup is included, or six nontrivial proper subgroups. These are the prime-sided case of the classification of subgroups of a dihedral group.