Since a group is a set of elements, a subgroup is really just a subset. This means that all of the elements of the subset will also be elements of the set. For example, {a, b} is a subset of {a, b, c}. Every group has two trivial subgroups: the order 1 group containing only the identity element and the group itself.
C2 (rotations by {0, 180}) is a subgroup of C4 (rotations by {0, 90, 180, 270}).
C2 ({identity transformation, reflection}) is a subgroup of the dihedral group
D4.
Page author: Sasie Sealy