  A gyrogroup is a nonassociative group-like structure modelled on the space of
relativistically admissible velocities with a binary operation given by
Einstein's velocity addition law. In this article, we present a few of groups
sitting inside a gyrogroup $G$, including the commutator subgyrogroup, the left
nucleus, and the radical of $G$. The normal closure of the commutator
subgyrogroup, the left nucleus, and the radical of $G$ are in particular normal
subgroups of $G$. We then give a criterion to determine when a subgyrogroup $H$
of a finite gyrogroup $G$, where the index $[G\colon H]$ is the smallest prime
dividing $|G|$, is normal in $G$.
