  In network coding, a flag code is a set of sequences of nested subspaces of
$\mathbb{F}_q^n$, being $\mathbb{F}_q$ the finite field with $q$ elements. Flag
codes defined as orbits of a cyclic subgroup of the general linear group acting
on flags of $\mathbb{F}_q^n$ are called cyclic orbit flag codes. Inspired by
the ideas in arXiv:1403.1218, we determine the cardinality of a cyclic orbit
flag code and provide bounds for its distance with the help of the largest
subfield over which all the subspaces of a flag are vector spaces (the best
friend of the flag). Special attention is paid to two specific families of
cyclic orbit flag codes attaining the extreme possible values of the distance:
Galois cyclic orbit flag codes and optimum distance cyclic orbit flag codes. We
study in detail both classes of codes and analyze the parameters of the
respective subcodes that still have a cyclic orbital structure.
