  A $k$-orbit map is a map with its automorphism group partitioning the set of
flags into $k$ orbits. Recently $k$-orbit maps were studied by Orbani\' c,
Pellicer and Weiss, for $k \leq 4$. In this paper we use symmetry type graphs
to extend such study and classify all the types of $5$-orbit maps, as well as
all self-dual, properly and improperly, symmetry type of $k$-orbit maps with
$k\leq 7$. Moreover, we determine, for small values of $k$, all types of
$k$-orbits maps that are medial maps. Self-dualities constitute an important
tool in this quest.
