  The Kautz digraphs $K(d,\ell)$ are a well-known family of dense digraphs,
widely studied as a good model for interconnection networks. Closely related to
these, the cyclic Kautz digraphs $CK(d,\ell)$ were recently introduced by
B\"ohmov\'a, Huemer and the author, and some of its distance-related parameters
were fixed. In this paper we propose a new approach to the cyclic Kautz
digraphs by introducing the family of the subKautz digraphs $sK(d,\ell)$, from
where the cyclic Kautz digraphs can be obtained as line digraphs. This allows
us to give exact formulas for the distance between any two vertices of both
$sK(d,\ell)$ and $CK(d,\ell)$. Moreover, we compute the diameter and the
semigirth of both families, also providing efficient routing algorithms to find
the shortest path between any pair of vertices. Using these parameters, we also
prove that $sK(d,\ell)$ and $CK(d,\ell)$ are maximally vertex-connected and
super-edge-connected. Whereas $K(d,\ell)$ are optimal with respect to the
diameter, we show that $sK(d,\ell)$ and $CK(d,\ell)$ are optimal with respect
to the mean distance, whose exact values are given for both families when
$\ell=3$. Finally, we provide a lower bound on the girth of $CK(d,\ell)$ and
$sK(d,\ell)$.
