This paper is concerned with holomorphic C-semigroups. The main purpose is to give a characterization of the C-complete infinitesimal generator of a holomorphic C-semigroup, which coincides with that of a holomorphic \((C_ 0)\)-semigroup in the case of \(C=I\). We also clarify the relationship between holomorphic C-semigroups and holomorphic semigroups of growth order \(\alpha\). Applying the theory for holomorphic C- semigroups, we deduce a characterization of the complete infinitesimal generator of a holomorphic semigroup of growth order \(\alpha\).