In this talk we discuss an efficient discretization method for the solution of the unsteady incompressible Navier-Stokes equations based on a high order (Hybrid) Discontinuous Galerkin (HDG) formulation. The discretization features exact incompressibility and a reduced stabilization in the HDG formulation. Efficiency is enhanced with the disctinction between stiff linear and less stiff non-linear parts with respect to their temporal and spatial treatment. We exploit the flexibility of operator-splitting time integration schemes to combine two spatial discretizations which are tailored for two simpler sub-problems: a corresponding hyperbolic transport problem and an unsteady Stokes problem. The method is presented, implementational aspects are discussed and the performance of the discretization is demonstrated on two and three dimensional benchmark problems.