This study compares various multigrid strategies for the fast solution of elliptic equations discretized by the Hybrid High-Order method. Combinations of h-, p-and hp-coarsening strategies are considered, combined with diverse intergrid transfer operators. Comparisons are made experimentally on 2D and 3D test cases, with structured and unstructured meshes, and with nested and non-nested hierarchies. Advantages and drawbacks of each strategy are discussed for each case to establish simplified guidelines for the optimization of the time to solution.