A novel inbred maize line designated PHC5H and seed, plants and plant parts thereof. Methods for producing a maize plant that comprise crossing inbred maize line PHC5H with another maize plant. Methods for producing a maize plant containing in its genetic material one or more traits introgressed into PHC5H through backcross conversion and/or transformation, and to the maize seed, plant and plant part produced thereby. Hybrid maize seed, plant or plant part produced by crossing the inbred line PHC5H or an introgressed trait conversion of PHC5H with another maize line. Inbred maize lines derived from inbred maize line PHC5H, methods for producing other inbred maize lines derived from inbred maize line PHC5H and the inbred maize lines and their parts derived by the use of those methods.