Cessna 172 Flight Instructor reported severe engine vibration during climb. Instructor returned to departure airport.

Narrative

During the preflight and run up; nothing unusual was found. After takeoff from runway XXL at ZZZ; at about 1200 ft; I heard a very loud bang followed immediately by severe engine vibrations and unusual engine sounds. I was still in a normal climb just before this happened. I took the controls from my student and started turning back to ZZZ and told ATC that I needed to land immediately. The engine was still producing some power. It was stable around 2000 rpm but running extremely rough. I did reduce the throttle slightly after the initial bang to reduce the vibration. Surprisingly; the oil pressure and temperature were normal even though it was clear something went catastrophically wrong. I did not declare an emergency because the engine was still producing some power; and I did not know the extent of the damage. I did not attempt any troubleshooting to make the engine run normally because of the limited time I had to set up for approach and landing. Also; I did not think the engine was in a recoverable state due to the vibration and sounds coming from it. I landed on runway XY at ZZZ and taxied back to parking without issue. After shutting down the engine; the most obvious damage was the front right engine cylinder. It was completely broken in half and pushed out about a half inch. I believe this was the cause of the engine roughness and partial loss of power. I believe the bang I heard was from the cylinder breaking. Also; oil was coating the right side of the engine and coming out of the exhaust.