As I recall the engines were made by Chrysler.
The engines were made by Rocketdyne, then part of Rockwell, more recently Aerojet Rocketdyne which per the internet is owned by L3Harris. Harris, of course, is best known for the development of offset printing presses and the Harris Intertype typesetting machine that competed with Linotype.
The Saturn’s first stage - the S-IC - was built by the Boeing Company at the Michoud Assembly Facility, New Orleans, where the Space Shuttle external tanks would later be built by Lockheed Martin. . The S-IC was powered by five Rocketdyne F-1 engines arrayed in a quincunx.
The S-II was built by North American Aviation at Seal Beach, California. Using liquid hydrogen and liquid oxygen, it had five Rocketdyne J-2 engines in a similar arrangement to the S-IC, and also used the four outer engines for control.
he S-IVB stage was built by the Douglas Aircraft Company at Huntington Beach, California. It had one J-2 engine and used the same fuel as the S-II.
The S-IVB burned for almost six minutes, giving the spacecraft a velocity close to the Earth’s escape velocity of 25,053 mph (40,319 km/h). This gave an energy-efficient transfer to lunar orbit, with the Moon helping to capture the spacecraft with a minimum of CSM fuel consumption.[11]
In 1968, Boeing studied another Saturn-V derivative, the Saturn C-5N, which included a nuclear thermal rocket engine for the third stage of the vehicle.[93] The Saturn C-5N would carry a considerably greater payload for interplanetary spaceflight. Work on the nuclear engines, along with all Saturn V ELVs, ended in 1973.[94]
https://en.wikipedia.org/wiki/Saturn_V#S-IC_first_stage