AMC Entertainment Holdings and Regal Entertainment Group are two separate entities, as of September 2015. The two companies have partnered on a joint venture to directly acquire movies for distribution, however.
Regal Entertainment Group operates Regal Cinemas, Edwards Theatres, United Artists Theatres and Hoyt's Theatres. AMC generally operates theaters under its own name.
AMC is mostly owned by Dalian Wanda Group, who also own Wanda Cinema Line Corp, which is the largest cinema chain in China. Regal Entertainment Group remains self-owned. The two are the largest theater companies in the United States, with a little over 900 locations between them.