To avoid AVIC as a monopoly, China breaks it up, and starts COMAC and AECC. Maybe that can help you understand the relationship among AVIC, COMAC, and AECC.
AVIC was split into AVIC I and AVIC II in July 1999.
AVIC I was historically focused on large planes such as bombers (Xian H-6, Xian JH-7), medium commercial planes (ARJ21), or fighter planes (J-7, J-8, J-10, J-11 and JF-17), while AVIC II was mainly focused on smaller planes, such as trainers (JL-8, utility aircraft (licensed version of Cessna 208 Caravan), L-15, and CJ-6), small passenger airliners (Harbin Y-12), medium range transport aircraft (Y-8), and helicopters (Z-8, Z-9, WZ-10 and Z-11).
However, the two corporation were merged in October 2008 because the previous separation resulted in split resources and led to redundant projects.
AVIC is a corporation having over 100 subsidiaries, 27 listed companies.
COMAC was established in an effort aimed at reducing China's dependency on
Boeing and
Airbus and eventually compete with
Airbus and
Boeing in the civilian airline industry. Companies dealing medium commercial planes and previously under AVIC I were transferred to COMAC.
AECC was established to focus on aeroengine and related technology development. The corporation consists of 46 affiliate companies, including 22 engine companies, several institutes, 3 aeroengine-repairing factories and some other small companies, with the majority of the affiliate companies having been split from AVIC.
In short, AVIC became major shareholder of both COMAC and AECC by transferring its assets and companies.
I think establishing of COMAC and AECC was not because of AVIC's monopoly in China's aviation industry. This was done only to have better and efficient control on the assets and companies.
Hierarchy and working of enterprises in China are completely different as compared to their western counterparts.
Almost all companies working under these corporations are listed in Chinese stock exchanges. These companies do compete with each other. Very prominent example is the manufacturers of Chinese AESA radars KLJ-7A and LKF601E both competing for JF-17. So monopoly is out of question.