WordNet records no synonym and no antonym for exarch. It is the only member of each of its synsets, which is common for technical, taxonomic and single-sense vocabulary — there is no other English word that means the same thing. The broader and narrower terms below are the useful relations in that case.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
a bishop in one of several Eastern Orthodox Churches in North America
a bishop in eastern Christendom who holds a place below a patriarch but above a metropolitan
a viceroy who governed a large province in the Roman Empire
Synonyms, antonyms and the broader and narrower relations for exarch come from Princeton University’s WordNet. Full sources and licences.