To answer your question. Yes, four of the five Great Patriarchal Sees do belong to the Eastern Orthodox Communion. Though, two of these Sees (Alexandria and Antioch) are divided between Chalcedonian and Non-Chalcedonian Patriarchates. Sadly, all four (six if we include the Non-Chalcedonian Patriarchs) of these Ancients Sees are out of communion with the See of Peter.
There are also Catholic Patriarchs in Antioch (actually three of them, Maronite, Melkite, and Syrian), Alexandria (Coptic and Melkite), and Jerusalem (Melkite). The Melkite Patriarch holds the title of Patriarch of Antioch, Alexandria, and Jerusalem.