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In 1965, Pope Paul VI decreed in his motu proprio Ad Purpuratorum Patrum that patriarchs of the Eastern Catholic Churches who were named cardinals (i.e., patriarch cardinals) would also be part of the episcopal order, ranking after the six cardinal bishops of the suburbicarian sees (who had been relieved of direct responsibilities for those sees by Pope John XXIII three years earlier). Patriarch cardinals do not receive title of a suburbicarian see, and as such they cannot elect the dean or become dean. There are currently three Eastern Patriarchs who are cardinal bishops:

Who ranked after the seven cardinal bishops of the suburbicarian sees?
unanswerable