Complex sentence: It does not however indicate that the Archbishop is the most senior clergyman of his Chistian denomination in Ireland but rather he is the second-most senior figure, the most senior figure in both denominations, the Archbishops of Armagh, possessing the title Primate of All Ireland.
Split into simpler sentences: It does not however indicate that the Archbishop is the most senior clergyman of his Chistian denomination in Ireland. It actually indicates that he is the second-most senior figure, the most senior figure in both denominations, the Archbishops of Armagh, possessing the title Primate of All Ireland.