Alyosha’s statement reveals Dostoevsky’s undeniable prejudice against the Catholic Church. While it’s certainly true that the Jesuits in particular have been guilty of seeking temporal power, this is by no means their avowed aim. Moreover, by this period in their history the order was in fact almost completely charitable and religious in nature (abuses were largely in the 16th–17th centuries, and even then were not in accord with the stated aims of the Society).
Additionally, as a rather secretive Society, the Jesuits are a favourite subject of conspiracy theorists.