Here's an excerpt from IVP's New Dictionary of Theology:
"Hippolytus led his followers into schism shortly after Callistus was elected Bishop of Rome in 217. The two men were personal rivals ... the two men clashed on church discipline: which sinners could be reconciled to the church and on what terms, and what would be the church's attitude in fellowship on social and moral questions. Callistus favoured taking a forgiving and moderate approach, willing to reconcile those guilty of sexual sins and to recognize marriages not sanctioned by Roman law. Since the church is a saving society it should be inclusive in membership. Hippolytus favoured keeping serious sinners under discipline until their deathbed, leaving forgiveness in the hands of God. He wanted a church of the pure. The two men also represented rival Christologies. Callistus emphasized the oneness of God, trying to walk a middle way between the modalism of Sabellius and what he called the ditheism of Hippolytus. The latter developed his doctrine of Christ from the Logos Christology of the Apologists and from Ireneaus."
That sounds more than a bit like current moves and motivations in the Anglican Communion today!
But the article goes on:
"Hippolytus and Pontianus, Callistus' second successor, were exiled at the same time. They apparently became reconciled, for the two parties reunited and commemorated both men as martyrs."
Which is a point I have made before: where possible avoid schism since the warring parties will only have to reconcile eventually!
(Excerpts from 'Hippolytus' by E. Ferguson, in New Dictionary of Theology, IVP, 1988, p. 304).