There is one Church, the Christian Church. It started as one and was referred to as Catholic. Error entered the Church and disagreements arose. Unity was soon lost as groups left or were ejected from the original Church. The Christian Church was no longer found in one organization. The largest part of the Church retained the name Catholic. But the fragments were also part of the actual universal Church. These fragments have continued until today in such forms as the Assyrian Church of the East, the Armenian Apostolic Church, the Coptic Church and various other groups that were thought to be monophysite. As such the claim of unity until the Great Schism is not accurate.
In 1054, the Great Schism resulted in the Catholic Church splitting again. One portion retained the name Catholic Church for itself while the other became the Orthodox. Both of these and the earlier fragments then made up the Christian Church.
Error continued to occur and opposition arose to it through such groups as the Waldenes, the Lollards and the Hussites. Eventually during the Reformation dissent erupted into further fractures in the Church as groups left or were expelled as a result of trying to correct the errors they perceived. As such there were many groups forming the Christian Church. The largest of these groups continued to claim the name Catholic.
All the groups contain parts of the true universal Christian Church. Splits do not deprive the parts of the heritage and history that they shared until the split.
Today we see the Christian Church fragmented into many parts, each of which is part of the Church started by Jesus in the beginning.
Various parts of the Church feel that they are incapable of error and that they are the only “true” church. The claim of infallibility is perhaps the greatest error in the Church and prevents the reunification of the one Christian Church into one organization.