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The Eastern Orthodox Church separated with the Roman Catholic Church in the 11th century (but the process was gradual and the schism was finalized, so to speak, in the 15th century, with the rejection of the Council of Florence).
The Eastern Orthodox Church recognizes 7 ecumenical councils (Nicea I, Constantinople I, Ephesus, Chalcedon, Constantinople II, Constantinople III, Nicea II). It consists today of 14 autocephalous churches:
- Ecumenical Patriarchate of Constantinople
- Patriarchate of Alexandria
- Patriarchate of Antioch
- Patriarchate of Jerusalem
- Patriarchate of Russia
- Patriarchate of Serbia
- Patriarchate of Bulgaria
- Patriarchate of Romania
- Patriarchate of Georgia
- Church of Cyprus
- Church of Greece
- Orthodox Church of Albania
- Orthodox Church of Poland
- Orthodox Church of the Czech lands and Slovakia
And two churches' autocephaly are disputed:
- Orthodox Church in America (everyone recognizes them as canonical but Moscow and Constantinople disagree on how much autonomy the OCA has and who it answers to)
- Orthodox Church of Ukraine (only recognized as canonical and autocephalous by Constantinople right now, this is a major ongoing dispute)
Each autocephalous Church has its own tradition, although generally the Church is distinguished in two major traditions: Greek and Slavic. This affects the tones used for hymns, the language used at the liturgy, etc.
>Or are they just for that ethnic group?
The different Churches represent different jurisdictions and traditions, that's it. Of course, Russians, for example, will prefer to frequent a church where Russian is used, but that doesn't even mean they will go to a church that's under the Patriarchate of Moscow (see the Exarchate for churches of Russian tradition in Western Europe, which is under the Ecumenical Patriarchate).
The Oriental Orthodox Church separated with the other churches in the 5th century, with the rejection of the Council of Chalcedon. It consists today of the following Churches:
- Coptic Orthodox Church of Alexandria
- Syriac Orthodox Church of Antioch
- Armenian Apostolic Church
- Malankara Orthodox Syrian Church
- Ethiopian Tewahedo Orthodox Church
- Eritrean Tewahedo Orthodox Church
The Assyrian Church of the East separated with the other churches in the 5th century, with the rejection of the Council of Ephesus. Although it was the largest church in the world once, it has now been reduced to a very small and fragmented church.