In the mirror universe, the Sato Dynasty was the ruling family of the Terran Empire from the mid-22nd century to the mid-23rd century. The line originated with Empress Sato I, who took power in 2155. The line included Empress Sato II and Sato III, the last of the Sato Dynasty, who was overthrown by Spock in 2277. (ENT - Mirror Universe novel: Age of the Empress; TOS - Mirror Universe novel: The Sorrows of Empire)

It is unclear whether the binary clone created from the DNA of Sato I and Shran was a part of this royal line, or if another of Sato's male consorts was the patriarch of the dynasty.
It is also unclear whether or not there were other Emperors and Empresses in the Sato Dynasty besides the three named. Given the 122-year duration of the family's rule, however, this would seem more likely than not.