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Regarding IAG’s structure, from Wikipedia:

Qatar Airways is a minority shareholder, and held 20.01% of the shares on 29 July 2016.

Divisions, subsidiaries and franchises:
The structure of the main operating companies is:

Republic of Ireland: Aer Lingus
Republic of Ireland: Aer Lingus Regional (franchise)
United Kingdom: British Airways
United Kingdom: BA CityFlyer
France: OpenSkies
South Africa: Comair (18% stake)
Denmark: SUN-AIR (franchise)
Spain/United Kingdom: IAG Cargo (merger between Iberia Cargo and British Airways World Cargo)
Spain: Iberia
Spain: Air Nostrum (franchise, trading as Iberia Regional)
Spain: Iberia Express
Spain: Level
Spain: Vueling
Avios Group (operates the IAG frequent-flyer programme)