The Flagship is the Boating nerve center at the center of Azys Lla, where research was once conducted on three captive deities from the southern continent, known as the "Warring Triad."
— In-game description
Aetherochemical Research Facility
This floating structure once served as a research facility of the ancient Allagan Empire, wherein studies in restraining dragons and mastering primals were conducted. Test subjects from failed experiments prowl its long-since-abandoned halls.
The Fractal Continuum
The Fractal Continuum appears to be the hull of a once mighty warship converted after its decommission into a museum, intended to preserve for eternity precious artifacts from the thousand-year history of the Allagan Empire. From everyday tools to advanced military technology, the objects within are a testament to the fallen civilization's surpassing might and prosperity.
This floating islet in a lonely corner of Azys Lla served as an isolation area for chimerical creations which failed to meet established safety parameters. Historical records indicate that the facility began its life as a research center for advanced agricultural crossbreeding.
Mobile Supply Cache
The mobile supply cache was, in essence, a traveling warehouse which could move from facility to facility providing researchers with their requested supplies.
Engineers surmise that the Hyperstellar Downconverter was developed to communicate with Dalamud, the false moon containing Bahamut which had been propelled into the heavens. It was later repurposed and improved upon to facilitate communication over the massive distances between stars.