>Monarchs
Diamond Authority
>Noble gems
Hessonites, emeralds. They usually act as high military commanders and one step above of the court gems.
>Court gems
Things we've seen in 4th era ballroom and Garnet's flashback when encountering with Rose Quartz at the first time. Sapphires and pearls are also included in this list.
>Administrative gems
Blue Agates, zircons and aquamarines.
>infrastructural gems
Peridots, lapises and bismuths.
>Solider gems
Amethysts, jaspers, quartzes, nephrites, rubies, topazes. They provde the military force of Gempire.

This is the most accurate information regarding the gem hierarchy. Feel free to ask anything.