here are all the group Faction with in the federation and there symboles

UNCF Cosmo Navy also known as the EDF aka Earth Defens force

they mostly protect earth and  the soler system 

they are the main part of the navy

they are hom to the shock cannon project 

.

UNSC

they protect all the colony worlds and supply lanes 

they are apart of the main army and navy

they are home to the spartan and ODST program and   MAC cannons and railguns

.

Republic

they are the main army along with the UNSC and  are also the same in Navy

they are also the group that make clone trooper and clone amour and weapons

there many Regiment  in there from some that help the navy to a few that go on commando Mission or  some that do scouting there even a Police Regiment 

.

Star fleet

they are the main  explorer and science group as they go off with  a few UNSC and Republic ship  as they help Exploer unknown space and  there main gole is "to seek out new life and new civilizations. To boldly go where no one has gone before!"

.

Three regiments ( even thow there four)

Cadet Corps

The Cadet Corps oversees the education and training of recruits. They assess the qualifications and eligibility of the cadets over the course of the training regime. Upon graduation, the cadets choose either of the three main branches to join and formally begin their careers in the military. A commandant of the Cadet Corps

.

 Scout Regiment

Ranked among the most elite soldiers, they are tasked with attempting to reclaim territory that has been lost to the Titans. Despite being the most well-trained, they often come under dispute from those within the Walls and are constantly under threat of being disbanded since they suffer high losses, and usually fail to supply results.

they also go off in space on a few of the exploer ship or supply ship to also do there job

.

 Military Police Regiment

The Military Police are in charge of keeping order inside the Walls  of each city and on there home island  and protecting the world Councal  as they help the  Clone shock troopers to keep the peace and due to their important function as guardians of the most important humans, only the top 10 cadets from a graduating class are permitted to join them

.

Garrison Regiment.

The Garrison is responsible for patrolling, maintaining, and protecting the Walls. They are set the task of evacuating citizens and dispatching Titans in the case of an emergency breach in the Walls. Falling intermediately between the Scouts and Military Police, the soldiers in the Garrison are second to the former in Titan combat and are less corrupt than the latter. Garrison soldiers assigned to the southern part of the walls are seen as the most experienced, since Titans have been noted to primarily come from the south

they also act as a 2nd line of defens in protecting earth

.

 international rescue

International Rescue use to be a secret, organisation before showing them selfs to the world in 2020 and now known tot he world but they are still the same as they were before a rescie group as they are a non-profit organisation dedicated to saving lives. Any who find themselves in danger, anywhere in the world, need only put out a call for help - on any frequency - and, from...somewhere...I.R. will hear it, and respond

wreath it is in space, on the grounder or under the sea IR would always go to help