Technically, the military part of Shinra was called SOLDIER.

Shinra encompassed a lot more than just a military during the events of the game, even though they started as just a weapons manufacturer. They were the government, a power plant, lead the way in scientific discovery/technology, and yeah, still did some weapons manufacturing as well.
So out of SOLDIER and SeeD, I'm gonna go with SOLDIER. SeeD was alright, but I don't think a bunch of kids could hold thier own against Mako-infused super soldiers. Plus, Shinra in it's glory would have just bought out the SeeDs, since money is a main priority for the Gardens.
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