Interestingly, some of those teams dedicated to hacking are either private sector or a branch that nobody has heard of. I once interviewed for a company whose pitch to me was basically "we get indemnity to hack foreign telcos" and "we develop ways to spy that nobody has thought of". That was 20 years ago
Some are specialized, some are diversified. Definitely public, I believe they all have to be listed on fedgov's contractor list? Some are obvious weapons contractors, some aren't (like extensions of big-name universities). If you see job listings for weapons development, cyber ops, secret-clearance software dev, cryptography, etc, that's a clue.
