Usually the owner or general contractor requests permits. If the owner is agreeable, the subcontractor can do so, but the owner must sign an authorization, just as with the general contractor.
You can easily check whether your contractor is licensed by visiting the Contractors' State License Board web site at http://www.cslb.ca.gov/ and clicking on the "Instant License Check" flag. If your contractor is not licensed, you should fire him and demand refund of everything already paid. You jeopardize yourself by hiring an unlicensed contractor, and the contractor is not entitled to do the work or charge you for it (over $500).
You can, of course, ask the contractor why he didn't pull the permit for you, and see whether he had a good reason for it. I'm guessing not. If the contractor is a corporation, then some authorized person must sign on its behalf, so you should check with the contractor on that issue, too.
Your issues are complicated by your specific situation which would probably require review of the contracts and permit application and other documents.
But generally speaking, subcontractors can pull permits. Generally they can only obtain permits for work that is within the scope of their license. When it comes right down to it, it really does not matter who pulls the permits, as long as the permits are obtained and cover the whole project.
There might be some legitimate reason the general asked the sub to do it, or there might be some improper reason. Check everyone's licensing, bond, workers comp and insurance records. If everyone is licensed, bonded, and insured, then you should not worry about who pulled the permits.