If you and the other parent live together, either one of you (but not both) may claim the child. You may decide between you which one will claim the child. Only if you can’t agree, do the IRS tie breaker rules apply, to see who has first choice (it's you because you have the higher income). It may be worthwhile to prepare trial returns, both ways, to see which way the family comes out best. This tool may be useful: https://turbotax.intuit.com/tax-tools/calculators/taxcaster/?s=1.
You must be claiming the child in order to claim Head of Household (HoH). If your GF claims the child, she cannot claim HoH, because she didn't pay more than half the household expenses.
