To start with, I am going to assume that the child is the biological child of both of you and that you live together. In that case, only one person can claim all the tax benefits -- dependent exemption, child tax credit, head of household filing status (sometimes), EIC (sometimes), and dependent care credit or tax-free FSA.
The Statute IRC 501(c)(4) provides for exemption of: Civic leagues or organizations not organized for profit but operated. exclusively for the promotion of social welfare. Local associations of employees, the membership of which is limited to the. employees of a designated person or persons in a particular municipality.
A 501(c)(3), or non-profit, is an IRS approved organization that has a charitable, religious, public safety, or educational purpose. The benefit of qualifying as a nonprofit is that the organization does not have to pay taxes on the money it receives and donors to a 501(c)(3) can deduct the amount of their gifts from their taxable income.