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.
Non-governmental organizations, nongovernmental organizations, or nongovernment organizations, commonly referred to as NGOs, are usually nonprofit and sometimes international organizations independent of governments and international governmental organizations (though often funded by governments) that are active in humanitarian, educational, healthcare, public policy, social, human rights, environmental, and other areas to effect changes according to their objectives.