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-profit organizations are not in business to make a profit. This simple but significant issue points to the absence of profits or losses in the non-profit world--a major difference as compared with the for-profit firms.