An organization that is purely social is not going to be tax exempt. To get the 401 (c) tax-exempt designation the group has to be primarily for some public purpose. American Mensa, for example, is NOT exempt from taxes. Its stated purpose is to give intelligent people a means of interacting. However, there is a Mensa Education and Research Foundation which is exempt.
The power to tax includes the power to destroy. Religious organizations are exempt so that the government has no way to interfere with them.
Organizations whose purpose is political – that is, to influence voting and governance – are not tax exempt. See ACLU, above.