Can a mother take her children out the state if she is an illegal immigrant and the father did not give her permission to?

It depends. If the parents were never married, and the father never asked for custody of the children, the mother does not need his consent to go anywhere with her children as the parent who has sole custody of the children. If the parents are husband and wife, they are presumed to have equal custody rights; and the mother cannot take the children out of the state against the father's will - unless a court awards the mother sole custody of the children. Same result if the father was given custody of the children by a judicial or administrative decision. In reality, one parent's illegal immigration status will, most likely, influence the outcome of a custody dispute between the parents. But, if the parents were never married, the mother can move out of the state with the children without asking for the non-custodial father's permission (he can try to block the moving, but that's another story).