Panpsychism has a problem. We are subjects of experience and, according to panpsychism, we are also somehow the combination of the smaller subjects of experience that we comprise. But we have a seemingly unshakeable intuition that we are irreducible. If this is so, then combination is impossible, and panpsychism fails. The question, then, is: why believe that subjects are irreducible? A number of arguments have been offered in defence of the irreducibility of subjects. I consider five. The primary purpose of this paper is to show that none of them clearly succeed. For each argument, at least one premise is either false, likely false, or in serious need of defence. I end the paper by attempting to nudge us away from the intuition that we are neither reducible nor combinable through the use of a fanciful thought experiment.