A recurring theme in the study of society is the concentration of influence and power that is driven through unequal membership of groups and associations. In some instances these bodies constitute a small world while in others they are fragmented into distinct cliques. This paper presents a new model of clubs and networks to understand the sources of individual marginalization and the origins of different club networks. In our model, individuals seek to become members of clubs while clubs wish to have members. Club value is increasing in its size and in the strength of ties with other clubs. We show that a stable membership profile exhibits marginalization of individuals and that this is generally not welfare maximizing. Our second result shows that if returns from strength of ties are convex (concave) then stable memberships support fragmented networks with strong ties (small worlds held together by weak ties). We illustrate the value of these theoretical results through case studies of inter-locking directorates and boards of editors of journals.