There are few studies devoted to analyze the relationships between the structure of the social network and performance in the online learning process as group. In this study, the Community of Inquiry (CoI) model is used as an analytical framework, along with quantitative content analysis and social network analysis, in order to identify the effect of density, centralization, and centrality--of the coordinators--in the different categories of CoI model in self-regulated groups, according to the type of task. Over a period of 3 academic years, a total of 96 discussion forums and 7,155 units of analysis were analyzed, focusing on two types of tasks. The results showed that the effects of the density, centralization, and centrality of the coordinators on the categories of social and cognitive presence were moderated by the type of task. The findings contribute to identify the adequate structure of the social network for the CoI model categories and the learning tasks.