Although we typically talk about attention as a single process, it comprises multiple independent components. But what are these components, and how are they represented in the functional organization of the brain? To investigate whether long-studied components of attention are reflected in the brain's intrinsic functional organization, here we apply connectome-based predictive modeling (CPM) to predict the components of Posner and Petersen's influential model of attention: alerting (preparing and maintaining alertness and vigilance), orienting (directing attention to a stimulus), and executive control (detecting and resolving cognitive conflict) [Posner, M. I., & Petersen, S. E. The attention system of the human brain. Annual Review of Neuroscience, 13, 25-42, 1990]. Participants performed the Attention Network Task (ANT), which measures these three factors, and rested during fMRI scanning. CPMs tested with leave-one-subject-out cross-validation successfully predicted novel individual's overall ANT accuracy, RT variability, and executive control scores from functional connectivity observed during ANT performance. CPMs also generalized to predict participants' alerting scores from their resting-state functional connectivity alone, demonstrating that connectivity patterns observed in the absence of an explicit task contain a signature of the ability to prepare for an upcoming stimulus. Suggesting that significant variance in ANT performance is also explained by an overall sustained attention factor, the sustained attention CPM, a model defined in prior work to predict sustained attentional abilities, predicted accuracy, RT variability, and executive control from task-based data and predicted RT variability from resting-state data. Our results suggest that, whereas executive control may be closely related to sustained attention, the infrastructure that supports alerting is distinct and can be measured at rest. In the future, CPM may be applied to elucidate additional independent components of attention and relationships between the functional brain networks that predict them.