Understanding brain dynamics during motor tasks is a significant challenge in neuroscience, often limited to studying pairwise interactions. This study provides a comprehensive hierarchical characterization of node-specific, pairwise and higher-order interactions within the human brain's motor network during handgrip task execution. Methods: The cortical activity was reconstructed from the scalp EEG signals of ten healthy subjects performing a motor task, identifying five brain regions within the contralateral and ipsilateral motor networks. Using the spectral entropy rate as the basis for the decomposition of dynamic information in the alpha and beta frequency bands, we assessed the predictability of the individual rhythms within each brain region, the information shared between the activity of pairs of regions, and the higher-order interactions among groups of signals from more than two regions. Results: An overall decrease in hierarchical interactions at various orders within the motor network is observed during motor task execution, primarily in the alpha frequency band, due to the well-known sensorimotor mu rhythm. Conclusions and Significance: This work emphasizes the importance of examining brain interactions at multiple levels within the frequency domain using a unified framework which fully captures the complex dynamics of the motor network highlighting the critical role of its hierarchical organization.