Complex sentence: Both the vassalage ties with the samurai, and control over "shugo" lords were stabilized after the regime was solidified in the 1350s: these two hierarchies were the most important connections that largely determined the shogun's power.
Split into simpler sentences: Both the vassalage ties with the samurai and control over "shugo" lords were established after the regime had solidified in the 1350s. These two hierarchies were the most important connections in determining the shogun's power.