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The main focus in this work is to establish that L-group theory, which uses the language of functions instead of formal set theoretic language, is capable of capturing most of the refined ideas and concepts of classical group theory. We demonstrate this by extending the notion of subnormality to the L-setting and investigating its properties. We develop a mechanism to tackle the join problem of subnormal L-subgroups. The conjugate L-subgroup as is defined in our previous paper [4] has been used to formulate the concept of normal closure and normal closure series of an L-subgroup which, in turn, is used to define subnormal L-subgroups. Further, the concept of subnormal series has been introduced in L-setting and utilized to establish the subnor-mality of L-subgroups. Also, several results pertaining to the notion of subnormality have been established. Lastly, the level subset characterization of a subnormal L-subgroup is provided after developing a necessary mechanism. Finally, we establish that every subgroup of a nilpotent L-group is subnormal. In fact, it has been exhibited through this work that L-group theory presents a modernized approach to study classical group theory.