Let G be a finite group and \(N \unlhd G\). The normal subgroup based power graph of G, \(\Gamma _N(G)\), is an undirected graph with vertex set \((G{\setminus }N) \cup \{e\}\) in which two distinct vertices a and b are adjacent if and only if \(aN = b^mN\) or \(bN = a^nN\), for some positive integers m and n. The aim of this paper is to characterize all pairs (G, N) of a finite group G and a proper non-trivial normal subgroup N of G such that \(\Gamma _{H}(G)\) is a split, bisplit and \((n-1)-\)bisplit. A classification of finite groups with unicyclic and tricyclic normal subgroup based power graph are also given and it is proved that there is no group containing a proper and non-trivial normal subgroup such that its normal subgroup based power graph is a bicyclic or tetracyclic graph.

令G为一个有限群，\（N \ unlhd G \）。的普通的基于亚组功率图表ģ，\（\伽玛_N（G）\） ，是一种无向图的顶点集\（（G {\ setminus} N）\杯\ {ë\} \），其中两个不同的对于且仅当\（aN = b ^ mN \）或\（bN = a ^ nN \）时，顶点a和b相邻，对于某些正整数m和n。本文的目的是表征所有对（g ^，  Ñ有限群）G ^和适当的非平凡的正常子群Ñ的ģ这样\（\ Gamma _ {H}（G）\）是分割的，双分割的，而\（（n-1）-\）是双分割的。还给出了基于单环和三环正态子群的幂图的有限群的分类，并证明不存在包含适当且非平凡的正态子群的群，因此其基于正子子集的幂图是双环或四环图。