Let G be a finite group. A subset X of G is called a trivial intersection set if for every \(g\in G\) , either \(X^{g} = X\) or \(X^{g}\cap X \subseteq \{1\}\) . A subgroup S of G is called a TI-subgroup if S is a trivial intersection set. Also, a subgroup K of G is termed an H-subgroup if \(N_{G}(K) \cap K^x\le K\) for all \(x\in G\) . In this paper, we introduce the concept of HI-subgroups. A subgroup T of G is said to be an HI-subgroup if \(N_{G}(T) \cap T^x\le T\) and \(N_{G}(T) \cap T^x\ne \{1\}\) for every \(x\in G\setminus N_{G}(T)\) . We describe the structure of finite groups G in which every subgroup is either a TI-subgroup or an HI-subgroup. We demonstrate that if G is a nilpotent TH-group, then it is either a Dedekind group or a p-group in which all non-normal subgroups of G have the same prime order p. Furthermore, we show that if G is a TH-group and every subgroup of order two in a non-abelian Sylow 2-subgroup P of G is normal in the normalizer of P in G, then G is 2-nilpotent.
