Employing a comprehensive structure search and high-throughput first-principles calculation method on 1561 compounds, this paper reveals the phase diagram of Lu-H-N. In detail, the formation energy landscape of Lu-H-N is derived and utilized to assess the thermodynamic stability of each compound that is created via element substitution. The result indicates that there is no stable ternary structure in the Lu-H-N chemical system, however, metastable ternary structures, such as Lu20H2N17(C2/m), Lu2H2N (P$\bar 3$m1), are observed to have small E hull (< 100 meV/atom). It is also found that the energy convex hull of the Lu-H-N system shifts its shape when applying hydrostatic pressure up to 10 GPa, and the external pressure stabilizes a couple of binary phases such as LuN9 and Lu10H21. Additionally, interstitial voids in LuH2 are observed, which may explain the formation of Lu10H21 and LuH3-δN$_ \in $. To provide a basis for comparison, X-ray diffraction patterns and electronic structures of some compounds are also presented.

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