The structural stability of nanoalloys is a challenging research subject due to the complexity of size, shape, composition, and chemical ordering. The genetic algorithm is a popular global optimization method that can efficiently search for the ground-state nanoalloy structure. However, the algorithm suffers from three significant limitations: the efficiency and accuracy of the energy evaluator and the algorithm's efficiency. Here we describe the construction of a neural network potential intended for rapid and accurate energy predictions of Pt-Ni nanoalloys of various sizes, shapes, and compositions. We further introduce a symmetry-constrained genetic algorithm that significantly improves the efficiency and viability of the algorithm for realistic size nanoalloys. The combination of the two allows us to explore the space of homotops and compositions of Pt-Ni nanoalloys consisting of up to 4033 atoms and quantitatively report the interplay of shape, size, and composition on the dominant chemical ordering patterns.