Magnetic skyrmions are swirl-like spin textures with intriguing topological properties. Topological transitions between skyrmions and other magnetic solitons have been explored to design emerging topological spintronics. In magnets with S4 and D2d symmetries, complex magnetic exchange interactions lead to diverse magnetic solitons, including two types of skyrmions and four types of bubbles, which could be applied for multiple-state information processing and storage. However, controlled topological transitions among two types of elliptic skyrmions and four types of bubbles in a controlled manner in strongly confined nanodisks remain elusive. Here, we demonstrate the stabilization and topological transformations of single magnetic solitons in a Fe1.96Ni0.84Pd0.2P nanodisk. Our results show reversible transitions between the two types of skyrmions and four types of bubbles by precisely controlling the in-plane magnetic fields. Further numerical simulations reproduce and agree well with our experiments.