A topological insulator is regarded as an ideal candidate for information storage and high-speed lossless electrical transmission devices due to robust topological protected boundary modes. Previous studies revealed that symmetry exerts an unbreakable constraint on the existence, classes, and orders of its boundary modes. It severely limits the controllability and application of a topological insulator. Here, we propose a Floquet-engineering method to break this symmetry-imposed constraint on a topological insulator. By applying periodic driving on a system belonging to a symmetry class that prohibits the existence of first-order topological phases, we find that rich first-order boundary modes are created. Interestingly, exotic hybrid-order topological insulators with coexisting first-order helical boundary modes and second-order corner modes not only in two different quasienergy gaps but also in one single gap are generated easily by periodic driving. Refreshing the prevailing understanding of symmetry constraint on topological phases, our result opens an avenue for the creation of exotic topological insulators without altering symmetries. It greatly expands the scope of the fabricated materials that host topological insulators.

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• Majorana bound states
• Symmetry protected topological states
• Topological superconductors
• Floquet systems