There are no strict rules here, of course. Basically, when you realize that a component became complex, it's advised to extract a meaningful part of it into a child component. Ideally that part should be self-contained, e.g. be easily extractable and decoupled as a piece of code. E.g. in a chat screen, the input bar is a good example of a candidate for extraction.
A big amount of methods in the component's class/interface is a good sign of complexity, however not always. If it's difficult to locate a meaningful part to extract, either the implementation details should be refactored. Or just extract the code from the component's class to separate internal classes, so the component class is just a thin proxy.