Nested Routes not rendering with React Router v4

I am trying to group some of my routes together with React Router v4 to clean up some of my components. For now I just want to have my non logged in routes group together and my admin routes grouped together but the following doens't work.

main.js

const Main = () => {
  return (
    <main>
      <Switch>
        <Route exact path='/' component={Public} />
        <Route path='/admin' component={Admin} />
      </Switch>
    </main>
  );
};

export default Main;

public.js

const Public = () => {
  return (
    <Switch>
      <Route exact path='/' component={Greeting} />
      <Route path='/signup' component={SignupPage} />
      <Route path='/login' component={LoginPage} />
    </Switch>
  );
};

export default Public;

The Greeting component shows at "localhost:3000/", but the SignupPage component does not show at "localhost:3000/signup" and the Login component doesn't show at "localhost:3000/signup". Looking at the React Dev Tools these two routes return Null.