I have a question... if for every finite simple group, we can construct a Galois extension over $\mathbb{Q}$ with that Galois group, does it follow that we can construct Galois extensions over $\mathbb{Q}$ with any finite group as Galois group?
Is the above known to be true or false? I was just using naive reasoning thinking finite simple groups are the "building blocks" of all finite groups (Jordan-Hölder theorem), so I'm just guessing that the inverse Galois problem reduces to the problem for finite simple groups.
Is there a proper investigation of the above?