Verifying political claims is a challenging task, as politicians can use various tactics to subtly misrepresent the facts for their agenda. Existing automatic fact-checking systems fall short here, and their predictions like “half-true” are not very useful in isolation, since it is unclear which parts of a claim are true and which are not. In this work, we focus on decomposing a complex claim into a comprehensive set of yes-no subquestions whose answers influence the veracity of the claim. We present CLAIMDECOMP, a dataset of decompositions for over 1000 claims. Given a claim and its verification paragraph written by fact-checkers, our trained annotators write subquestions covering both explicit propositions of the original claim and its implicit facets, such as asking about additional political context that changes our view of the claim’s veracity. We study whether state-of-the-art models can generate such subquestions, showing that these models generate reasonable questions to ask, but predicting the comprehensive set of subquestions from the original claim without evidence remains challenging. We further show that these subquestions can help identify relevant evidence to fact-check the full claim and derive the veracity through their answers, suggesting that they can be useful pieces of a fact-checking pipeline.