For this problem, I'd like to know how I can solve this problem using Lagrange's Theorem.
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You can't solve it using Lagrange's theorem. Lagrange's theorem limits the possible subgroups and elements you can have, but it doesn't say anything about whether anything actually exists. Sylow's first theorem and then Lagrange's theorem would help, but that's probably overkill. – Arthur Sep 20 '17 at 07:10
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One feature of finite groups is that you can count the elements - so when finiteness is specified, counting arguments come into the frame as a strategy. – Mark Bennet Sep 20 '17 at 07:15
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This question has really many duplicates on this site, e.g., also this one. Please have a look before posting your question. – Dietrich Burde Sep 20 '17 at 09:05