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Let $G$ be a finite group on $84$ elements. The size of a largest possible proper subgroup of $G$ is _____
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Order of a Subgroup always divides the order of Group.
Proper Subgroup of Group having order $84$ would have one of the order (proper factors of $84)$ $2,3, 4,6,7,12,14, 21,28, 42$.

So the largest order would be $42$.
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Lagrange's theorem states that order of every subgroup of G, it must be the divisor of G.

So the largest subgroup will be 84 which is trivial, but in the question it is asking for the proper subgroup hence it will be 42.

Reference: https://en.wikipedia.org/wiki/Subgroup

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