Let $G$ be a finite group on 84 elements. The size of a largest possible proper subgroup of $G$ is ________.

According to Lagrange's Theorem, the order (size) of any subgroup $H$ of a finite group $G$ must divide the order of $G$, denoted by $|G|$.

Given $|G| = 84$:

The divisors of 84 are: $1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84$.

A proper subgroup of $G$ is any subgroup other than $G$ itself, meaning its order must be strictly less than $|G| = 84$.

The largest proper divisor of $84$ is:

Therefore, the size of the largest possible proper subgroup of $G$ is 42.