Let $G$ be a group with 15 elements. Let $L$ be a subgroup of $G$. It is known that $L \neq G$ and that the size of $L$ is at least 4. The size of $L$ is _______.

By Lagrange's Theorem, the order (size) of any subgroup $L$ of a finite group $G$ must divide the order of $G$.

Given:

• $|G| = 15$

• The divisors of $15$ are $1, 3, 5,$ and $15$.

We are given the following conditions on $|L|$:

1. $L$ is a proper subgroup ($L \neq G$), so $|L| < 15$.

2. The size of $L$ is at least $4$, so $|L| \ge 4$.

The only divisor of $15$ strictly less than $15$ and greater than or equal to $4$ is 5.

Therefore, the size of $L$ is 5.