What is the law of a piece of a log-normal distribution?

We know that log-normals are infinitely divisible. What would be the law of a root of log-normal?

More specifically, suppose that $X$ is a log-normal. Given an integer $k\geq 2$, we know that there exist $X_1,...,X_k$ i.i.d such that:

$$\mathcal{L}(X) = \mathcal{L}\left(\sum\limits_{i=1}^{k} X_i\right),$$

where $\mathcal{L}$ denotes the distribution of a random variable.

Question: Is there a way to simulate directly from $X_i$'s law?