Maximization problem on finite collection of finite sets Problem
I am considering the following maximization problem:

Input is a finite collection of finite sets $\mathcal{F} = \{ X_1, X_2, \ldots, X_n \}$.
Goal is to find a subset $G \subseteq \mathcal{F}$ that maximizes $|G| \times |\bigcap G|$ where

$|G|$ is the cardinality of the set $G$, and
$\bigcap G = \bigcap \{X_{i_1}, X_{i_2}, \ldots, X_{i_m} \} = X_{i_1} \cap X_{i_2} \cap \cdots \cap X_{i_m}$.



As an example, for the collection
$$
\mathcal{F} = \{ \{a, b, c\}, \{a, b, c, x\}, \{b, c, y\}, \{a, b, c, z\} \},
$$
the maximizing subset is $G = \{ \{a, b, c\}, \{a, b, c, x\}, \{a, b, c, z\} \}$
and the score is $3 \times |\{a, b, c\}| = 9$.
Note: the score of $\mathcal{F}$ itself is $4 \times |\{b, c\}| = 8$.
Question
I am planning to use a procedure of this problem for compressing data (represented by finite collections of finite sets).
However, I don't have any good idea to solve this problem efficiently.
As yow know, we can solve this by enumerating all the collections of $\mathcal{F}$; but, it's too slow for practical use.
Is there a polynomial-time or some kind of efficient algorithm for this problem?
Or, does this problem belong to the complexity class that cannot be solved in polynomial time?
This problem is NP-complete. Let's reformulate it first: we have a bipartite graph, where

The left side corresponds to elements
The right side corresponds to sets
The edge $(u,v)$ means that $u \in v$.

Our goal is to find the bipartite clique with the maximum number of edges. As stated in Rene Peeters, &quot;The maximum edge biclique problem is NP-complete&quot;, the decision problem is NP-complete.