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?