Vaughan Jones and I showed in the late 1990's that whenever a subfactor has an intermediate subfactor, the fundamental subfactor symmetries are captured by what we called the ``Fuss-Catalan algebras''. If the subfactor has two intermediate subfactors in general position, i.e. we are given a quadrilateral of subfactors, describing the quantum symmetries is more complicated and not much progress had been made since the work of Grossman, Izumi and Jones some 20 years ago. I will discuss recent work with Junhwi Lim in which we determine the quantum symmetries of a quadrilateral of subfactors when the four algebras form a co-commuting square and satisfy some additional conditions. The subfactor symmetries we find are related to partition algebras and Bell numbers.