Maximal connected algebraic subgroups of the group of birational transformations Bir(X) of a variety $X$ appear as automorphism groups of Mori fiber spaces birational to $X$. When $X = \mathbb{P}^n$, for $n = 2$ or $3$ we have a classification of such groups and the models they act on, while in dimension 4 and more this is an open problem. In this talk we will explore the equivariant geometry of a class of quadric fibrations over $\mathbb{P}^1$. As a result, we will see how their automorphism groups gives us families of maximal subgroups of $Bir(\mathbb{P}^n)$ for any $n$ at least 4. This is work in progress, joint with Enrica Floris.
