The flow category of a Morse-Bott-Smale function $f_{A}:G_{n}(\mathbb{C}^{\infty}) \rightarrow \mathbb{R}$ is shown to be related to the flow category of the action functional on the universal cover of $\mathcal{L}G_{n,n+k}(\mathbb{C})$ via a group action. The Floer homotopy type and the associated cohomology ring of $f_{A}:G_{n}(\mathbb{C}) \rightarrow \mathbb{R}$ are computed. When $n = 1$ this cohomology ring is the Floer cohomology of $G_{1,1+k}(\mathbb{C})$.