Complexity of determining feasibilty of a generalization of the interval scheduling problem

Assume there are n groups and each group has at most k intervals. We can sort the all the groups by the finish time of it's last interval, call the sorted order $g_1, g_2, \dots , g_n$. For each group, we also compute the list of incompatible groups (e.g. $g_1$ and $g_2$ are incompatible if there is a non-zero overlap across their intervals). We then compute the array $v$, where $v[k]$ represents the maximum number of groups that can be scheduled in $\lbrace{g_1, g_2, \dots, g_{k} \rbrace}$ with $g_k$ included. This computation is easy. For example, if $g_k$ is only compatible with the groups $g_1, g_2, g_5$ in the set $\lbrace{g_1, g_2, \dots, g_{k-1} \rbrace}$, then $v[k] = (1 + max(v[1], v[2], v[5]))$. This is correct since, either none of the compatible groups can be scheduled with $g_k$, or at least one of the compatible groups is present when scheduling along with $g_k$.

The overall complexity is $ O\left( {{n}\choose{2}} k + n \, k \, log (k) \right)$ time and O(n + k) space.