We need to find the number of ways to form a team of 7 people with exactly 1 swimmer, at least 4 cyclists, and at most 2 runners.

Consider the following scenarios:

1. 1 swimmer, 4 cyclists, 2 runners:
   The number of ways to choose 1 swimmer from 6 is \(\binom{6}{1}\).
   The number of ways to choose 4 cyclists from 8 is \(\binom{8}{4}\).
   The number of ways to choose 2 runners from 11 is \(\binom{11}{2}\).
   Total for this scenario: \(\binom{6}{1} \times \binom{8}{4} \times \binom{11}{2} = 6 \times 70 \times 55 = 23100\).
2. 1 swimmer, 5 cyclists, 1 runner:
   The number of ways to choose 1 swimmer from 6 is \(\binom{6}{1}\).
   The number of ways to choose 5 cyclists from 8 is \(\binom{8}{5}\).
   The number of ways to choose 1 runner from 11 is \(\binom{11}{1}\).
   Total for this scenario: \(\binom{6}{1} \times \binom{8}{5} \times \binom{11}{1} = 6 \times 56 \times 11 = 3696\).
3. 1 swimmer, 6 cyclists, 0 runners:
   The number of ways to choose 1 swimmer from 6 is \(\binom{6}{1}\).
   The number of ways to choose 6 cyclists from 8 is \(\binom{8}{6}\).
   The number of ways to choose 0 runners from 11 is \(\binom{11}{0}\).
   Total for this scenario: \(\binom{6}{1} \times \binom{8}{6} \times \binom{11}{0} = 6 \times 28 \times 1 = 168\).

Adding these scenarios together gives the total number of ways:
\(23100 + 3696 + 168 = 26964\).