Finite Group DatabasebetaFinite Group DatabasebetaOrder 17216.
32 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #22575503 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C2×#16864377 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #16864377 | C4 | 1 | 1 | yes | characteristic |
| 4 | #16864377 | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×C8×C269 | C4 | 1 | 1 | yes | characteristic |
| 6 | C2×C4×C269 | C8 | 1 | 1 | yes | characteristic |
| 7 | C8×C269 | C8 | 1 | 1 | yes | characteristic |
| 8 | C8×C269 | C2×C4 | 1 | 1 | yes | characteristic |
| 9 | C22×C269 | C16 | 1 | 1 | yes | characteristic |
| 10 | C4×C269 | C16 | 1 | 1 | yes | characteristic |
| 11 | C4×C269 | C2×C8 | 1 | 1 | yes | characteristic |
| 12 | C2×C269 | — | 2 | 1 | no | — |
| 13 | C2×C269 | C2×C16 | 1 | 1 | yes | characteristic |
| 14 | C269 | M64 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 15 | M64 | — | 269 | 1 | no | maximal, Sylow |
| 16 | C2×C16 | — | 269 | 1 | no | — |
| 17 | C32 | — | 269 | 2 | no | — |
| 18 | C2×C8 | #11761640 | 1 | 1 | yes | characteristic |
| 19 | C16 | — | 269 | 1 | no | — |
| 20 | C16 | — | 269 | 1 | no | — |
| 21 | C2×C4 | #14524137 | 1 | 1 | yes | characteristic |
| 22 | C8 | #14524137 | 1 | 1 | yes | characteristic |
| 23 | C8 | C2×#11761640 | 1 | 1 | yes | characteristic |
| 24 | C22 | #16864405 | 1 | 1 | yes | characteristic |
| 25 | C4 | #16864405 | 1 | 1 | yes | characteristic |
| 26 | C4 | C2×#14524137 | 1 | 1 | yes | characteristic |
| 27 | C2 | — | 2 | 1 | no | — |
| 28 | C2 | C2×#16864405 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #22575503 | C2 | semidirect |
| 14. C269 | M64 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #22575503 | 2 |
| C2×#16864377 | 2 |
| M64 | 269 |