Finite Group DatabasebetaFinite Group DatabasebetaOrder 17216.
334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D269 |
| commutator, socle, cosocle | C22×C269 | C24 |
| Fitting | #117×C269 | C2 |
| hypercenter | #117 | D269 |
| hypocenter | C269 | #535 |
| LCS | G ▷ C22×C269 ▷ C269 | |
| UCS | C1 ◁ C22 ◁ #117 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#16864402 | C2 | semidirect |
| 2. C22×D1076 | C2 | semidirect |
| 3. #22575450 | C2 | semidirect |
| 4. #22575441 | C2 | semidirect |
| 5. #22575428 | C2 | semidirect |
| 6. #22575423 | C2 | semidirect |
| 7. #117×C269 | C2 | semidirect |
| 10. C2×Dic538 | C22 | semidirect |
| 14. C2×D1076 | C22 | semidirect |
| 15. C2×D1076 | C22 | semidirect |
| 17. C2×C4×D269 | C22 | semidirect |
| 18. #16864379 | C22 | semidirect |
| 20. C4⋊C4×C269 | C22 | semidirect |
| 22. C42×C269 | C22 | semidirect |
| 36. C2×C4×C269 | D4 | semidirect |
| 57. C269 | #535 | semidirect |
| 65. #117 | D269 | semidirect |
| 77. C4⋊C4 | C2×D269 | semidirect |
| 80. C42 | C2×D269 | semidirect |
| 94. C2×C4 | D1076 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#16864402 | 2 |
| C22×D1076 | 2 |
| #22575450 | 2 |
| #22575441 | 2 |
| #22575428 | 2 |
| #22575423 | 2 |
| #117×C269 | 2 |
| #535 | 269 |