Finite Group DatabasebetaFinite Group DatabasebetaOrder 17536.
840 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 840 conjugacy classes of subgroups — too many to draw. Only its 163 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C23×D137 |
| commutator, socle, cosocle | C23×C137 | C24 |
| Fitting | #399×C137 | C2 |
| hypercenter | #399 | D137 |
| hypocenter | C137 | #2397 |
| LCS | G ▷ C23×C137 ▷ C137 | |
| UCS | C1 ◁ C23 ◁ #399 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#17984050 | C2 | semidirect |
| 2. C2×#17984048 | C2 | semidirect |
| 3. #22577338 | C2 | semidirect |
| 4. #22577313 | C2 | semidirect |
| 5. #22577312 | C2 | semidirect |
| 6. #22577248 | C2 | semidirect |
| 7. #399×C137 | C2 | semidirect |
| 37. C4×Dic137 | D4 | semidirect |
| 86. C137 | #2397 | semidirect |
| 91. #399 | D137 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#17984050 | 2 |
| C2×#17984048 | 2 |
| #22577338 | 2 |
| #22577313 | 2 |
| #22577312 | 2 |
| #22577248 | 2 |
| #399×C137 | 2 |
| #2397 | 137 |