Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
532 conjugacy classes of subgroups; only the 153 normal subgroups are shown.
This group has 532 conjugacy classes of subgroups — too many to draw. Only its 153 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×D17 |
| commutator, socle, cosocle | C23×C17 | C24 |
| Fitting | C4×C4⋊C4×C17 | C2 |
| prime radical | C22×D68 | C22 |
| hypercenter | C4×C4⋊C4 | D17 |
| hypocenter | C17 | #2562 |
| LCS | G ▷ C23×C17 ▷ C17 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#10587556 | C2 | semidirect |
| 3. #11762749 | C2 | semidirect |
| 5. #11762700 | C2 | semidirect |
| 6. #11762666 | C2 | semidirect |
| 7. #11762666 | C2 | semidirect |
| 8. C4×C4⋊C4×C17 | C2 | semidirect |
| 106. C4×C4⋊C4 | D17 | semidirect |
| 131. C17 | #2562 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10587556 | 2 |
| C2×C4×D68 | 2 |
| #11762749 | 2 |
| #11762727 | 2 |
| #11762700 | 2 |
| #11762666 | 2 |
| C4×C4⋊C4×C17 | 2 |
| #2562 | 17 |