Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
500 conjugacy classes of subgroups; only the 141 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 141 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 | C2×C4×Q8×C17 | C2 |
| prime radical | C22×D68 | C22 |
| hypercenter | C2×C4×Q8 | D17 |
| hypocenter | C17 | #2632 |
| LCS | G ▷ C23×C17 ▷ C17 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11762744 | C2 | semidirect |
| 3. #11762729 | C2 | semidirect |
| 5. #11762720 | C2 | semidirect |
| 6. #11762700 | C2 | semidirect |
| 7. #11762677 | C2 | semidirect |
| 8. C2×C4×Q8×C17 | C2 | semidirect |
| 91. C2×C4×Q8 | D17 | semidirect |
| 121. C17 | #2632 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10587561 | 2 |
| #11762744 | 2 |
| #11762729 | 2 |
| #11762726 | 2 |
| #11762720 | 2 |
| #11762700 | 2 |
| #11762677 | 2 |
| C2×C4×Q8×C17 | 2 |
| #2632 | 17 |