Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
676 conjugacy classes of subgroups; only the 189 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 189 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×C17×C4⋊Q8 | C2 |
| prime radical | C22×D68 | C22 |
| hypercenter | C2×C4⋊Q8 | D17 |
| hypocenter | C17 | #2542 |
| 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 |
|---|---|---|
| 1. C2×#10587632 | C2 | semidirect |
| 3. #11762744 | C2 | semidirect |
| 5. #11762678 | C2 | semidirect |
| 6. C4×C4⋊Dic17 | C2 | semidirect |
| 7. C2×C17×C4⋊Q8 | C2 | semidirect |
| 35. C2×D68 | Q8 | semidirect |
| 44. C4⋊Dic17 | D4 | semidirect |
| 83. C4×C17 | C22⋊Q8 | semidirect |
| 87. C2×C4⋊Q8 | D17 | semidirect |
| 116. C17 | #2542 | semidirect |
| 178. C4 | #10587632 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10587632 | 2 |
| C2×C4×D68 | 2 |
| #11762744 | 2 |
| #11762710 | 2 |
| #11762678 | 2 |
| C4×C4⋊Dic17 | 2 |
| C2×C17×C4⋊Q8 | 2 |
| #2542 | 17 |