Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
760 conjugacy classes of subgroups; only the 418 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 418 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C8 | #595 |
| commutator, hypocenter | C17 | C22×C4×C8 |
| Frattini | C4 | C23×#595 |
| Fitting | C22×C8×C17 | C4 |
| socle | C23×C17 | C42 |
| prime radical | C23×D17 | C2×C4 |
| cosocle | C4×D17 | C24 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C22×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C8×#595 | C2 | direct (N × H) |
| 7. C22×#3624 | C4 | semidirect |
| 10. C2×#63242 | C4 | semidirect |
| 12. C22×C8×C17 | C4 | semidirect |
| 13. C22×#3622 | C4 | semidirect |
| 14. C2×C8×D17 | C4 | semidirect |
| 16. C8×#595 | C22 | direct (N × H) |
| 18. C22×#595 | C8 | direct (N × H) |
| 26. C2×#3624 | C2×C4 | semidirect |
| 29. #63242 | C2×C4 | semidirect |
| 31. C2×C8×C17 | C2×C4 | semidirect |
| 32. C2×#3622 | C2×C4 | semidirect |
| 33. C8×D17 | C2×C4 | semidirect |
| 39. C2×#595 | C2×C8 | direct (N × H) |
| 50. #3624 | C22×C4 | semidirect |
| 51. C8×C17 | C22×C4 | semidirect |
| 52. #3622 | C22×C4 | semidirect |
| 54. C22×C17 | C4×C8 | semidirect |
| 56. C2×D17 | C4×C8 | semidirect |
| 60. #595 | C22×C8 | direct (N × H) |
| 69. C2×C17 | C2×C4×C8 | semidirect |
| 71. D17 | C2×C4×C8 | semidirect |
| 81. C22×C8 | #595 | direct (N × H) |
| 85. C17 | C22×C4×C8 | semidirect |
| 101. C2×C8 | C2×#595 | direct (N × H) |
| 121. C8 | C22×#595 | direct (N × H) |
| 126. C22 | C8×#595 | direct (N × H) |
| 135. C2 | C2×C8×#595 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#63242 | 2 |
| C22×C4×#595 | 2 |
| C2×C8×#595 | 2 |
| C22×C8×D17 | 2 |
| C22×C4×C8 | 17 |