Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
216 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C16 | #595 |
| commutator, hypocenter | C17 | C2×C4×C16 |
| Frattini | C8 | C22×#595 |
| Fitting | C2×C16×C17 | C4 |
| socle | C22×C17 | C4×C8 |
| prime radical | C22×D17 | C2×C8 |
| cosocle | C8×D17 | C23 |
| LCS | G ▷ C17 | |
| UCS | C1 ◁ C2×C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C16×#595 | C2 | direct (N × H) |
| 8. C2×#63216 | C4 | semidirect |
| 11. #10587529 | C4 | semidirect |
| 13. C2×C16×C17 | C4 | semidirect |
| 14. C2×#63214 | C4 | semidirect |
| 15. C16×D17 | C4 | semidirect |
| 30. #63216 | C2×C4 | semidirect |
| 31. C16×C17 | C2×C4 | semidirect |
| 32. #63214 | C2×C4 | semidirect |
| 34. C2×#595 | C16 | direct (N × H) |
| 52. #595 | C2×C16 | direct (N × H) |
| 61. C2×C17 | C4×C16 | semidirect |
| 63. D17 | C4×C16 | semidirect |
| 74. C2×C16 | #595 | direct (N × H) |
| 77. C17 | C2×C4×C16 | semidirect |
| 93. C16 | C2×#595 | direct (N × H) |
| 119. C2 | C16×#595 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10587529 | 2 |
| C2×C8×#595 | 2 |
| C16×#595 | 2 |
| C2×C16×D17 | 2 |
| C2×C4×C16 | 17 |