Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
160 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C211 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C211 |
| prime radical | #493×C211 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C211 ◁ C42×C211 ◁ C4×C8×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #500×C211 | C2 | semidirect |
| 3. #357×C211 | C2 | semidirect |
| 4. C4×C16×C211 | C2 | semidirect |
| 50. C211 | #2240 | direct (N × H) |
| 51. #2240 | C211 | direct (N × H) |
| 52. #500 | C2×C211 | semidirect |
| 54. #357 | C2×C211 | semidirect |
| 55. C4×C16 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #500×C211 | 2 |
| #493×C211 | 2 |
| #357×C211 | 2 |
| C4×C16×C211 | 2 |
| #2240 | 211 |