Finite Group DatabasebetaFinite Group DatabasebetaOrder 27008.
154 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C211 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C23×C211 |
| prime radical | #509×C211 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C211 ◁ C2×C4×C211 ◁ C4×C8×C211 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #510×C211 | C2 | semidirect |
| 3. #495×C211 | C2 | semidirect |
| 4. #443×C211 | C2 | semidirect |
| 5. #364×C211 | C2 | semidirect |
| 6. #362×C211 | C2 | semidirect |
| 7. #361×C211 | C2 | semidirect |
| 74. C211 | #2212 | direct (N × H) |
| 75. #2212 | C211 | direct (N × H) |
| 76. #510 | C2×C211 | semidirect |
| 78. #495 | C2×C211 | semidirect |
| 79. #443 | C2×C211 | semidirect |
| 80. #364 | C2×C211 | semidirect |
| 81. #362 | C2×C211 | semidirect |
| 82. #361 | C2×C211 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #510×C211 | 2 |
| #509×C211 | 2 |
| #495×C211 | 2 |
| #443×C211 | 2 |
| #364×C211 | 2 |
| #362×C211 | 2 |
| #361×C211 | 2 |
| #2212 | 211 |