Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
468 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 468 conjugacy classes of subgroups — too many to draw. Only its 94 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×C71 | #353 |
| commutator | C42 | C22×C4×C71 |
| Frattini, cosocle | #124 | C23×C71 |
| prime radical | C2×#496×C71 | C2 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C23×C71 ◁ C22×C4×C71 ◁ C2×#124×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#355×C71 | C2 | semidirect |
| 3. C2×#354×C71 | C2 | semidirect |
| 4. C71×#1406 | C2 | direct (N × H) |
| 7. C2×#127×C71 | C4 | semidirect |
| 11. #496×C71 | C4 | semidirect |
| 15. C2×#97×C71 | C4 | semidirect |
| 17. #355×C71 | C22 | semidirect |
| 19. #354×C71 | C22 | semidirect |
| 36. #127×C71 | C2×C4 | semidirect |
| 50. #97×C71 | C2×C4 | semidirect |
| 153. C2×#1406 | C71 | direct (N × H) |
| 159. C2×C71 | #1406 | direct (N × H) |
| 162. C2×#355 | C2×C71 | semidirect |
| 163. C2×#354 | C2×C71 | semidirect |
| 164. #1406 | C2×C71 | direct (N × H) |
| 165. C71 | C2×#1406 | direct (N × H) |
| 168. C2×#127 | C4×C71 | semidirect |
| 172. #496 | C4×C71 | semidirect |
| 176. C2×#97 | C4×C71 | semidirect |
| 178. #355 | C22×C71 | semidirect |
| 180. #354 | C22×C71 | semidirect |
| 197. #127 | C2×C4×C71 | semidirect |
| 211. #97 | C2×C4×C71 | semidirect |
| 319. C2 | C71×#1406 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#496×C71 | 2 |
| C2×#355×C71 | 2 |
| C2×#354×C71 | 2 |
| C71×#1406 | 2 |
| C2×#1406 | 71 |