Finite Group DatabasebetaFinite Group DatabasebetaOrder 18176.
484 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 484 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 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C22×C4×C71 |
| Frattini, cosocle | C4⋊Q8 | C23×C71 |
| prime radical | C2×2−1+4×C71 | C4 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C71 ◁ C23×C71 ◁ C22×C4×C71 ◁ C2×C4⋊Q8×C71 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#356×C71 | C2 | semidirect |
| 3. C2×#333×C71 | C2 | semidirect |
| 4. C71×#1405 | C2 | direct (N × H) |
| 17. #356×C71 | C22 | semidirect |
| 19. #333×C71 | C22 | semidirect |
| 165. C2×#1405 | C71 | direct (N × H) |
| 169. C2×C71 | #1405 | direct (N × H) |
| 172. C2×#356 | C2×C71 | semidirect |
| 173. C2×#333 | C2×C71 | semidirect |
| 174. #1405 | C2×C71 | direct (N × H) |
| 175. C71 | C2×#1405 | direct (N × H) |
| 188. #356 | C22×C71 | semidirect |
| 190. #333 | C22×C71 | semidirect |
| 339. C2 | C71×#1405 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#456×C71 | 2 |
| C2×#356×C71 | 2 |
| C2×#333×C71 | 2 |
| C71×#1405 | 2 |
| C2×#1405 | 71 |