Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
670 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C79 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C79 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C79 ◁ C23×C4×C79 ◁ C23×C8×C79 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×D16×C79 | C2 | semidirect |
| 2. C2×#466×C79 | C2 | semidirect |
| 3. C2×#357×C79 | C2 | semidirect |
| 4. C2×C22⋊C16×C79 | C2 | semidirect |
| 5. C79×#1380 | C2 | semidirect |
| 27. C2×D8×C79 | D4 | semidirect |
| 42. C2×C16×C79 | D4 | semidirect |
| 44. #107×C79 | D4 | semidirect |
| 93. C23×C79 | D16 | semidirect |
| 138. #13304 | C79 | direct (N × H) |
| 146. C22×D16 | C2×C79 | semidirect |
| 147. C2×#466 | C2×C79 | semidirect |
| 148. C2×#357 | C2×C79 | semidirect |
| 149. C2×C22⋊C16 | C2×C79 | semidirect |
| 150. #1380 | C2×C79 | semidirect |
| 151. C79 | #13304 | direct (N × H) |
| 173. C2×D8 | D4×C79 | semidirect |
| 188. C2×C16 | D4×C79 | semidirect |
| 190. #107 | D4×C79 | semidirect |
| 239. C23 | D16×C79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D16×C79 | 2 |
| C2×#466×C79 | 2 |
| C2×#357×C79 | 2 |
| C2×C22⋊C16×C79 | 2 |
| C79×#1380 | 2 |
| #13304 | 79 |