Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
726 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 726 conjugacy classes of subgroups — too many to draw. Only its 74 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×C79 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23×C79 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C79 ◁ C2×C4⋊C4×C79 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C79×#2679 | C2 | semidirect |
| 2. C79×#2006 | C2 | semidirect |
| 3. C79×#1598 | C2 | semidirect |
| 4. C79×#1345 | C2 | semidirect |
| 5. C79×#1318 | C2 | semidirect |
| 7. #537×C79 | C22 | semidirect |
| 16. #343×C79 | C22 | semidirect |
| 18. #324×C79 | C22 | semidirect |
| 75. C2×D4×C79 | D8 | semidirect |
| 175. #12096 | C79 | direct (N × H) |
| 184. #2679 | C2×C79 | semidirect |
| 185. #2006 | C2×C79 | semidirect |
| 186. #1598 | C2×C79 | semidirect |
| 187. #1345 | C2×C79 | semidirect |
| 188. #1318 | C2×C79 | semidirect |
| 189. C79 | #12096 | direct (N × H) |
| 191. #537 | C22×C79 | semidirect |
| 200. #343 | C22×C79 | semidirect |
| 202. #324 | C22×C79 | semidirect |
| 259. C2×D4 | D8×C79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C79×#2679 | 2 |
| C79×#2006 | 2 |
| C79×#1598 | 2 |
| C79×#1345 | 2 |
| C79×#1318 | 2 |
| #12096 | 79 |