Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
522 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 522 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×#2733 | C2 | semidirect |
| 2. C79×#2007 | C2 | semidirect |
| 3. C79×#1598 | C2 | semidirect |
| 4. C79×#1345 | C2 | semidirect |
| 5. C79×#1319 | C2 | semidirect |
| 57. C2×Q8×C79 | D8 | semidirect |
| 157. #12100 | C79 | direct (N × H) |
| 165. #2733 | C2×C79 | semidirect |
| 166. #2007 | C2×C79 | semidirect |
| 167. #1598 | C2×C79 | semidirect |
| 168. #1345 | C2×C79 | semidirect |
| 169. #1319 | C2×C79 | semidirect |
| 170. C79 | #12100 | direct (N × H) |
| 222. C2×Q8 | D8×C79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C79×#2733 | 2 |
| C79×#2007 | 2 |
| C79×#1598 | 2 |
| C79×#1345 | 2 |
| C79×#1319 | 2 |
| #12100 | 79 |