Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
44 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | D5056 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C79⋊C128 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C79×Q128 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | D2528 | — | 2 | 1 | no | — |
| 5 | Q64×C79 | — | 2 | 1 | no | — |
| 6 | C64×C79 | C22 | 1 | 1 | yes | characteristic |
| 7 | D1264 | — | 4 | 1 | no | — |
| 8 | Q32×C79 | — | 4 | 1 | no | — |
| 9 | C32×C79 | D4 | 1 | 1 | yes | characteristic |
| 10 | D632 | — | 8 | 1 | no | — |
| 11 | Q16×C79 | — | 8 | 1 | no | — |
| 12 | C16×C79 | D8 | 1 | 1 | yes | characteristic |
| 13 | Q8×C79 | — | 16 | 1 | no | — |
| 14 | D316 | — | 16 | 1 | no | — |
| 15 | C8×C79 | D16 | 1 | 1 | yes | characteristic |
| 16 | C2×D79 | — | 32 | 1 | no | — |
| 17 | C4×C79 | — | 32 | 1 | no | — |
| 18 | C4×C79 | D32 | 1 | 1 | yes | characteristic |
| 19 | SD256 | — | 79 | 1 | no | maximal, Sylow |
| 20 | C2×C79 | D64 | 1 | 1 | yes | characteristic |
| 21 | D79 | — | 64 | 1 | no | — |
| 22 | Q128 | D79 | 1 | 1 | yes | characteristic, complemented |
| 23 | D64 | — | 79 | 1 | no | — |
| 24 | C128 | — | 79 | 1 | no | — |
| 25 | C79 | SD256 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 26 | Q64 | — | 2 | 1 | no | — |
| 27 | D32 | — | 158 | 1 | no | — |
| 28 | C64 | C2×D79 | 1 | 1 | yes | characteristic |
| 29 | Q32 | — | 4 | 1 | no | — |
| 30 | D16 | — | 316 | 1 | no | — |
| 31 | C32 | C22⋊D79 | 1 | 1 | yes | characteristic |
| 32 | Q16 | — | 8 | 1 | no | — |
| 33 | D8 | — | 632 | 1 | no | — |
| 34 | C16 | #11929609 | 1 | 1 | yes | characteristic |
| 35 | Q8 | — | 16 | 1 | no | — |
| 36 | D4 | — | 1264 | 1 | no | — |
| 37 | C8 | #14579102 | 1 | 1 | yes | characteristic |
| 38 | C22 | — | 2528 | 1 | no | — |
| 39 | C4 | — | 32 | 1 | no | — |
| 40 | C4 | #19098398 | 1 | 1 | yes | characteristic |
| 41 | C2 | — | 5056 | 1 | no | — |
| 42 | C2 | #23761612 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #23761612 |
| commutator, cosocle | C64×C79 | C22 |
| Frattini | C64 | C2×D79 |
| Fitting | C79×Q128 | C2 |
| socle | C2×C79 | D64 |
| prime radical | D5056 | C2 |
| hypercenter | Q128 | D79 |
| hypocenter | C79 | SD256 |
| LCS | G ▷ C64×C79 ▷ C32×C79 ▷ C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ C64 ◁ Q128 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C79⋊C128 | C2 | semidirect |
| 3. C79×Q128 | C2 | semidirect |
| 22. Q128 | D79 | semidirect |
| 25. C79 | SD256 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D5056 | 2 |
| C79⋊C128 | 2 |
| C79×Q128 | 2 |
| SD256 | 79 |