Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
128 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 39 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D32×D79 |
| commutator, cosocle | C32×C79 | C23 |
| Frattini | C32 | C22×D79 |
| Fitting | C79×SD128 | C2 |
| socle | C2×C79 | C2×D32 |
| hypercenter | SD128 | D79 |
| hypocenter | C79 | #13741 |
| LCS | G ▷ C32×C79 ▷ C16×C79 ▷ C8×C79 ▷ C4×C79 ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 ◁ C32 ◁ SD128 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #23765710 | C2 | semidirect |
| 2. #23765706 | C2 | semidirect |
| 3. #23761617 | C2 | semidirect |
| 4. #23761616 | C2 | semidirect |
| 5. #23761612 | C2 | semidirect |
| 6. C64×D79 | C2 | semidirect |
| 7. C79×SD128 | C2 | semidirect |
| 68. SD128 | D79 | semidirect |
| 71. C79 | #13741 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #23765710 | 2 |
| #23765706 | 2 |
| #23761617 | 2 |
| #23761616 | 2 |
| #23761612 | 2 |
| C64×D79 | 2 |
| C79×SD128 | 2 |
| #13741 | 79 |