Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
134 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 134 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C32 | C22×D79 |
| commutator, socle | C2×C79 | C22×C32 |
| Fitting | C79×M128 | C2 |
| prime radical | C22×D79 | C32 |
| cosocle | C32×C79 | C23 |
| hypercenter | M128 | D79 |
| hypocenter | C79 | C2×M128 |
| LCS | G ▷ C2×C79 ▷ C79 | |
| UCS | C1 ◁ C32 ◁ M128 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23761611 | C2 | semidirect |
| 3. #23761610 | C2 | semidirect |
| 4. C64×D79 | C2 | semidirect |
| 5. C79×M128 | C2 | semidirect |
| 12. C79⋊C64 | C22 | semidirect |
| 14. C64×C79 | C22 | semidirect |
| 60. D79 | M128 | direct (N × H) |
| 64. M128 | D79 | direct (N × H) |
| 66. C79 | C2×M128 | semidirect |
| 75. C64 | C2×D79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×D79 | 2 |
| #23761611 | 2 |
| #23761610 | 2 |
| C64×D79 | 2 |
| C79×M128 | 2 |
| C2×M128 | 79 |