Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
990 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 990 conjugacy classes of subgroups — too many to draw. Only its 150 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 | C2×#99 |
| commutator | C22×C4 | C22×C4×C79 |
| Frattini, cosocle | C22×D4 | C23×C79 |
| prime radical | C22×#41×C79 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C22×C4×C79 ◁ C22×#41×C79 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#354×C79 | C2 | semidirect |
| 5. C23×Q8×C79 | C4 | semidirect |
| 7. C2×C4⋊Q8×C79 | C4 | semidirect |
| 11. C2×#117×C79 | C4 | semidirect |
| 141. #12698 | C79 | direct (N × H) |
| 149. C2×#354 | C2×C79 | semidirect |
| 151. C79 | #12698 | direct (N × H) |
| 153. C23×Q8 | C4×C79 | semidirect |
| 155. C2×C4⋊Q8 | C4×C79 | semidirect |
| 159. C2×#117 | C4×C79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#536×C79 | 2 |
| C2×#354×C79 | 2 |
| C79×#1782 | 2 |
| #12698 | 79 |