Finite Group DatabasebetaFinite Group DatabasebetaOrder 20224.
1118 conjugacy classes of subgroups; only the 366 normal subgroups are shown.
This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 366 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×#120 |
| commutator, Frattini, cosocle | C24 | C24×C79 |
| prime radical | C24×C79 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C79 ◁ C23×C4×C79 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 29. #117×C79 | Q8 | semidirect |
| 102. #23566 | C79 | direct (N × H) |
| 112. C79 | #23566 | direct (N × H) |
| 137. #117 | Q8×C79 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C79×#3458 | 2 |
| C79×#2317 | 2 |
| C79×#2062 | 2 |
| C79×#1740 | 2 |
| #23566 | 79 |