Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
516 conjugacy classes of subgroups; only the 106 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 106 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×C109 | C2×#99 |
| commutator | C22×C4 | C22×C4×C109 |
| Frattini, cosocle | C2×C22⋊C4 | C23×C109 |
| prime radical | C22×D4×C109 | D4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C2×C4×D4×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C109×#1352 | C2 | semidirect |
| 5. C2×C4⋊Q8×C109 | C4 | semidirect |
| 11. #399×C109 | C4 | semidirect |
| 15. C4×C4⋊C4×C109 | C4 | semidirect |
| 155. #11788 | C109 | direct (N × H) |
| 166. #1352 | C2×C109 | semidirect |
| 167. C109 | #11788 | direct (N × H) |
| 168. C2×C4⋊Q8 | C4×C109 | semidirect |
| 174. #399 | C4×C109 | semidirect |
| 178. C4×C4⋊C4 | C4×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2542 | 2 |
| C4×#99×C109 | 2 |
| C109×#1465 | 2 |
| C109×#1352 | 2 |
| #11788 | 109 |