Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
584 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 584 conjugacy classes of subgroups — too many to draw. Only its 166 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C109 | #99 |
| commutator | C2×C4 | C2×C42×C109 |
| Frattini, cosocle | C22×D4 | C23×C109 |
| socle | C22×C109 | C2×#99 |
| prime radical | C22×D4×C109 | C23 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C109 ◁ C22×C4×C109 ◁ C2×C4×D4×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C2×C4×Q8×C109 | C4 | semidirect |
| 9. #391×C109 | C4 | semidirect |
| 12. #382×C109 | C4 | semidirect |
| 15. C43×C109 | C4 | semidirect |
| 16. #354×C109 | C4 | direct (N × H) |
| 50. C2×Q8×C109 | C42 | semidirect |
| 84. C42×C109 | C42 | semidirect |
| 142. C4×C109 | #354 | direct (N × H) |
| 143. C4×#354 | C109 | direct (N × H) |
| 154. C109 | C4×#354 | direct (N × H) |
| 156. C2×C4×Q8 | C4×C109 | semidirect |
| 159. #391 | C4×C109 | semidirect |
| 162. #382 | C4×C109 | semidirect |
| 165. C43 | C4×C109 | semidirect |
| 166. #354 | C4×C109 | direct (N × H) |
| 200. C2×Q8 | C42×C109 | semidirect |
| 234. C42 | C42×C109 | semidirect |
| 292. C4 | #354×C109 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#124×C109 | 2 |
| C2×#354×C109 | 2 |
| C4×#99×C109 | 2 |
| C109×#1393 | 2 |
| C4×#354 | 109 |