Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
324 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 324 conjugacy classes of subgroups — too many to draw. Only its 78 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 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C23×C109 |
| prime radical | C109×#2250 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C42×C109 ◁ #446×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C109×#3402 | C2 | semidirect |
| 3. C109×#2207 | C2 | semidirect |
| 4. C109×#1688 | C2 | semidirect |
| 5. C109×#1360 | C2 | semidirect |
| 6. C109×#1338 | C2 | semidirect |
| 7. C109×#1334 | C2 | semidirect |
| 150. #12337 | C109 | direct (N × H) |
| 156. #3402 | C2×C109 | semidirect |
| 158. #2207 | C2×C109 | semidirect |
| 159. #1688 | C2×C109 | semidirect |
| 160. #1360 | C2×C109 | semidirect |
| 161. #1338 | C2×C109 | semidirect |
| 162. #1334 | C2×C109 | semidirect |
| 163. C109 | #12337 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#3402 | 2 |
| C109×#2250 | 2 |
| C109×#2207 | 2 |
| C109×#1688 | 2 |
| C109×#1360 | 2 |
| C109×#1338 | 2 |
| C109×#1334 | 2 |
| #12337 | 109 |