Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
348 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 348 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 | C22×C8 | C23×C109 |
| prime radical | C109×#3191 | C2 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C2×C22⋊C8×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C109×#2235 | C2 | semidirect |
| 3. C109×#2232 | C2 | semidirect |
| 4. C109×#2075 | C2 | semidirect |
| 5. C109×#1348 | C2 | semidirect |
| 6. C109×#1347 | C2 | semidirect |
| 7. C109×#1314 | C2 | semidirect |
| 161. #12280 | C109 | direct (N × H) |
| 170. #2235 | C2×C109 | semidirect |
| 171. #2232 | C2×C109 | semidirect |
| 172. #2075 | C2×C109 | semidirect |
| 173. #1348 | C2×C109 | semidirect |
| 174. #1347 | C2×C109 | semidirect |
| 175. #1314 | C2×C109 | semidirect |
| 176. C109 | #12280 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#3191 | 2 |
| C109×#2235 | 2 |
| C109×#2232 | 2 |
| C109×#2075 | 2 |
| C109×#1348 | 2 |
| C109×#1347 | 2 |
| C109×#1314 | 2 |
| #12280 | 109 |