Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
328 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C109 | #1613 |
| commutator, Frattini, cosocle | C2×M16 | C23×C109 |
| prime radical | C109×#3197 | C2 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C2×M16×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C109×#2239 | C2 | semidirect |
| 3. C109×#2238 | C2 | semidirect |
| 4. C109×#2091 | C2 | semidirect |
| 5. C109×#1342 | C2 | semidirect |
| 6. C109×#1341 | C2 | semidirect |
| 7. C109×#1320 | C2 | semidirect |
| 149. #12300 | C109 | direct (N × H) |
| 156. #2239 | C2×C109 | semidirect |
| 157. #2238 | C2×C109 | semidirect |
| 158. #2091 | C2×C109 | semidirect |
| 159. #1342 | C2×C109 | semidirect |
| 160. #1341 | C2×C109 | semidirect |
| 161. #1320 | C2×C109 | semidirect |
| 162. C109 | #12300 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#3197 | 2 |
| C109×#2239 | 2 |
| C109×#2238 | 2 |
| C109×#2091 | 2 |
| C109×#1342 | 2 |
| C109×#1341 | 2 |
| C109×#1320 | 2 |
| #12300 | 109 |