Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
906 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 906 conjugacy classes of subgroups — too many to draw. Only its 206 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C109 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C109 |
| prime radical | C109×#2615 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C109 ◁ C22×#41×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C109×#2616 | C2 | semidirect |
| 3. C109×#2077 | C2 | semidirect |
| 4. C109×#2035 | C2 | semidirect |
| 5. C109×#2034 | C2 | semidirect |
| 6. C109×#2000 | C2 | semidirect |
| 7. C109×#1780 | C2 | semidirect |
| 149. #32716 | C109 | direct (N × H) |
| 155. #2616 | C2×C109 | semidirect |
| 157. #2077 | C2×C109 | semidirect |
| 158. #2035 | C2×C109 | semidirect |
| 159. #2034 | C2×C109 | semidirect |
| 160. #2000 | C2×C109 | semidirect |
| 161. #1780 | C2×C109 | semidirect |
| 162. C109 | #32716 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2616 | 2 |
| C109×#2615 | 2 |
| C109×#2077 | 2 |
| C109×#2035 | 2 |
| C109×#2034 | 2 |
| C109×#2000 | 2 |
| C109×#1780 | 2 |
| #32716 | 109 |