Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
1140 conjugacy classes of subgroups; only the 226 normal subgroups are shown.
This group has 1140 conjugacy classes of subgroups — too many to draw. Only its 226 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C109 ◁ C24×C4×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C109×#2847 | C2 | semidirect |
| 2. C109×#2075 | C2 | semidirect |
| 3. C109×#2033 | C2 | semidirect |
| 4. C109×#1999 | C2 | semidirect |
| 5. C109×#1779 | C2 | semidirect |
| 7. C22×D8×C109 | C22 | semidirect |
| 10. C2×#107×C109 | C22 | semidirect |
| 11. C2×#102×C109 | C22 | semidirect |
| 16. #340×C109 | C22 | semidirect |
| 17. #340×C109 | C22 | semidirect |
| 42. C24×C109 | D8 | semidirect |
| 112. #32706 | C109 | direct (N × H) |
| 118. #2847 | C2×C109 | semidirect |
| 119. #2075 | C2×C109 | semidirect |
| 120. #2033 | C2×C109 | semidirect |
| 121. #1999 | C2×C109 | semidirect |
| 122. #1779 | C2×C109 | semidirect |
| 123. C109 | #32706 | direct (N × H) |
| 125. C22×D8 | C22×C109 | semidirect |
| 128. C2×#107 | C22×C109 | semidirect |
| 129. C2×#102 | C22×C109 | semidirect |
| 134. #340 | C22×C109 | semidirect |
| 135. #340 | C22×C109 | semidirect |
| 160. C24 | D8×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2847 | 2 |
| C109×#2075 | 2 |
| C109×#2033 | 2 |
| C109×#1999 | 2 |
| C109×#1779 | 2 |
| #32706 | 109 |