Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
256 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 86 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 | #459 |
| commutator, Frattini, cosocle | C4×C8 | C23×C109 |
| prime radical | C109×#2206 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C42×C109 ◁ C82×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C109×#2224 | C2 | semidirect |
| 3. C109×#1706 | C2 | semidirect |
| 4. C8×D8×C109 | C2 | semidirect |
| 5. C109×#1367 | C2 | semidirect |
| 6. C109×#1364 | C2 | semidirect |
| 7. C109×#1361 | C2 | semidirect |
| 57. D8×C109 | SD16 | semidirect |
| 94. C8×C109 | D16 | semidirect |
| 111. #12416 | C109 | direct (N × H) |
| 117. #2224 | C2×C109 | semidirect |
| 119. #1706 | C2×C109 | semidirect |
| 120. C8×D8 | C2×C109 | semidirect |
| 121. #1367 | C2×C109 | semidirect |
| 122. #1364 | C2×C109 | semidirect |
| 123. #1361 | C2×C109 | semidirect |
| 124. C109 | #12416 | direct (N × H) |
| 174. D8 | SD16×C109 | semidirect |
| 211. C8 | D16×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2224 | 2 |
| C109×#2206 | 2 |
| C109×#1706 | 2 |
| C8×D8×C109 | 2 |
| C109×#1367 | 2 |
| C109×#1364 | 2 |
| C109×#1361 | 2 |
| #12416 | 109 |