Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
284 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 110 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 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C109 |
| prime radical | #493×C109 | C22 |
| 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×#2254 | C2 | semidirect |
| 3. C109×#2241 | C2 | semidirect |
| 5. C109×#1707 | C2 | semidirect |
| 7. C8×C16×C109 | C2 | semidirect |
| 68. C16×C109 | SD16 | semidirect |
| 85. C8×C109 | SD32 | semidirect |
| 99. #12505 | C109 | direct (N × H) |
| 104. #2254 | C2×C109 | semidirect |
| 106. #2241 | C2×C109 | semidirect |
| 108. #1707 | C2×C109 | semidirect |
| 110. C8×C16 | C2×C109 | semidirect |
| 111. C109 | #12505 | direct (N × H) |
| 172. C16 | SD16×C109 | semidirect |
| 189. C8 | SD32×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2254 | 2 |
| C109×#2248 | 2 |
| C109×#2241 | 2 |
| C109×#2240 | 2 |
| C109×#1707 | 2 |
| C109×#1706 | 2 |
| C8×C16×C109 | 2 |
| #12505 | 109 |