Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
348 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C109 | C2×D8 |
| commutator | C2×C8 | C22×C4×C109 |
| Frattini, cosocle | C4×C8 | C23×C109 |
| socle | C22×C109 | C2×#102 |
| prime radical | C109×#3169 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C109 ◁ C2×C42×C109 ◁ C2×C4×C8×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C109×#2162 | C2 | semidirect |
| 3. C109×#1564 | C2 | semidirect |
| 4. C109×#1331 | C2 | semidirect |
| 113. #11649 | C109 | direct (N × H) |
| 120. #2162 | C2×C109 | semidirect |
| 121. #1564 | C2×C109 | semidirect |
| 122. #1331 | C2×C109 | semidirect |
| 123. C109 | #11649 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#3169 | 2 |
| C109×#2162 | 2 |
| C109×#1564 | 2 |
| C109×#1331 | 2 |
| #11649 | 109 |