Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
596 conjugacy classes of subgroups; only the 206 normal subgroups are shown.
This group has 596 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 | C22×D4×C109 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C109 ◁ #376×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C109×#2645 | C2 | semidirect |
| 3. C109×#2088 | C2 | semidirect |
| 4. C109×#2066 | C2 | semidirect |
| 7. C109×#1843 | C2 | semidirect |
| 112. #32868 | C109 | direct (N × H) |
| 117. #2645 | C2×C109 | semidirect |
| 119. #2088 | C2×C109 | semidirect |
| 120. #2066 | C2×C109 | semidirect |
| 123. #1843 | C2×C109 | semidirect |
| 124. C109 | #32868 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2645 | 2 |
| C109×#2644 | 2 |
| C109×#2088 | 2 |
| C109×#2066 | 2 |
| C109×#2065 | 2 |
| C109×#2055 | 2 |
| C109×#1843 | 2 |
| #32868 | 109 |