Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
412 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 412 conjugacy classes of subgroups — too many to draw. Only its 70 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 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C23×C109 |
| prime radical | C109×#2828 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C109 ◁ C23×C109 ◁ C2×C4⋊C4×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C109×#2027 | C2 | semidirect |
| 3. C109×#1617 | C2 | semidirect |
| 4. C109×#1599 | C2 | semidirect |
| 5. C109×#1353 | C2 | semidirect |
| 6. C109×#1345 | C2 | semidirect |
| 7. C109×#1318 | C2 | semidirect |
| 189. #12274 | C109 | direct (N × H) |
| 198. #2027 | C2×C109 | semidirect |
| 199. #1617 | C2×C109 | semidirect |
| 200. #1599 | C2×C109 | semidirect |
| 201. #1353 | C2×C109 | semidirect |
| 202. #1345 | C2×C109 | semidirect |
| 203. #1318 | C2×C109 | semidirect |
| 204. C109 | #12274 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2828 | 2 |
| C109×#2027 | 2 |
| C109×#1617 | 2 |
| C109×#1599 | 2 |
| C109×#1353 | 2 |
| C109×#1345 | 2 |
| C109×#1318 | 2 |
| #12274 | 109 |