Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
3798 conjugacy classes of subgroups; only the 438 normal subgroups are shown.
This group has 3798 conjugacy classes of subgroups — too many to draw. Only its 438 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 | C25×C109 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C109 ◁ C24×C4×C109 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#451×C109 | C2 | semidirect |
| 4. C109×#1787 | C2 | semidirect |
| 11. C22×Q16×C109 | C22 | semidirect |
| 14. C2×#103×C109 | C22 | semidirect |
| 46. C24×C109 | Q16 | semidirect |
| 104. C22×C109 | #451 | semidirect |
| 113. #32882 | C109 | direct (N × H) |
| 119. C2×#451 | C2×C109 | semidirect |
| 121. #1787 | C2×C109 | semidirect |
| 123. C109 | #32882 | direct (N × H) |
| 129. C22×Q16 | C22×C109 | semidirect |
| 132. C2×#103 | C22×C109 | semidirect |
| 164. C24 | Q16×C109 | semidirect |
| 222. C22 | #451×C109 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q8×C109 | 2 |
| C2×#451×C109 | 2 |
| C109×#2848 | 2 |
| C109×#1787 | 2 |
| C109×#1779 | 2 |
| #32882 | 109 |