Finite Group DatabasebetaFinite Group DatabasebetaOrder 27904.
334 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 334 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 | C23×C109 | #120 |
| 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 |
|---|---|---|
| 151. #12237 | C109 | direct (N × H) |
| 164. C109 | #12237 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C109×#2789 | 2 |
| C109×#2061 | 2 |
| C109×#1611 | 2 |
| C109×#1605 | 2 |
| C109×#1354 | 2 |
| C109×#1346 | 2 |
| C109×#1324 | 2 |
| #12237 | 109 |