Finite Group DatabasebetaFinite Group DatabasebetaOrder 23168.
122 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 28 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C181 | AΣL(1,𝔽16) |
| commutator | C2×D4 | C2×C4×C181 |
| Frattini, cosocle | #127 | C22×C181 |
| prime radical | #453×C181 | C2 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C181 ◁ C22×C181 ◁ C2×C4×C181 ◁ #127×C181 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #353×C181 | C2 | semidirect |
| 3. #331×C181 | C2 | semidirect |
| 4. H(5,𝔽2)×C181 | C4 | semidirect |
| 9. #100×C181 | C4 | semidirect |
| 59. C181 | #1402 | direct (N × H) |
| 60. #1402 | C181 | direct (N × H) |
| 62. #353 | C2×C181 | semidirect |
| 63. #331 | C2×C181 | semidirect |
| 64. H(5,𝔽2) | C4×C181 | semidirect |
| 69. #100 | C4×C181 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #453×C181 | 2 |
| #353×C181 | 2 |
| #331×C181 | 2 |
| #1402 | 181 |