Finite Group DatabasebetaFinite Group DatabasebetaOrder 22144.
248 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 56 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×C173 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C23×C173 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C173 ◁ C22×C173 ◁ C2×Q8×C173 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(3,𝔽4)×C173 | C2 | semidirect |
| 2. #454×C173 | C2 | semidirect |
| 3. #355×C173 | C2 | semidirect |
| 4. 2−1+4×C173 | C22 | semidirect |
| 10. #101×C173 | C22 | semidirect |
| 51. C173 | #2202 | direct (N × H) |
| 52. #2202 | C173 | direct (N × H) |
| 53. H(3,𝔽4) | C2×C173 | semidirect |
| 54. #454 | C2×C173 | semidirect |
| 55. #355 | C2×C173 | semidirect |
| 56. 2−1+4 | C22×C173 | semidirect |
| 62. #101 | C22×C173 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(3,𝔽4)×C173 | 2 |
| #454×C173 | 2 |
| #355×C173 | 2 |
| #2202 | 173 |