Finite Group DatabasebetaFinite Group DatabasebetaOrder 22784.
734 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 734 conjugacy classes of subgroups — too many to draw. Only its 90 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×C89 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C23×C89 |
| prime radical | C89×#2669 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C89 ◁ C23×C89 ◁ C22×D4×C89 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#353×C89 | C2 | semidirect |
| 3. C2×AΣL(1,𝔽16)×C89 | C2 | semidirect |
| 4. C89×#1889 | C2 | semidirect |
| 5. C89×#1887 | C2 | semidirect |
| 6. C89×#1867 | C2 | semidirect |
| 7. C89×#1390 | C2 | semidirect |
| 76. #99×C89 | D4 | semidirect |
| 288. #13422 | C89 | direct (N × H) |
| 301. C2×#353 | C2×C89 | semidirect |
| 302. C2×AΣL(1,𝔽16) | C2×C89 | semidirect |
| 303. #1889 | C2×C89 | semidirect |
| 304. #1887 | C2×C89 | semidirect |
| 305. #1867 | C2×C89 | semidirect |
| 306. #1390 | C2×C89 | semidirect |
| 307. C89 | #13422 | direct (N × H) |
| 376. #99 | D4×C89 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C89×#2669 | 2 |
| C2×#353×C89 | 2 |
| C2×AΣL(1,𝔽16)×C89 | 2 |
| C89×#1889 | 2 |
| C89×#1887 | 2 |
| C89×#1867 | 2 |
| C89×#1390 | 2 |
| #13422 | 89 |