Finite Group DatabasebetaFinite Group DatabasebetaOrder 14272.
236 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D223 |
| commutator, socle, cosocle | C22×C223 | C24 |
| Fitting | #123×C223 | C2 |
| prime radical | #20297300 | C2 |
| hypercenter | #123 | D223 |
| hypocenter | C223 | #559 |
| LCS | G ▷ C22×C223 ▷ C223 | |
| UCS | C1 ◁ C22 ◁ #123 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #20297290 | C2 | semidirect |
| 3. #20297290 | C2 | semidirect |
| 4. #20297289 | C2 | semidirect |
| 5. #20297285 | C2 | semidirect |
| 6. #20297269 | C2 | semidirect |
| 7. C4×C22⋊D223 | C2 | semidirect |
| 8. (C22⋊D223)⋊C4 | C2 | semidirect |
| 9. #123×C223 | C2 | semidirect |
| 67. C223 | #559 | semidirect |
| 74. #123 | D223 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20297300 | 2 |
| #20297290 | 2 |
| #20297289 | 2 |
| #20297285 | 2 |
| #20297269 | 2 |
| C4×C22⋊D223 | 2 |
| (C22⋊D223)⋊C4 | 2 |
| #123×C223 | 2 |
| #559 | 223 |