Finite Group DatabasebetaFinite Group DatabasebetaOrder 14272.
250 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 250 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 | C22⋊Q8×C223 | C2 |
| hypercenter | C22⋊Q8 | D223 |
| hypocenter | C223 | #538 |
| LCS | G ▷ C22×C223 ▷ C223 | |
| UCS | C1 ◁ C22 ◁ C22⋊Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20297299 | C2 | semidirect |
| 2. #20297293 | C2 | semidirect |
| 3. #20297289 | C2 | semidirect |
| 4. #20297279 | C2 | semidirect |
| 5. #20297278 | C2 | semidirect |
| 6. #20297274 | C2 | semidirect |
| 7. (C22⋊D223)⋊C4 | C2 | semidirect |
| 8. C2×(C2×D223)⋊C4 | C2 | semidirect |
| 9. C22⋊Q8×C223 | C2 | semidirect |
| 13. C2×D892 | C22 | semidirect |
| 16. C2×C4×D223 | C22 | semidirect |
| 17. C4⋊Dic223 | C22 | semidirect |
| 18. Dic223⋊C4 | C22 | semidirect |
| 23. (C2×D223)⋊C4 | C22 | semidirect |
| 24. (C2×D223)⋊C4 | C22 | semidirect |
| 25. C4×Dic223 | C22 | semidirect |
| 26. C2×Q8×C223 | C22 | semidirect |
| 28. C4⋊C4×C223 | C22 | semidirect |
| 29. C4⋊C4×C223 | C22 | semidirect |
| 74. C223 | #538 | semidirect |
| 79. C22⋊Q8 | D223 | semidirect |
| 86. C2×Q8 | C2×D223 | semidirect |
| 96. C4⋊C4 | C2×D223 | semidirect |
| 97. C4⋊C4 | C2×D223 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20297299 | 2 |
| #20297293 | 2 |
| #20297289 | 2 |
| #20297279 | 2 |
| #20297278 | 2 |
| #20297274 | 2 |
| (C22⋊D223)⋊C4 | 2 |
| C2×(C2×D223)⋊C4 | 2 |
| C22⋊Q8×C223 | 2 |
| #538 | 223 |