Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
1362 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C25×C7 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C23×Dic7 | C2 | 1 | 30 | yes | maximal, direct product factor |
| 3 | C24×C7 | C4 | 1 | 16 | yes | complemented |
| 4 | C24×C7 | C22 | 1 | 15 | yes | — |
| 5 | C22×Dic7 | C22 | 1 | 140 | yes | direct product factor |
| 6 | C24×C4 | — | 7 | 1 | no | maximal, Sylow |
| 7 | C23×C7 | C2×C4 | 1 | 120 | yes | complemented |
| 8 | C23×C7 | C23 | 1 | 35 | yes | — |
| 9 | C2×Dic7 | C23 | 1 | 120 | yes | direct product factor |
| 10 | C25 | D7 | 1 | 1 | yes | characteristic |
| 11 | C23×C4 | — | 7 | 30 | no | — |
| 12 | C22×C7 | C22×C4 | 1 | 140 | yes | complemented |
| 13 | C22×C7 | C24 | 1 | 15 | yes | — |
| 14 | Dic7 | C24 | 1 | 16 | yes | direct product factor |
| 15 | C24 | Dic7 | 1 | 16 | yes | direct product factor |
| 16 | C24 | C2×D7 | 1 | 15 | yes | — |
| 17 | C22×C4 | — | 7 | 140 | no | — |
| 18 | C2×C7 | C23×C4 | 1 | 30 | yes | complemented |
| 19 | C2×C7 | C25 | 1 | 1 | yes | characteristic |
| 20 | C23 | C2×Dic7 | 1 | 120 | yes | direct product factor |
| 21 | C23 | C22×D7 | 1 | 35 | yes | — |
| 22 | C2×C4 | — | 7 | 120 | no | — |
| 23 | C7 | C24×C4 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 24 | C22 | C22×Dic7 | 1 | 140 | yes | direct product factor |
| 25 | C22 | C23×D7 | 1 | 15 | yes | — |
| 26 | C4 | — | 7 | 16 | no | — |
| 27 | C2 | C23×Dic7 | 1 | 30 | yes | direct product factor |
| 28 | C2 | C24×D7 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25 | D7 |
| commutator, hypocenter | C7 | C24×C4 |
| Frattini | C2 | C24×D7 |
| Fitting, socle, prime radical | C25×C7 | C2 |
| cosocle | C2×C7 | C25 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C25 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×Dic7 | C2 | direct (N × H) |
| 3. C24×C7 | C4 | semidirect |
| 5. C22×Dic7 | C22 | direct (N × H) |
| 7. C23×C7 | C2×C4 | semidirect |
| 9. C2×Dic7 | C23 | direct (N × H) |
| 12. C22×C7 | C22×C4 | semidirect |
| 14. Dic7 | C24 | direct (N × H) |
| 15. C24 | Dic7 | direct (N × H) |
| 18. C2×C7 | C23×C4 | semidirect |
| 20. C23 | C2×Dic7 | direct (N × H) |
| 23. C7 | C24×C4 | semidirect |
| 24. C22 | C22×Dic7 | direct (N × H) |
| 27. C2 | C23×Dic7 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C7 | 2 |
| C23×Dic7 | 2 |
| C24×C4 | 7 |