Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
186 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×C42×C7 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×C4⋊Dic7 | C2 | 1 | 6 | yes | maximal |
| 3 | C22×C4×C7 | C22 | 1 | 3 | yes | — |
| 4 | C22×Dic7 | C22 | 1 | 4 | yes | — |
| 5 | C42×C7 | C4 | 1 | 4 | yes | complemented |
| 6 | C4⋊Dic7 | — | 2 | 12 | no | — |
| 7 | #384 | — | 7 | 1 | no | maximal, Sylow |
| 8 | C23×C7 | C23 | 1 | 1 | yes | characteristic |
| 9 | C2×C4×C7 | C2×C4 | 1 | 6 | yes | — |
| 10 | C2×C4×C7 | D4 | 1 | 6 | yes | — |
| 11 | C2×C4×C7 | Q8 | 1 | 6 | yes | — |
| 12 | C2×Dic7 | — | 2 | 12 | no | — |
| 13 | C2×C4⋊C4 | — | 7 | 6 | no | — |
| 14 | C2×C42 | D7 | 1 | 1 | yes | characteristic |
| 15 | C22×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 16 | C22×C7 | C2×D4 | 1 | 3 | yes | — |
| 17 | C22×C7 | C2×Q8 | 1 | 3 | yes | — |
| 18 | C4×C7 | C4⋊C4 | 1 | 12 | yes | complemented |
| 19 | Dic7 | — | 4 | 4 | no | — |
| 20 | C22×C4 | — | 7 | 4 | no | — |
| 21 | C22×C4 | C2×D7 | 1 | 3 | yes | — |
| 22 | C4⋊C4 | — | 14 | 12 | no | — |
| 23 | C42 | Dic7 | 1 | 4 | yes | complemented |
| 24 | C2×C7 | C2×C4⋊C4 | 1 | 3 | yes | — |
| 25 | C2×C7 | #127 | 1 | 1 | yes | characteristic |
| 26 | C2×C7 | C4⋊Q8 | 1 | 3 | yes | — |
| 27 | C23 | C22×D7 | 1 | 1 | yes | characteristic |
| 28 | C2×C4 | — | 14 | 12 | no | — |
| 29 | C2×C4 | Dic14 | 1 | 6 | yes | — |
| 30 | C2×C4 | D28 | 1 | 6 | yes | — |
| 31 | C2×C4 | C2×Dic7 | 1 | 6 | yes | — |
| 32 | C7 | #384 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 33 | C22 | C2×Dic14 | 1 | 3 | yes | — |
| 34 | C22 | C2×D28 | 1 | 3 | yes | — |
| 35 | C22 | C22×Dic7 | 1 | 1 | yes | characteristic |
| 36 | C4 | — | 28 | 4 | no | — |
| 37 | C4 | C4⋊Dic7 | 1 | 12 | yes | complemented |
| 38 | C2 | #6432 | 1 | 3 | yes | — |
| 39 | C2 | #6437 | 1 | 1 | yes | characteristic |
| 40 | C2 | C2×C4⋊Dic7 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D7 |
| commutator | C22×C7 | C22×C4 |
| Fitting | C2×C42×C7 | C2 |
| socle, prime radical, cosocle | C23×C7 | C23 |
| hypercenter | C2×C42 | D7 |
| hypocenter | C7 | #384 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ C2×C42 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 5. C42×C7 | C4 | semidirect |
| 18. C4×C7 | C4⋊C4 | semidirect |
| 23. C42 | Dic7 | semidirect |
| 32. C7 | #384 | semidirect |
| 37. C4 | C4⋊Dic7 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C42×C7 | 2 |
| C2×C4⋊Dic7 | 2 |
| #384 | 7 |