Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
118 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×#95 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7⋊#95 | C2 | 1 | 3 | yes | maximal |
| 3 | C7⋊#95 | C2 | 1 | 3 | yes | maximal |
| 4 | C22×C4×C7 | C22 | 1 | 3 | yes | — |
| 5 | C22×Dic7 | C22 | 1 | 1 | yes | characteristic |
| 6 | C22×Dic7 | C22 | 1 | 3 | yes | — |
| 7 | #401 | — | 7 | 1 | no | maximal, Sylow |
| 8 | C23×C7 | C23 | 1 | 1 | yes | characteristic |
| 9 | C2×C4×C7 | — | 2 | 3 | no | — |
| 10 | C2×C4×C7 | — | 2 | 6 | no | — |
| 11 | C2×Dic7 | — | 2 | 3 | no | — |
| 12 | C2×Dic7 | — | 2 | 3 | no | — |
| 13 | C2×Dic7 | — | 2 | 3 | no | — |
| 14 | C2×Dic7 | — | 2 | 3 | no | — |
| 15 | #95 | — | 7 | 3 | no | — |
| 16 | #95 | — | 7 | 3 | no | — |
| 17 | #95 | D7 | 1 | 1 | yes | characteristic |
| 18 | C22×C7 | #41 | 1 | 3 | yes | — |
| 19 | C22×C7 | #41 | 1 | 3 | yes | — |
| 20 | C22×C7 | #41 | 1 | 1 | yes | characteristic |
| 21 | C4×C7 | — | 4 | 3 | no | — |
| 22 | Dic7 | — | 4 | 1 | no | — |
| 23 | Dic7 | — | 4 | 3 | no | — |
| 24 | C22×C4 | — | 7 | 1 | no | — |
| 25 | C22×C4 | — | 7 | 3 | no | — |
| 26 | C22×C4 | C2×D7 | 1 | 3 | yes | — |
| 27 | C2×C7 | #126 | 1 | 1 | yes | characteristic |
| 28 | C2×C7 | #126 | 1 | 3 | yes | — |
| 29 | C2×C7 | #126 | 1 | 3 | yes | — |
| 30 | C23 | C22×D7 | 1 | 1 | yes | characteristic |
| 31 | C2×C4 | — | 2 | 3 | no | — |
| 32 | C2×C4 | — | 2 | 6 | no | — |
| 33 | C2×C4 | — | 14 | 3 | no | — |
| 34 | C2×C4 | — | 14 | 3 | no | — |
| 35 | C2×C4 | — | 14 | 3 | no | — |
| 36 | C2×C4 | — | 14 | 3 | no | — |
| 37 | C7 | #401 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 38 | C22 | #1157 | 1 | 3 | yes | — |
| 39 | C22 | #1159 | 1 | 3 | yes | — |
| 40 | C22 | #1161 | 1 | 1 | yes | characteristic |
| 41 | C4 | — | 4 | 3 | no | — |
| 42 | C4 | — | 28 | 1 | no | — |
| 43 | C4 | — | 28 | 3 | no | — |
| 44 | C2 | #6439 | 1 | 1 | yes | characteristic |
| 45 | C2 | #6442 | 1 | 3 | yes | — |
| 46 | C2 | #6461 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C23 | C22×D7 |
| commutator, socle, prime radical, cosocle | C23×C7 | C23 |
| Fitting | C7×#95 | C2 |
| hypercenter | #95 | D7 |
| hypocenter | C7 | #401 |
| LCS | G ▷ C23×C7 ▷ C7 | |
| UCS | C1 ◁ C23 ◁ #95 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 37. C7 | #401 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#95 | 2 |
| C7⋊#95 | 2 |
| #401 | 7 |