Finite Group DatabasebetaFinite Group DatabasebetaOrder 448.
84 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×#131 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #6386 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×C7⋊C16 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C7×#41 | C4 | 1 | 1 | yes | characteristic |
| 5 | C7×M16 | C4 | 1 | 3 | yes | — |
| 6 | C2×C7×C8 | C22 | 1 | 3 | yes | — |
| 7 | C7⋊C16 | C22 | 1 | 1 | yes | characteristic |
| 8 | C7⋊C16 | C22 | 1 | 3 | yes | — |
| 9 | #504 | — | 7 | 1 | no | maximal, Sylow |
| 10 | C7×Q8 | C8 | 1 | 1 | yes | characteristic |
| 11 | C7×D4 | C8 | 1 | 3 | yes | — |
| 12 | C2×C4×C7 | C2×C4 | 1 | 3 | yes | — |
| 13 | C7×C8 | C2×C4 | 1 | 3 | yes | — |
| 14 | C7×C8 | C23 | 1 | 1 | yes | characteristic |
| 15 | #131 | D7 | 1 | 1 | yes | characteristic |
| 16 | M32 | — | 7 | 3 | no | — |
| 17 | C2×C16 | — | 7 | 3 | no | — |
| 18 | C22×C7 | C2×C8 | 1 | 3 | yes | — |
| 19 | C4×C7 | C2×C8 | 1 | 3 | yes | — |
| 20 | C4×C7 | C22×C4 | 1 | 1 | yes | characteristic |
| 21 | #41 | Dic7 | 1 | 1 | yes | characteristic |
| 22 | M16 | Dic7 | 1 | 3 | yes | — |
| 23 | C2×C8 | C2×D7 | 1 | 3 | yes | — |
| 24 | C16 | — | 7 | 1 | no | — |
| 25 | C16 | — | 7 | 3 | no | — |
| 26 | C2×C7 | — | 2 | 3 | no | — |
| 27 | C2×C7 | C22×C8 | 1 | 1 | yes | characteristic |
| 28 | Q8 | C7⋊C8 | 1 | 1 | yes | characteristic |
| 29 | D4 | C7⋊C8 | 1 | 3 | yes | — |
| 30 | C2×C4 | C2×Dic7 | 1 | 3 | yes | — |
| 31 | C8 | C2×Dic7 | 1 | 3 | yes | — |
| 32 | C8 | C22×D7 | 1 | 1 | yes | characteristic |
| 33 | C7 | #504 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 34 | C22 | C2×C7⋊C8 | 1 | 3 | yes | — |
| 35 | C4 | C2×C7⋊C8 | 1 | 3 | yes | — |
| 36 | C4 | C22×Dic7 | 1 | 1 | yes | characteristic |
| 37 | C2 | — | 2 | 3 | no | — |
| 38 | C2 | C22×C7⋊C8 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×D7 |
| commutator, socle | C2×C7 | C22×C8 |
| Fitting | C7×#131 | C2 |
| prime radical | C7×#41 | C4 |
| cosocle | C7×C8 | C23 |
| hypercenter | #131 | D7 |
| hypocenter | C7 | #504 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C8 ◁ #131 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #6386 | C2 | semidirect |
| 3. C2×C7⋊C16 | C2 | semidirect |
| 33. C7 | #504 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#131 | 2 |
| #6386 | 2 |
| C2×C7⋊C16 | 2 |
| #504 | 7 |