Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
196 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×#997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #90016 | — | 3 | 1 | no | maximal, Hall |
| 3 | C2×C7⋊S4 | — | 4 | 3 | no | maximal |
| 4 | C7×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 5 | #6500 | — | 3 | 3 | no | — |
| 6 | #6408 | — | 3 | 3 | no | — |
| 7 | #5934 | — | 7 | 1 | no | maximal, Hall |
| 8 | C2×C7×A4 | — | 4 | 3 | no | — |
| 9 | C7⋊S4 | — | 4 | 6 | no | — |
| 10 | C7×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 11 | C23×D7 | — | 3 | 1 | no | — |
| 12 | C7×#41 | — | 6 | 1 | no | — |
| 13 | C2×C7×D4 | — | 3 | 3 | no | — |
| 14 | C2×C22⋊D7 | — | 6 | 3 | no | — |
| 15 | C22⋊Dic7 | — | 3 | 3 | no | — |
| 16 | (C2×D7)⋊C4 | — | 6 | 3 | no | — |
| 17 | #997 | D7 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×D21 | — | 16 | 1 | no | — |
| 19 | C7×A4 | — | 4 | 3 | no | — |
| 20 | B(4,𝔽2) | — | 21 | 1 | no | Sylow |
| 21 | C23×C7 | — | 3 | 1 | no | — |
| 22 | C23×C7 | S4 | 1 | 3 | yes | complemented |
| 23 | C22×D7 | — | 6 | 3 | no | — |
| 24 | C22×D7 | — | 12 | 2 | no | — |
| 25 | C7×Q8 | — | 2 | 1 | no | — |
| 26 | C7×D4 | — | 6 | 3 | no | — |
| 27 | C2×C4×C7 | — | 3 | 3 | no | — |
| 28 | C22⋊D7 | — | 12 | 6 | no | — |
| 29 | C2×Dic7 | — | 6 | 3 | no | — |
| 30 | C2×S4 | — | 28 | 3 | no | — |
| 31 | C2×C3×C7 | — | 16 | 1 | no | — |
| 32 | D21 | — | 16 | 2 | no | — |
| 33 | H(5,𝔽2) | D21 | 1 | 1 | yes | characteristic, complemented |
| 34 | #120 | — | 21 | 3 | no | — |
| 35 | #99 | — | 21 | 3 | no | — |
| 36 | C22×C7 | — | 3 | 3 | no | — |
| 37 | C22×C7 | — | 4 | 3 | no | — |
| 38 | C22×C7 | — | 12 | 1 | no | — |
| 39 | C2×D7 | — | 6 | 6 | no | — |
| 40 | C2×D7 | — | 12 | 3 | no | — |
| 41 | C2×D7 | — | 12 | 1 | no | — |
| 42 | C4×C7 | — | 6 | 1 | no | — |
| 43 | Dic7 | — | 12 | 3 | no | — |
| 44 | C2×A4 | — | 4 | 3 | no | — |
| 45 | S4 | — | 28 | 6 | no | — |
| 46 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 47 | C3×C7 | — | 16 | 1 | no | Hall |
| 48 | C24 | — | 21 | 1 | no | — |
| 49 | #41 | — | 6 | 1 | no | — |
| 50 | C2×D4 | — | 3 | 3 | no | — |
| 51 | C2×D4 | — | 42 | 3 | no | — |
| 52 | C22⋊C4 | — | 21 | 3 | no | — |
| 53 | C22⋊C4 | — | 42 | 3 | no | — |
| 54 | C2×C7 | — | 6 | 3 | no | — |
| 55 | C2×C7 | #1020 | 1 | 1 | yes | characteristic |
| 56 | D7 | — | 12 | 2 | no | — |
| 57 | C2×D3 | — | 112 | 1 | no | — |
| 58 | A4 | — | 4 | 3 | no | — |
| 59 | C23 | — | 3 | 1 | no | — |
| 60 | C23 | — | 42 | 3 | no | — |
| 61 | C23 | — | 84 | 2 | no | — |
| 62 | C23 | C7⋊S4 | 1 | 3 | yes | complemented |
| 63 | Q8 | — | 2 | 1 | no | — |
| 64 | D4 | — | 6 | 3 | no | — |
| 65 | D4 | — | 84 | 6 | no | — |
| 66 | C2×C4 | — | 3 | 3 | no | — |
| 67 | C2×C4 | — | 42 | 3 | no | — |
| 68 | C7 | #5934 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 69 | C2×C3 | — | 16 | 1 | no | — |
| 70 | D3 | — | 112 | 2 | no | — |
| 71 | C22 | — | 3 | 3 | no | — |
| 72 | C22 | — | 4 | 3 | no | — |
| 73 | C22 | — | 12 | 1 | no | — |
| 74 | C22 | — | 42 | 6 | no | — |
| 75 | C22 | — | 84 | 3 | no | — |
| 76 | C22 | — | 84 | 1 | no | — |
| 77 | C4 | — | 6 | 1 | no | — |
| 78 | C4 | — | 84 | 3 | no | — |
| 79 | C3 | — | 16 | 1 | no | Sylow |
| 80 | C2 | — | 6 | 3 | no | — |
| 81 | C2 | — | 84 | 2 | no | — |
| 82 | C2 | C7⋊#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C7⋊#1020 |
| commutator, cosocle, hypocenter | C7×#997 | C2 |
| Fitting | C7×H(5,𝔽2) | D3 |
| socle | C2×C7 | #1020 |
| LCS | G ▷ C7×#997 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#997 | C2 | semidirect |
| 4. C7×H(5,𝔽2) | D3 | semidirect |
| 17. #997 | D7 | semidirect |
| 22. C23×C7 | S4 | semidirect |
| 33. H(5,𝔽2) | D21 | semidirect |
| 62. C23 | C7⋊S4 | semidirect |
| 68. C7 | #5934 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#997 | 2 |
| #90016 | 3 |
| C2×C7⋊S4 | 4 |
| #5934 | 7 |