Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
156 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C7×#997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C7×#453 | — | 3 | 1 | no | maximal, Hall |
| 3 | C2×C7×S4 | — | 4 | 1 | no | maximal |
| 4 | C7×GL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 5 | C7×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C7×AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 7 | C7×#127 | — | 3 | 1 | no | — |
| 8 | C7×#104 | — | 3 | 2 | no | — |
| 9 | C7×#100 | — | 3 | 1 | no | — |
| 10 | #5935 | C7 | 1 | 1 | yes | maximal, characteristic, direct product factor, Hall |
| 11 | C2×C7×A4 | — | 4 | 1 | no | — |
| 12 | C2×C7×A4 | — | 8 | 1 | no | — |
| 13 | C7×S4 | — | 4 | 2 | no | — |
| 14 | C7×SL(2,𝔽3) | — | 4 | 2 | no | — |
| 15 | C7×#41 | — | 3 | 2 | no | — |
| 16 | C2×C7×D4 | — | 3 | 1 | no | — |
| 17 | C2×C7×D4 | — | 3 | 1 | no | — |
| 18 | C2×C7×D4 | — | 6 | 1 | no | — |
| 19 | C2×C7×D4 | — | 6 | 1 | no | — |
| 20 | C2×C7×D4 | — | 6 | 1 | no | — |
| 21 | C7×SD16 | — | 6 | 2 | no | — |
| 22 | C7×D8 | — | 6 | 2 | no | — |
| 23 | C7×M16 | — | 3 | 2 | no | — |
| 24 | C42×C7 | — | 3 | 1 | no | — |
| 25 | #997 | C2×C7 | 1 | 1 | yes | characteristic, complemented |
| 26 | C2×D3×C7 | — | 16 | 1 | no | — |
| 27 | C7×A4 | — | 4 | 1 | no | — |
| 28 | C7×A4 | — | 8 | 1 | no | — |
| 29 | #453 | — | 3 | 1 | no | Sylow |
| 30 | C23×C7 | — | 2 | 1 | no | — |
| 31 | C23×C7 | — | 3 | 1 | no | — |
| 32 | C23×C7 | — | 6 | 1 | no | — |
| 33 | C23×C7 | S4 | 1 | 1 | yes | characteristic |
| 34 | C7×Q8 | S4 | 1 | 2 | yes | complemented |
| 35 | C7×D4 | — | 3 | 2 | no | — |
| 36 | C7×D4 | — | 6 | 2 | no | — |
| 37 | C7×D4 | — | 6 | 2 | no | — |
| 38 | C7×D4 | — | 12 | 2 | no | — |
| 39 | C7×D4 | — | 12 | 2 | no | — |
| 40 | C2×C4×C7 | — | 3 | 1 | no | — |
| 41 | C2×C4×C7 | — | 6 | 1 | no | — |
| 42 | C2×C4×C7 | — | 6 | 1 | no | — |
| 43 | C7×C8 | — | 6 | 2 | no | — |
| 44 | C2×S4 | — | 4 | 1 | no | — |
| 45 | GL(2,𝔽3) | — | 4 | 2 | no | — |
| 46 | C2×C3×C7 | — | 16 | 1 | no | — |
| 47 | D3×C7 | — | 16 | 2 | no | — |
| 48 | H(5,𝔽2) | D3×C7 | 1 | 1 | yes | characteristic, complemented |
| 49 | AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 50 | #127 | — | 3 | 1 | no | — |
| 51 | #104 | — | 3 | 2 | no | — |
| 52 | #100 | — | 3 | 1 | no | — |
| 53 | C22×C7 | — | 3 | 1 | no | — |
| 54 | C22×C7 | — | 4 | 1 | no | — |
| 55 | C22×C7 | — | 6 | 1 | no | — |
| 56 | C22×C7 | — | 8 | 1 | no | — |
| 57 | C22×C7 | — | 12 | 1 | no | — |
| 58 | C22×C7 | — | 12 | 1 | no | — |
| 59 | C22×C7 | — | 12 | 1 | no | — |
| 60 | C22×C7 | — | 12 | 1 | no | — |
| 61 | C4×C7 | — | 3 | 2 | no | — |
| 62 | C4×C7 | — | 6 | 2 | no | — |
| 63 | C2×A4 | — | 4 | 1 | no | — |
| 64 | C2×A4 | — | 8 | 1 | no | — |
| 65 | S4 | — | 4 | 2 | no | — |
| 66 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 67 | C3×C7 | — | 16 | 1 | no | Hall |
| 68 | #41 | — | 3 | 2 | no | — |
| 69 | C2×D4 | — | 3 | 1 | no | — |
| 70 | C2×D4 | — | 3 | 1 | no | — |
| 71 | C2×D4 | — | 6 | 1 | no | — |
| 72 | C2×D4 | — | 6 | 1 | no | — |
| 73 | C2×D4 | — | 6 | 1 | no | — |
| 74 | SD16 | — | 6 | 2 | no | — |
| 75 | D8 | — | 6 | 2 | no | — |
| 76 | M16 | — | 3 | 2 | no | — |
| 77 | C42 | — | 3 | 1 | no | — |
| 78 | C2×C7 | — | 6 | 1 | no | — |
| 79 | C2×C7 | — | 12 | 1 | no | — |
| 80 | C2×C7 | — | 24 | 1 | no | — |
| 81 | C2×C7 | #1020 | 1 | 1 | yes | characteristic |
| 82 | C2×D3 | — | 16 | 1 | no | — |
| 83 | A4 | — | 4 | 1 | no | — |
| 84 | A4 | — | 8 | 1 | no | — |
| 85 | C23 | — | 2 | 1 | no | — |
| 86 | C23 | — | 3 | 1 | no | — |
| 87 | C23 | — | 6 | 1 | no | — |
| 88 | C23 | C7×S4 | 1 | 1 | yes | characteristic |
| 89 | Q8 | C7×S4 | 1 | 2 | yes | complemented |
| 90 | D4 | — | 3 | 2 | no | — |
| 91 | D4 | — | 6 | 2 | no | — |
| 92 | D4 | — | 6 | 2 | no | — |
| 93 | D4 | — | 12 | 2 | no | — |
| 94 | D4 | — | 12 | 2 | no | — |
| 95 | C2×C4 | — | 3 | 1 | no | — |
| 96 | C2×C4 | — | 6 | 1 | no | — |
| 97 | C2×C4 | — | 6 | 1 | no | — |
| 98 | C8 | — | 6 | 2 | no | — |
| 99 | C7 | #5935 | 1 | 1 | yes | characteristic, direct product factor, Sylow |
| 100 | C2×C3 | — | 16 | 1 | no | — |
| 101 | D3 | — | 16 | 2 | no | — |
| 102 | C22 | — | 3 | 1 | no | — |
| 103 | C22 | — | 4 | 1 | no | — |
| 104 | C22 | — | 6 | 1 | no | — |
| 105 | C22 | — | 8 | 1 | no | — |
| 106 | C22 | — | 12 | 1 | no | — |
| 107 | C22 | — | 12 | 1 | no | — |
| 108 | C22 | — | 12 | 1 | no | — |
| 109 | C22 | — | 12 | 1 | no | — |
| 110 | C4 | — | 3 | 2 | no | — |
| 111 | C4 | — | 6 | 2 | no | — |
| 112 | C3 | — | 16 | 1 | no | Sylow |
| 113 | C2 | — | 6 | 1 | no | — |
| 114 | C2 | — | 12 | 1 | no | — |
| 115 | C2 | — | 24 | 1 | no | — |
| 116 | C2 | C7×#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C7 | #1020 |
| commutator, cosocle, hypocenter | #997 | C2×C7 |
| Frattini | C2 | C7×#1020 |
| Fitting | C7×H(5,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C7 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C7×#997 | C2 | semidirect |
| 5. C7×H(5,𝔽2) | D3 | semidirect |
| 10. #5935 | C7 | direct (N × H) |
| 25. #997 | C2×C7 | semidirect |
| 34. C7×Q8 | S4 | semidirect |
| 48. H(5,𝔽2) | D3×C7 | semidirect |
| 89. Q8 | C7×S4 | semidirect |
| 99. C7 | #5935 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C7×#997 | 2 |
| C7×#453 | 3 |
| C2×C7×S4 | 4 |
| C7×GL(2,𝔽3) | 4 |
| #5935 | 7 |