Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
156 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 21 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 | #10621528 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C7×#986 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C7⋊GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #89970 | — | 3 | 1 | no | maximal, Hall |
| 5 | C7×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | #66662 | — | 2 | 1 | no | — |
| 7 | C7×GL(2,𝔽3) | — | 2 | 1 | no | — |
| 8 | #66564 | — | 4 | 1 | no | maximal |
| 9 | #6553 | D3 | 1 | 1 | yes | characteristic, complemented |
| 10 | C7×AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 11 | #6503 | — | 3 | 1 | no | — |
| 12 | #6495 | — | 3 | 1 | no | — |
| 13 | #6411 | — | 3 | 1 | no | — |
| 14 | #6399 | — | 3 | 1 | no | — |
| 15 | #6397 | — | 3 | 1 | no | — |
| 16 | #5429 | — | 7 | 1 | no | maximal, Hall |
| 17 | C7×D12 | — | 4 | 1 | no | — |
| 18 | C3×Dic14 | — | 4 | 1 | no | — |
| 19 | C7×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic, complemented |
| 20 | C3⋊(C7⋊C8) | — | 4 | 1 | no | — |
| 21 | C7×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 22 | C2×C7×D4 | — | 3 | 1 | no | — |
| 23 | C2×C22⋊D7 | — | 6 | 2 | no | — |
| 24 | #1161 | — | 2 | 1 | no | — |
| 25 | D4×D7 | — | 6 | 1 | no | — |
| 26 | #1157 | — | 3 | 1 | no | — |
| 27 | C2×D28 | — | 3 | 1 | no | — |
| 28 | C7×SD16 | — | 6 | 1 | no | — |
| 29 | C7×D8 | — | 6 | 1 | no | — |
| 30 | C7×M16 | — | 3 | 1 | no | — |
| 31 | #1142 | — | 6 | 1 | no | — |
| 32 | #1141 | — | 6 | 1 | no | — |
| 33 | C4×Dic7 | — | 3 | 1 | no | — |
| 34 | #1136 | — | 3 | 1 | no | — |
| 35 | #994 | — | 7 | 1 | no | — |
| 36 | #986 | D7 | 1 | 1 | yes | characteristic, complemented |
| 37 | GU(2,𝔽3) | — | 7 | 1 | no | — |
| 38 | C2×D3×C7 | — | 8 | 1 | no | — |
| 39 | C3×C4×C7 | — | 4 | 1 | no | — |
| 40 | C3×Dic7 | — | 8 | 1 | no | — |
| 41 | #453 | — | 21 | 1 | no | Sylow |
| 42 | C23×C7 | — | 6 | 1 | no | — |
| 43 | C22×D7 | — | 3 | 2 | no | — |
| 44 | C7×Q8 | #67 | 1 | 1 | yes | characteristic |
| 45 | C7×D4 | — | 3 | 1 | no | — |
| 46 | C7×D4 | — | 6 | 1 | no | — |
| 47 | C7×D4 | — | 6 | 1 | no | — |
| 48 | C2×C4×C7 | — | 3 | 1 | no | — |
| 49 | C22⋊D7 | — | 6 | 1 | no | — |
| 50 | C22⋊D7 | — | 12 | 4 | no | — |
| 51 | C2×Dic7 | — | 6 | 1 | no | — |
| 52 | D28 | — | 3 | 1 | no | — |
| 53 | D28 | — | 6 | 1 | no | — |
| 54 | C4×D7 | — | 6 | 1 | no | — |
| 55 | Dic14 | S4 | 1 | 1 | yes | characteristic |
| 56 | C7×C8 | — | 6 | 1 | no | — |
| 57 | C7⋊C8 | — | 6 | 1 | no | — |
| 58 | #41⋊C3 | — | 14 | 1 | no | — |
| 59 | #41⋊C3 | C2×D7 | 1 | 1 | yes | characteristic |
| 60 | GL(2,𝔽3) | — | 2 | 1 | no | — |
| 61 | #215 | — | 28 | 1 | no | — |
| 62 | C2×C3×C7 | — | 4 | 1 | no | — |
| 63 | D3×C7 | — | 16 | 1 | no | — |
| 64 | H(5,𝔽2) | — | 7 | 1 | no | — |
| 65 | AGL(1,ℤ/8) | — | 3 | 1 | no | — |
| 66 | AGL(1,ℤ/8) | — | 21 | 1 | no | — |
| 67 | #127 | — | 21 | 1 | no | — |
| 68 | #104 | — | 21 | 1 | no | — |
| 69 | #104 | — | 21 | 1 | no | — |
| 70 | #100 | — | 21 | 1 | no | — |
| 71 | C22×C7 | — | 3 | 1 | no | — |
| 72 | C22×C7 | — | 12 | 2 | no | — |
| 73 | C22×C7 | — | 12 | 1 | no | — |
| 74 | C2×D7 | — | 6 | 1 | no | — |
| 75 | C2×D7 | — | 12 | 2 | no | — |
| 76 | C4×C7 | — | 3 | 1 | no | — |
| 77 | C4×C7 | C2×S4 | 1 | 1 | yes | characteristic |
| 78 | Dic7 | — | 2 | 1 | no | — |
| 79 | Dic7 | — | 6 | 2 | no | — |
| 80 | C3×Q8 | — | 28 | 1 | no | — |
| 81 | D12 | — | 4 | 1 | no | — |
| 82 | SL(2,𝔽3) | C22⋊D7 | 1 | 1 | yes | characteristic, complemented |
| 83 | C3⋊C8 | — | 28 | 1 | no | — |
| 84 | C3×C7 | — | 4 | 1 | no | Hall |
| 85 | #41 | — | 14 | 1 | no | — |
| 86 | #41 | — | 21 | 1 | no | — |
| 87 | #41 | D3×D7 | 1 | 1 | yes | characteristic |
| 88 | C2×D4 | — | 3 | 1 | no | — |
| 89 | C2×D4 | — | 21 | 1 | no | — |
| 90 | C2×D4 | — | 42 | 1 | no | — |
| 91 | C2×D4 | — | 42 | 2 | no | — |
| 92 | SD16 | — | 6 | 1 | no | — |
| 93 | SD16 | — | 42 | 1 | no | — |
| 94 | D8 | — | 6 | 1 | no | — |
| 95 | D8 | — | 42 | 1 | no | — |
| 96 | M16 | — | 3 | 1 | no | — |
| 97 | M16 | — | 21 | 1 | no | — |
| 98 | C42 | — | 21 | 1 | no | — |
| 99 | C2×C7 | — | 6 | 1 | no | — |
| 100 | C2×C7 | — | 24 | 1 | no | — |
| 101 | C2×C7 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 102 | D7 | — | 12 | 1 | no | — |
| 103 | C2×D3 | — | 8 | 1 | no | — |
| 104 | C3×C4 | — | 4 | 1 | no | — |
| 105 | C3×C4 | — | 56 | 1 | no | — |
| 106 | C23 | — | 6 | 1 | no | — |
| 107 | C23 | — | 21 | 2 | no | — |
| 108 | Q8 | — | 7 | 1 | no | — |
| 109 | Q8 | #4245 | 1 | 1 | yes | characteristic |
| 110 | D4 | — | 3 | 1 | no | — |
| 111 | D4 | — | 6 | 1 | no | — |
| 112 | D4 | — | 6 | 1 | no | — |
| 113 | D4 | — | 21 | 1 | no | — |
| 114 | D4 | — | 42 | 1 | no | — |
| 115 | D4 | — | 42 | 1 | no | — |
| 116 | D4 | — | 84 | 4 | no | — |
| 117 | C2×C4 | — | 3 | 1 | no | — |
| 118 | C2×C4 | — | 42 | 1 | no | — |
| 119 | C2×C4 | — | 42 | 1 | no | — |
| 120 | C8 | — | 6 | 1 | no | — |
| 121 | C8 | — | 42 | 1 | no | — |
| 122 | C7 | #5429 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 123 | C2×C3 | — | 4 | 1 | no | — |
| 124 | D3 | — | 16 | 1 | no | — |
| 125 | C22 | — | 3 | 1 | no | — |
| 126 | C22 | — | 12 | 2 | no | — |
| 127 | C22 | — | 12 | 1 | no | — |
| 128 | C22 | — | 42 | 1 | no | — |
| 129 | C22 | — | 84 | 2 | no | — |
| 130 | C4 | — | 3 | 1 | no | — |
| 131 | C4 | — | 14 | 1 | no | — |
| 132 | C4 | — | 42 | 2 | no | — |
| 133 | C4 | D7×S4 | 1 | 1 | yes | characteristic |
| 134 | C3 | — | 4 | 1 | no | Sylow |
| 135 | C2 | — | 6 | 1 | no | — |
| 136 | C2 | — | 24 | 1 | no | — |
| 137 | C2 | — | 84 | 1 | no | — |
| 138 | C2 | #10621482 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10621482 |
| commutator, cosocle | C7×#41⋊C3 | C22 |
| Frattini, hypercenter | C4 | D7×S4 |
| Fitting | C7×#41 | C2×D3 |
| socle | C2×C7 | GL(2,ℤ/4) |
| hypocenter | C7×SL(2,𝔽3) | D4 |
| LCS | G ▷ C7×#41⋊C3 ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10621528 | C2 | semidirect |
| 2. C7×#986 | C2 | semidirect |
| 3. C7⋊GU(2,𝔽3) | C2 | semidirect |
| 9. #6553 | D3 | semidirect |
| 19. C7×SL(2,𝔽3) | D4 | semidirect |
| 36. #986 | D7 | semidirect |
| 82. SL(2,𝔽3) | C22⋊D7 | semidirect |
| 122. C7 | #5429 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10621528 | 2 |
| C7×#986 | 2 |
| C7⋊GU(2,𝔽3) | 2 |
| #89970 | 3 |
| #66564 | 4 |
| #5429 | 7 |