Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
300 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 25 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 | C2×C7×C22⋊A4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #90019 | — | 3 | 1 | no | maximal, Hall |
| 3 | C7×C22⋊A4 | C4 | 1 | 1 | yes | characteristic, complemented |
| 4 | C7⋊SL(2,ℤ/4) | — | 4 | 3 | no | maximal |
| 5 | C25×C7 | D3 | 1 | 1 | yes | characteristic |
| 6 | C2×C22⋊Dic7 | — | 3 | 3 | no | — |
| 7 | C2×C22⋊Dic7 | — | 3 | 3 | no | — |
| 8 | #5936 | — | 7 | 1 | no | maximal, Hall |
| 9 | C2×C7×A4 | — | 4 | 3 | no | — |
| 10 | C2×C7×A4 | — | 8 | 1 | no | — |
| 11 | C24×C7 | — | 3 | 3 | no | — |
| 12 | C24×C7 | — | 3 | 3 | no | — |
| 13 | C24×C7 | — | 6 | 1 | no | — |
| 14 | C24×C7 | — | 6 | 1 | no | — |
| 15 | C24×C7 | Dic3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C22×Dic7 | — | 3 | 3 | no | — |
| 17 | C22×Dic7 | — | 3 | 1 | no | — |
| 18 | C22⋊Dic7 | — | 6 | 3 | no | — |
| 19 | C22⋊Dic7 | — | 6 | 3 | no | — |
| 20 | C22⋊Dic7 | — | 6 | 3 | no | — |
| 21 | C22⋊Dic7 | — | 6 | 3 | no | — |
| 22 | C2×C22⋊A4 | D7 | 1 | 1 | yes | characteristic |
| 23 | C7×A4 | — | 4 | 3 | no | — |
| 24 | C7×A4 | — | 8 | 1 | no | — |
| 25 | Dic21 | — | 16 | 1 | no | — |
| 26 | #379 | — | 21 | 1 | no | Sylow |
| 27 | C23×C7 | — | 2 | 1 | no | — |
| 28 | C23×C7 | — | 3 | 1 | no | — |
| 29 | C23×C7 | — | 3 | 3 | no | — |
| 30 | C23×C7 | — | 3 | 3 | no | — |
| 31 | C23×C7 | — | 3 | 3 | no | — |
| 32 | C23×C7 | — | 3 | 3 | no | — |
| 33 | C23×C7 | — | 3 | 3 | no | — |
| 34 | C23×C7 | — | 6 | 1 | no | — |
| 35 | C23×C7 | — | 6 | 1 | no | — |
| 36 | C23×C7 | — | 6 | 6 | no | — |
| 37 | C23×C7 | — | 6 | 3 | no | — |
| 38 | C23×C7 | — | 6 | 3 | no | — |
| 39 | C23×C7 | — | 6 | 3 | no | — |
| 40 | C23×C7 | S4 | 1 | 3 | yes | — |
| 41 | C2×Dic7 | — | 6 | 6 | no | — |
| 42 | C2×Dic7 | — | 6 | 3 | no | — |
| 43 | C2×Dic7 | — | 6 | 3 | no | — |
| 44 | C22⋊A4 | Dic7 | 1 | 1 | yes | characteristic, complemented |
| 45 | SL(2,ℤ/4) | — | 28 | 3 | no | — |
| 46 | C2×C3×C7 | — | 16 | 1 | no | — |
| 47 | C25 | D21 | 1 | 1 | yes | characteristic |
| 48 | C2×C22⋊C4 | — | 21 | 3 | no | — |
| 49 | C2×C22⋊C4 | — | 21 | 3 | no | — |
| 50 | C22×C7 | — | 2 | 1 | no | — |
| 51 | C22×C7 | — | 3 | 3 | no | — |
| 52 | C22×C7 | — | 3 | 3 | no | — |
| 53 | C22×C7 | — | 3 | 1 | no | — |
| 54 | C22×C7 | — | 3 | 3 | no | — |
| 55 | C22×C7 | — | 3 | 3 | no | — |
| 56 | C22×C7 | — | 3 | 3 | no | — |
| 57 | C22×C7 | — | 6 | 3 | no | — |
| 58 | C22×C7 | — | 6 | 6 | no | — |
| 59 | C22×C7 | — | 6 | 3 | no | — |
| 60 | C22×C7 | — | 6 | 3 | no | — |
| 61 | C22×C7 | — | 6 | 1 | no | — |
| 62 | C22×C7 | — | 6 | 1 | no | — |
| 63 | C22×C7 | SL(2,ℤ/4) | 1 | 3 | yes | complemented |
| 64 | Dic7 | — | 12 | 3 | no | — |
| 65 | Dic7 | — | 12 | 1 | no | — |
| 66 | C2×A4 | — | 4 | 3 | no | — |
| 67 | C2×A4 | — | 8 | 1 | no | — |
| 68 | C3×C7 | — | 16 | 1 | no | Hall |
| 69 | C24 | — | 3 | 3 | no | — |
| 70 | C24 | — | 3 | 3 | no | — |
| 71 | C24 | — | 6 | 1 | no | — |
| 72 | C24 | — | 6 | 1 | no | — |
| 73 | C24 | Dic21 | 1 | 1 | yes | characteristic, complemented |
| 74 | C22×C4 | — | 21 | 3 | no | — |
| 75 | C22×C4 | — | 21 | 1 | no | — |
| 76 | C22⋊C4 | — | 42 | 3 | no | — |
| 77 | C22⋊C4 | — | 42 | 3 | no | — |
| 78 | C22⋊C4 | — | 42 | 3 | no | — |
| 79 | C22⋊C4 | — | 42 | 3 | no | — |
| 80 | C2×C7 | — | 3 | 3 | no | — |
| 81 | C2×C7 | — | 3 | 3 | no | — |
| 82 | C2×C7 | — | 6 | 1 | no | — |
| 83 | C2×C7 | — | 6 | 1 | no | — |
| 84 | C2×C7 | #1020 | 1 | 1 | yes | characteristic |
| 85 | A4 | — | 4 | 3 | no | — |
| 86 | A4 | — | 8 | 1 | no | — |
| 87 | Dic3 | — | 112 | 1 | no | — |
| 88 | C23 | — | 2 | 1 | no | — |
| 89 | C23 | — | 3 | 1 | no | — |
| 90 | C23 | — | 3 | 3 | no | — |
| 91 | C23 | — | 3 | 3 | no | — |
| 92 | C23 | — | 3 | 3 | no | — |
| 93 | C23 | — | 3 | 3 | no | — |
| 94 | C23 | — | 3 | 3 | no | — |
| 95 | C23 | — | 6 | 1 | no | — |
| 96 | C23 | — | 6 | 1 | no | — |
| 97 | C23 | — | 6 | 6 | no | — |
| 98 | C23 | — | 6 | 3 | no | — |
| 99 | C23 | — | 6 | 3 | no | — |
| 100 | C23 | — | 6 | 3 | no | — |
| 101 | C23 | C7⋊S4 | 1 | 3 | yes | — |
| 102 | C2×C4 | — | 42 | 6 | no | — |
| 103 | C2×C4 | — | 42 | 3 | no | — |
| 104 | C2×C4 | — | 42 | 3 | no | — |
| 105 | C7 | #5936 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 106 | C2×C3 | — | 16 | 1 | no | — |
| 107 | C22 | — | 2 | 1 | no | — |
| 108 | C22 | — | 3 | 3 | no | — |
| 109 | C22 | — | 3 | 3 | no | — |
| 110 | C22 | — | 3 | 1 | no | — |
| 111 | C22 | — | 3 | 3 | no | — |
| 112 | C22 | — | 3 | 3 | no | — |
| 113 | C22 | — | 3 | 3 | no | — |
| 114 | C22 | — | 6 | 3 | no | — |
| 115 | C22 | — | 6 | 6 | no | — |
| 116 | C22 | — | 6 | 3 | no | — |
| 117 | C22 | — | 6 | 3 | no | — |
| 118 | C22 | — | 6 | 1 | no | — |
| 119 | C22 | — | 6 | 1 | no | — |
| 120 | C22 | C7⋊SL(2,ℤ/4) | 1 | 3 | yes | complemented |
| 121 | C4 | — | 84 | 3 | no | — |
| 122 | C4 | — | 84 | 1 | no | — |
| 123 | C3 | — | 16 | 1 | no | Sylow |
| 124 | C2 | — | 3 | 3 | no | — |
| 125 | C2 | — | 3 | 3 | no | — |
| 126 | C2 | — | 6 | 1 | no | — |
| 127 | C2 | — | 6 | 1 | no | — |
| 128 | C2 | C7⋊#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C7⋊#1020 |
| commutator, hypocenter | C7×C22⋊A4 | C4 |
| Fitting, socle | C25×C7 | D3 |
| prime radical, cosocle | C2×C7×C22⋊A4 | C2 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C7×C22⋊A4 | C4 | semidirect |
| 15. C24×C7 | Dic3 | semidirect |
| 44. C22⋊A4 | Dic7 | semidirect |
| 63. C22×C7 | SL(2,ℤ/4) | semidirect |
| 73. C24 | Dic21 | semidirect |
| 105. C7 | #5936 | semidirect |
| 120. C22 | C7⋊SL(2,ℤ/4) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C7×C22⋊A4 | 2 |
| #90019 | 3 |
| C7⋊SL(2,ℤ/4) | 4 |
| #5936 | 7 |