Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
138 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 138 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 | #10621530 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C7×#984 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C7⋊GU(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #89977 | — | 3 | 1 | no | maximal, Hall |
| 5 | C7×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 6 | D7×SL(2,𝔽3) | — | 2 | 1 | no | — |
| 7 | C7×#226 | — | 2 | 1 | no | — |
| 8 | #66563 | — | 4 | 1 | no | maximal |
| 9 | #6554 | D3 | 1 | 1 | yes | characteristic |
| 10 | C7×#137 | — | 3 | 1 | no | — |
| 11 | #6507 | — | 3 | 1 | no | — |
| 12 | #6505 | — | 3 | 1 | no | — |
| 13 | #6411 | — | 3 | 1 | no | — |
| 14 | #6399 | — | 3 | 1 | no | — |
| 15 | #6398 | — | 3 | 1 | no | — |
| 16 | #5430 | — | 7 | 1 | no | maximal, Hall |
| 17 | C7×Dic6 | — | 4 | 1 | no | — |
| 18 | C3×D28 | — | 4 | 1 | no | — |
| 19 | C7×SL(2,𝔽3) | D4 | 1 | 1 | yes | characteristic |
| 20 | C3⋊(C7⋊C8) | — | 4 | 1 | no | — |
| 21 | C7×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 22 | C2×C7×Q8 | — | 3 | 1 | no | — |
| 23 | Q8×D7 | — | 2 | 1 | no | — |
| 24 | #1159 | — | 6 | 1 | no | — |
| 25 | #1157 | — | 3 | 1 | no | — |
| 26 | C2×Dic14 | — | 3 | 1 | no | — |
| 27 | C7×Q16 | — | 6 | 1 | no | — |
| 28 | C7×SD16 | — | 6 | 1 | no | — |
| 29 | C7×M16 | — | 3 | 1 | no | — |
| 30 | #1144 | — | 6 | 1 | no | — |
| 31 | #1143 | — | 6 | 1 | no | — |
| 32 | Dic7⋊C4 | — | 6 | 2 | no | — |
| 33 | C4×Dic7 | — | 3 | 1 | no | — |
| 34 | #1136 | — | 3 | 1 | no | — |
| 35 | 2−1+4⋊C3 | — | 7 | 1 | no | — |
| 36 | #984 | D7 | 1 | 1 | yes | characteristic, complemented |
| 37 | GU(2,𝔽3) | — | 7 | 1 | no | — |
| 38 | C2×C3×D7 | — | 8 | 1 | no | — |
| 39 | C3×C4×C7 | — | 4 | 1 | no | — |
| 40 | C7×Dic3 | — | 8 | 1 | no | — |
| 41 | #456 | — | 21 | 1 | no | Sylow |
| 42 | C7×Q8 | — | 6 | 1 | no | — |
| 43 | C7×Q8 | — | 6 | 1 | no | — |
| 44 | C7×Q8 | #67 | 1 | 1 | yes | characteristic |
| 45 | C7×D4 | — | 3 | 1 | no | — |
| 46 | C2×C4×C7 | — | 3 | 1 | no | — |
| 47 | C2×C4×C7 | — | 6 | 1 | no | — |
| 48 | C22⋊D7 | — | 6 | 1 | no | — |
| 49 | C2×Dic7 | — | 3 | 2 | no | — |
| 50 | C2×Dic7 | — | 6 | 1 | no | — |
| 51 | D28 | S4 | 1 | 1 | yes | characteristic |
| 52 | C4×D7 | — | 6 | 1 | no | — |
| 53 | Dic14 | — | 3 | 1 | no | — |
| 54 | Dic14 | — | 6 | 1 | no | — |
| 55 | C7×C8 | — | 6 | 1 | no | — |
| 56 | C7⋊C8 | — | 6 | 1 | no | — |
| 57 | #41⋊C3 | C2×D7 | 1 | 1 | yes | characteristic |
| 58 | C2×SL(2,𝔽3) | — | 14 | 1 | no | — |
| 59 | #226 | — | 2 | 1 | no | — |
| 60 | #214 | — | 28 | 1 | no | — |
| 61 | C2×C3×C7 | — | 4 | 1 | no | — |
| 62 | C3×D7 | — | 16 | 1 | no | — |
| 63 | 2−1+4 | — | 7 | 1 | no | — |
| 64 | #137 | — | 3 | 1 | no | — |
| 65 | #137 | — | 21 | 1 | no | — |
| 66 | C4⋊Q8 | — | 21 | 1 | no | — |
| 67 | #104 | — | 21 | 1 | no | — |
| 68 | #104 | — | 21 | 1 | no | — |
| 69 | #101 | — | 21 | 1 | no | — |
| 70 | C22×C7 | — | 3 | 1 | no | — |
| 71 | C2×D7 | — | 2 | 1 | no | — |
| 72 | C4×C7 | — | 3 | 1 | no | — |
| 73 | C4×C7 | — | 12 | 1 | no | — |
| 74 | C4×C7 | C2×S4 | 1 | 1 | yes | characteristic |
| 75 | Dic7 | — | 6 | 2 | no | — |
| 76 | Dic7 | — | 6 | 1 | no | — |
| 77 | C3×D4 | — | 28 | 1 | no | — |
| 78 | Dic6 | — | 4 | 1 | no | — |
| 79 | SL(2,𝔽3) | C22⋊D7 | 1 | 1 | yes | characteristic |
| 80 | C3⋊C8 | — | 28 | 1 | no | — |
| 81 | C3×C7 | — | 4 | 1 | no | Hall |
| 82 | #41 | — | 21 | 1 | no | — |
| 83 | #41 | — | 42 | 1 | no | — |
| 84 | #41 | D3×D7 | 1 | 1 | yes | characteristic |
| 85 | C2×Q8 | — | 3 | 1 | no | — |
| 86 | C2×Q8 | — | 14 | 1 | no | — |
| 87 | C2×Q8 | — | 21 | 1 | no | — |
| 88 | Q16 | — | 6 | 1 | no | — |
| 89 | Q16 | — | 42 | 1 | no | — |
| 90 | SD16 | — | 6 | 1 | no | — |
| 91 | SD16 | — | 42 | 1 | no | — |
| 92 | M16 | — | 3 | 1 | no | — |
| 93 | M16 | — | 21 | 1 | no | — |
| 94 | C4⋊C4 | — | 42 | 2 | no | — |
| 95 | C42 | — | 21 | 1 | no | — |
| 96 | C2×C7 | — | 6 | 1 | no | — |
| 97 | C2×C7 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 98 | D7 | — | 4 | 1 | no | — |
| 99 | C22×C3 | — | 56 | 1 | no | — |
| 100 | C3×C4 | — | 4 | 1 | no | — |
| 101 | Dic3 | — | 8 | 1 | no | — |
| 102 | Q8 | — | 6 | 1 | no | — |
| 103 | Q8 | — | 6 | 1 | no | — |
| 104 | Q8 | — | 21 | 1 | no | — |
| 105 | Q8 | — | 42 | 1 | no | — |
| 106 | Q8 | #4245 | 1 | 1 | yes | characteristic |
| 107 | D4 | — | 3 | 1 | no | — |
| 108 | D4 | — | 7 | 1 | no | — |
| 109 | D4 | — | 42 | 1 | no | — |
| 110 | C2×C4 | — | 3 | 1 | no | — |
| 111 | C2×C4 | — | 6 | 1 | no | — |
| 112 | C2×C4 | — | 21 | 2 | no | — |
| 113 | C2×C4 | — | 42 | 1 | no | — |
| 114 | C2×C4 | — | 42 | 1 | no | — |
| 115 | C8 | — | 6 | 1 | no | — |
| 116 | C8 | — | 42 | 1 | no | — |
| 117 | C7 | #5430 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 118 | C2×C3 | — | 4 | 1 | no | — |
| 119 | C2×C3 | — | 112 | 1 | no | — |
| 120 | C22 | — | 3 | 1 | no | — |
| 121 | C22 | — | 14 | 1 | no | — |
| 122 | C4 | — | 3 | 1 | no | — |
| 123 | C4 | — | 12 | 1 | no | — |
| 124 | C4 | — | 42 | 2 | no | — |
| 125 | C4 | — | 42 | 1 | no | — |
| 126 | C4 | D7×S4 | 1 | 1 | yes | characteristic |
| 127 | C3 | — | 4 | 1 | no | Sylow |
| 128 | C2 | — | 6 | 1 | no | — |
| 129 | C2 | — | 28 | 1 | no | — |
| 130 | C2 | #10621482 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×#984 | C2 | semidirect |
| 3. C7⋊GU(2,𝔽3) | C2 | semidirect |
| 36. #984 | D7 | semidirect |
| 117. C7 | #5430 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10621530 | 2 |
| C7×#984 | 2 |
| C7⋊GU(2,𝔽3) | 2 |
| #89977 | 3 |
| #66563 | 4 |
| #5430 | 7 |