Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 79 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 | #11728256 | C2 | 1 | 3 | yes | maximal |
| 2 | #11728249 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C7×#1309 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×#6437 | C22 | 1 | 1 | yes | characteristic |
| 5 | C7×#423 | C22 | 1 | 3 | yes | — |
| 6 | #89697 | C22 | 1 | 3 | yes | — |
| 7 | #89280 | — | 2 | 3 | no | — |
| 8 | #89280 | — | 2 | 3 | no | — |
| 9 | #89275 | — | 2 | 3 | no | — |
| 10 | #89274 | — | 2 | 3 | no | — |
| 11 | #10611 | — | 7 | 1 | no | maximal, Sylow |
| 12 | C22×D28 | — | 2 | 3 | no | — |
| 13 | C2×C7×M16 | — | 2 | 3 | no | — |
| 14 | C2×C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 15 | #6437 | — | 2 | 3 | no | — |
| 16 | #6437 | Q8 | 1 | 2 | yes | — |
| 17 | C7×C4⋊C8 | — | 2 | 3 | no | — |
| 18 | C7×C4⋊C8 | — | 2 | 3 | no | — |
| 19 | C22⋊(C7⋊C8) | — | 2 | 3 | no | — |
| 20 | C4⋊(C7⋊C8) | D4 | 1 | 3 | yes | — |
| 21 | C4⋊(C7⋊C8) | D4 | 1 | 3 | yes | — |
| 22 | C4×C7⋊C8 | — | 2 | 3 | no | — |
| 23 | #1574 | — | 7 | 3 | no | — |
| 24 | #1545 | — | 7 | 3 | no | — |
| 25 | #1309 | D7 | 1 | 1 | yes | characteristic, complemented |
| 26 | C23×D7 | — | 4 | 1 | no | — |
| 27 | C22×C4×C7 | #41 | 1 | 3 | yes | — |
| 28 | C2×D28 | — | 2 | 6 | no | — |
| 29 | C2×D28 | — | 4 | 3 | no | — |
| 30 | C2×D28 | — | 8 | 3 | no | — |
| 31 | C2×D28 | — | 8 | 3 | no | — |
| 32 | C7×M16 | — | 8 | 3 | no | — |
| 33 | C2×C7×C8 | — | 4 | 3 | no | — |
| 34 | C42×C7 | C2×D4 | 1 | 3 | yes | — |
| 35 | C42×C7 | C2×Q8 | 1 | 1 | yes | characteristic |
| 36 | C2×C7⋊C8 | — | 2 | 3 | no | — |
| 37 | C2×C7⋊C8 | — | 2 | 3 | no | — |
| 38 | C2×#127 | — | 7 | 1 | no | — |
| 39 | #431 | — | 7 | 3 | no | — |
| 40 | #423 | C2×D7 | 1 | 3 | yes | — |
| 41 | #335 | — | 14 | 3 | no | — |
| 42 | #334 | — | 14 | 3 | no | — |
| 43 | #331 | — | 14 | 3 | no | — |
| 44 | #331 | — | 14 | 3 | no | — |
| 45 | C23×C7 | #126 | 1 | 1 | yes | characteristic |
| 46 | C22×D7 | — | 4 | 2 | no | — |
| 47 | C22×D7 | — | 8 | 6 | no | — |
| 48 | C2×C4×C7 | — | 2 | 3 | no | — |
| 49 | C2×C4×C7 | — | 2 | 3 | no | — |
| 50 | C2×C4×C7 | #121 | 1 | 3 | yes | — |
| 51 | C2×C4×C7 | C22⋊Q8 | 1 | 3 | yes | — |
| 52 | D28 | — | 8 | 6 | no | — |
| 53 | D28 | — | 8 | 3 | no | — |
| 54 | D28 | — | 8 | 3 | no | — |
| 55 | C7×C8 | — | 8 | 3 | no | — |
| 56 | C7⋊C8 | — | 4 | 3 | no | — |
| 57 | C7⋊C8 | — | 4 | 3 | no | — |
| 58 | C22×D4 | — | 14 | 3 | no | — |
| 59 | C2×M16 | — | 2 | 3 | no | — |
| 60 | #127 | — | 7 | 2 | no | — |
| 61 | #127 | — | 14 | 3 | no | — |
| 62 | C2×C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 63 | C4⋊C8 | — | 2 | 3 | no | — |
| 64 | C4⋊C8 | — | 2 | 3 | no | — |
| 65 | C4⋊C8 | — | 7 | 3 | no | — |
| 66 | C4⋊C8 | — | 7 | 3 | no | — |
| 67 | C22⋊C8 | — | 14 | 3 | no | — |
| 68 | C4×C8 | — | 14 | 3 | no | — |
| 69 | C22×C7 | — | 2 | 3 | no | — |
| 70 | C22×C7 | #396 | 1 | 1 | yes | characteristic |
| 71 | C2×D7 | — | 8 | 6 | no | — |
| 72 | C2×D7 | — | 16 | 4 | no | — |
| 73 | C4×C7 | — | 2 | 3 | no | — |
| 74 | C4×C7 | #471 | 1 | 3 | yes | — |
| 75 | C4×C7 | #472 | 1 | 3 | yes | — |
| 76 | C24 | — | 28 | 1 | no | — |
| 77 | C2×D4 | — | 14 | 6 | no | — |
| 78 | C2×D4 | — | 28 | 3 | no | — |
| 79 | C2×D4 | — | 56 | 3 | no | — |
| 80 | C2×D4 | — | 56 | 3 | no | — |
| 81 | C22×C4 | #1157 | 1 | 3 | yes | — |
| 82 | M16 | — | 8 | 3 | no | — |
| 83 | C2×C8 | — | 4 | 3 | no | — |
| 84 | C2×C8 | — | 14 | 3 | no | — |
| 85 | C2×C8 | — | 14 | 3 | no | — |
| 86 | C42 | D4×D7 | 1 | 3 | yes | — |
| 87 | C42 | Q8×D7 | 1 | 1 | yes | characteristic |
| 88 | C2×C7 | — | 4 | 1 | no | — |
| 89 | C2×C7 | #2063 | 1 | 3 | yes | — |
| 90 | D7 | — | 16 | 2 | no | — |
| 91 | C23 | — | 28 | 2 | no | — |
| 92 | C23 | — | 56 | 6 | no | — |
| 93 | C23 | #6439 | 1 | 1 | yes | characteristic |
| 94 | D4 | — | 56 | 6 | no | — |
| 95 | D4 | — | 56 | 3 | no | — |
| 96 | D4 | — | 56 | 3 | no | — |
| 97 | C2×C4 | — | 2 | 3 | no | — |
| 98 | C2×C4 | — | 2 | 3 | no | — |
| 99 | C2×C4 | #6447 | 1 | 3 | yes | — |
| 100 | C2×C4 | #6459 | 1 | 3 | yes | — |
| 101 | C8 | — | 8 | 3 | no | — |
| 102 | C8 | — | 28 | 3 | no | — |
| 103 | C8 | — | 28 | 3 | no | — |
| 104 | C7 | #10611 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 105 | C22 | — | 2 | 3 | no | — |
| 106 | C22 | — | 56 | 6 | no | — |
| 107 | C22 | — | 112 | 4 | no | — |
| 108 | C22 | #89448 | 1 | 1 | yes | characteristic |
| 109 | C4 | — | 2 | 3 | no | — |
| 110 | C4 | #89521 | 1 | 3 | yes | — |
| 111 | C4 | #89526 | 1 | 3 | yes | — |
| 112 | C2 | — | 4 | 1 | no | — |
| 113 | C2 | — | 112 | 2 | no | — |
| 114 | C2 | #11726750 | 1 | 3 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11728249 | C2 | semidirect |
| 3. C7×#1309 | C2 | semidirect |
| 25. #1309 | D7 | semidirect |
| 104. C7 | #10611 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11728256 | 2 |
| #11728249 | 2 |
| C7×#1309 | 2 |
| #10611 | 7 |