Finite Group DatabasebetaFinite Group DatabasebetaOrder 2176.
122 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 20 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 | #11763713 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #11763667 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11762979 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #10587691 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | #10587626 | C22 | 1 | 1 | yes | characteristic |
| 6 | #10587594 | — | 2 | 1 | no | — |
| 7 | #10587544 | — | 2 | 1 | no | — |
| 8 | #10587539 | — | 2 | 1 | no | — |
| 9 | C17×#100 | C4 | 1 | 1 | yes | characteristic, complemented |
| 10 | #10587496 | — | 2 | 1 | no | — |
| 11 | C2×D4×C17 | C2×C4 | 1 | 1 | yes | characteristic |
| 12 | C2×C22⋊D17 | — | 4 | 1 | no | — |
| 13 | C2×C22⋊D17 | — | 4 | 1 | no | — |
| 14 | #63254 | — | 4 | 1 | no | — |
| 15 | D4×D17 | — | 4 | 1 | no | — |
| 16 | #63252 | — | 2 | 1 | no | — |
| 17 | C2×D68 | D4 | 1 | 1 | yes | characteristic |
| 18 | C22⋊#595 | — | 4 | 1 | no | — |
| 19 | C2×#3624 | — | 4 | 1 | no | — |
| 20 | M16×C17 | — | 2 | 1 | no | — |
| 21 | C4×Dic17 | D4 | 1 | 1 | yes | characteristic |
| 22 | D136 | — | 4 | 1 | no | — |
| 23 | #63219 | — | 4 | 1 | no | — |
| 24 | C23×C17 | — | 2 | 1 | no | — |
| 25 | C22×D17 | — | 2 | 1 | no | — |
| 26 | C22×D17 | — | 4 | 1 | no | — |
| 27 | C2×#595 | — | 8 | 1 | no | — |
| 28 | D4×C17 | — | 4 | 1 | no | — |
| 29 | C2×C4×C17 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 30 | C22⋊D17 | — | 4 | 2 | no | — |
| 31 | C22⋊D17 | — | 4 | 1 | no | — |
| 32 | C22⋊D17 | — | 8 | 1 | no | — |
| 33 | C22⋊D17 | — | 8 | 1 | no | — |
| 34 | C2×Dic17 | — | 2 | 1 | no | — |
| 35 | C2×Dic17 | — | 4 | 1 | no | — |
| 36 | D68 | — | 2 | 1 | no | — |
| 37 | D68 | — | 4 | 1 | no | — |
| 38 | C4×D17 | — | 4 | 1 | no | — |
| 39 | Dic34 | — | 2 | 1 | no | — |
| 40 | #3624 | — | 8 | 1 | no | — |
| 41 | C8×C17 | — | 4 | 1 | no | — |
| 42 | #1402 | — | 17 | 1 | no | maximal, Sylow |
| 43 | C22×C17 | — | 4 | 1 | no | — |
| 44 | C22×C17 | — | 8 | 1 | no | — |
| 45 | C22×C17 | #99 | 1 | 1 | yes | characteristic |
| 46 | C2×D17 | — | 4 | 1 | no | — |
| 47 | C2×D17 | — | 4 | 1 | no | — |
| 48 | C2×D17 | — | 8 | 1 | no | — |
| 49 | C2×D17 | — | 16 | 1 | no | — |
| 50 | #595 | — | 16 | 1 | no | — |
| 51 | C4×C17 | — | 2 | 1 | no | — |
| 52 | Dic17 | — | 2 | 1 | no | — |
| 53 | Dic17 | — | 2 | 1 | no | — |
| 54 | Dic17 | — | 4 | 1 | no | — |
| 55 | #453 | — | 17 | 1 | no | — |
| 56 | #353 | — | 17 | 1 | no | — |
| 57 | #331 | — | 17 | 1 | no | — |
| 58 | C2×C17 | — | 2 | 1 | no | — |
| 59 | C2×C17 | — | 8 | 1 | no | — |
| 60 | C2×C17 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 61 | D17 | — | 8 | 1 | no | — |
| 62 | D17 | — | 8 | 1 | no | — |
| 63 | H(5,𝔽2) | — | 17 | 1 | no | — |
| 64 | AGL(1,ℤ/8) | — | 34 | 1 | no | — |
| 65 | #127 | — | 17 | 1 | no | — |
| 66 | C4⋊C8 | — | 34 | 1 | no | — |
| 67 | #104 | — | 34 | 1 | no | — |
| 68 | #100 | #595 | 1 | 1 | yes | characteristic, complemented |
| 69 | #99 | — | 34 | 1 | no | — |
| 70 | C17 | #1402 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 71 | #41 | — | 34 | 1 | no | — |
| 72 | #41 | — | 68 | 1 | no | — |
| 73 | C2×D4 | — | 17 | 1 | no | — |
| 74 | C2×D4 | — | 68 | 1 | no | — |
| 75 | C2×D4 | — | 68 | 1 | no | — |
| 76 | C2×D4 | — | 68 | 1 | no | — |
| 77 | C2×D4 | C2×#595 | 1 | 1 | yes | characteristic |
| 78 | SD16 | — | 68 | 1 | no | — |
| 79 | D8 | — | 68 | 1 | no | — |
| 80 | M16 | — | 2 | 1 | no | — |
| 81 | C2×C8 | — | 68 | 1 | no | — |
| 82 | C22⋊C4 | — | 68 | 1 | no | — |
| 83 | C42 | — | 17 | 1 | no | — |
| 84 | C23 | — | 2 | 1 | no | — |
| 85 | C23 | — | 34 | 1 | no | — |
| 86 | C23 | — | 68 | 1 | no | — |
| 87 | Q8 | — | 34 | 1 | no | — |
| 88 | D4 | — | 4 | 1 | no | — |
| 89 | D4 | — | 34 | 1 | no | — |
| 90 | D4 | — | 68 | 1 | no | — |
| 91 | D4 | — | 68 | 2 | no | — |
| 92 | D4 | — | 68 | 1 | no | — |
| 93 | D4 | — | 136 | 1 | no | — |
| 94 | D4 | — | 136 | 1 | no | — |
| 95 | C2×C4 | — | 34 | 1 | no | — |
| 96 | C2×C4 | — | 68 | 1 | no | — |
| 97 | C2×C4 | — | 68 | 1 | no | — |
| 98 | C2×C4 | — | 136 | 1 | no | — |
| 99 | C2×C4 | C22⋊#595 | 1 | 1 | yes | characteristic |
| 100 | C8 | — | 4 | 1 | no | — |
| 101 | C8 | — | 136 | 1 | no | — |
| 102 | C22 | — | 4 | 1 | no | — |
| 103 | C22 | — | 8 | 1 | no | — |
| 104 | C22 | — | 272 | 1 | no | — |
| 105 | C22 | — | 68 | 1 | no | — |
| 106 | C22 | — | 68 | 1 | no | — |
| 107 | C22 | — | 136 | 1 | no | — |
| 108 | C22 | #10587539 | 1 | 1 | yes | characteristic |
| 109 | C4 | — | 2 | 1 | no | — |
| 110 | C4 | — | 272 | 1 | no | — |
| 111 | C4 | — | 34 | 1 | no | — |
| 112 | C4 | — | 34 | 1 | no | — |
| 113 | C4 | — | 68 | 1 | no | — |
| 114 | C2 | — | 2 | 1 | no | — |
| 115 | C2 | — | 8 | 1 | no | — |
| 116 | C2 | — | 136 | 1 | no | — |
| 117 | C2 | — | 136 | 1 | no | — |
| 118 | C2 | #11763710 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #11763713 | C2 | semidirect |
| 2. #11763667 | C2 | semidirect |
| 4. #10587691 | C4 | semidirect |
| 9. C17×#100 | C4 | semidirect |
| 68. #100 | #595 | semidirect |
| 70. C17 | #1402 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #11763713 | 2 |
| #11763667 | 2 |
| #11762979 | 2 |
| #1402 | 17 |