Finite Group DatabasebetaFinite Group DatabasebetaOrder 896.
444 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 444 conjugacy classes of subgroups — too many to draw. Only its 117 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 | #90457 | C2 | 1 | 4 | yes | maximal, complemented |
| 2 | C7×#494 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #89964 | C2 | 1 | 4 | yes | maximal, complemented |
| 4 | #89867 | C2 | 1 | 4 | yes | maximal, complemented |
| 5 | #89472 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #6551 | C22 | 1 | 4 | yes | — |
| 7 | C2×C7×Q16 | C22 | 1 | 4 | yes | — |
| 8 | C7×C4⋊Q8 | C22 | 1 | 2 | yes | — |
| 9 | #6505 | C22 | 1 | 8 | yes | — |
| 10 | #6482 | — | 2 | 4 | no | — |
| 11 | #6481 | — | 2 | 8 | no | — |
| 12 | D7×Q16 | — | 2 | 4 | no | — |
| 13 | #6467 | C22 | 1 | 2 | yes | — |
| 14 | #6464 | C22 | 1 | 4 | yes | — |
| 15 | C4×C7×C8 | C22 | 1 | 1 | yes | characteristic |
| 16 | #6410 | C22 | 1 | 4 | yes | — |
| 17 | #6393 | C22 | 1 | 2 | yes | — |
| 18 | #6379 | C22 | 1 | 4 | yes | — |
| 19 | #3293 | — | 7 | 1 | no | maximal, Sylow |
| 20 | C2×C7×Q8 | C23 | 1 | 4 | yes | — |
| 21 | #1161 | — | 2 | 8 | no | — |
| 22 | Q8×D7 | — | 2 | 8 | no | — |
| 23 | #1157 | — | 4 | 4 | no | — |
| 24 | #1157 | C23 | 1 | 4 | yes | — |
| 25 | C7×Q16 | — | 2 | 4 | no | — |
| 26 | C7×Q16 | — | 2 | 4 | no | — |
| 27 | C2×C7×C8 | C23 | 1 | 2 | yes | — |
| 28 | C7×C4⋊C4 | — | 4 | 2 | no | — |
| 29 | C42×C7 | C23 | 1 | 1 | yes | characteristic |
| 30 | #1144 | — | 2 | 8 | no | — |
| 31 | #1143 | — | 2 | 8 | no | — |
| 32 | #1136 | C23 | 1 | 4 | yes | — |
| 33 | Dic28 | D4 | 1 | 2 | yes | — |
| 34 | D56 | D4 | 1 | 2 | yes | — |
| 35 | #1132 | D4 | 1 | 4 | yes | — |
| 36 | #1131 | — | 2 | 4 | no | — |
| 37 | C8×D7 | — | 2 | 4 | no | — |
| 38 | #578 | — | 7 | 4 | no | — |
| 39 | #494 | D7 | 1 | 1 | yes | characteristic, complemented |
| 40 | #473 | — | 7 | 4 | no | — |
| 41 | #456 | — | 7 | 4 | no | — |
| 42 | #443 | — | 7 | 2 | no | — |
| 43 | C7×Q8 | — | 2 | 8 | no | — |
| 44 | C2×C4×C7 | — | 2 | 1 | no | — |
| 45 | C2×C4×C7 | — | 2 | 4 | no | — |
| 46 | C2×C4×C7 | C24 | 1 | 1 | yes | characteristic |
| 47 | C22⋊D7 | — | 2 | 4 | no | — |
| 48 | D28 | — | 4 | 4 | no | — |
| 49 | D28 | C2×D4 | 1 | 4 | yes | — |
| 50 | C4×D7 | — | 2 | 4 | no | — |
| 51 | C4×D7 | — | 4 | 8 | no | — |
| 52 | Dic14 | — | 4 | 4 | no | — |
| 53 | Dic14 | C2×D4 | 1 | 4 | yes | — |
| 54 | C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 55 | C7×C8 | C2×D4 | 1 | 2 | yes | — |
| 56 | C7⋊C8 | — | 2 | 4 | no | — |
| 57 | 2−1+4 | — | 7 | 4 | no | — |
| 58 | #137 | — | 7 | 8 | no | — |
| 59 | #137 | — | 14 | 8 | no | — |
| 60 | #135 | — | 7 | 2 | no | — |
| 61 | #135 | — | 14 | 4 | no | — |
| 62 | C2×Q16 | — | 14 | 4 | no | — |
| 63 | C2×Q16 | C2×D7 | 1 | 4 | yes | — |
| 64 | #131 | — | 7 | 4 | no | — |
| 65 | C4⋊Q8 | C2×D7 | 1 | 2 | yes | — |
| 66 | #108 | — | 7 | 2 | no | — |
| 67 | #104 | — | 7 | 4 | no | — |
| 68 | #101 | — | 7 | 4 | no | — |
| 69 | C4×C8 | C2×D7 | 1 | 1 | yes | characteristic |
| 70 | C22×C7 | H(5,𝔽2) | 1 | 1 | yes | characteristic |
| 71 | C2×D7 | — | 2 | 4 | no | — |
| 72 | C4×C7 | — | 2 | 2 | no | — |
| 73 | C4×C7 | — | 4 | 4 | no | — |
| 74 | C4×C7 | C22×D4 | 1 | 2 | yes | — |
| 75 | Dic7 | — | 2 | 4 | no | — |
| 76 | #41 | — | 7 | 4 | no | — |
| 77 | #41 | — | 14 | 8 | no | — |
| 78 | #41 | — | 28 | 4 | no | — |
| 79 | C2×Q8 | — | 14 | 8 | no | — |
| 80 | C2×Q8 | C22×D7 | 1 | 4 | yes | — |
| 81 | Q16 | — | 2 | 4 | no | — |
| 82 | Q16 | — | 2 | 4 | no | — |
| 83 | Q16 | — | 7 | 2 | no | — |
| 84 | Q16 | — | 14 | 8 | no | — |
| 85 | SD16 | — | 7 | 4 | no | — |
| 86 | SD16 | — | 14 | 8 | no | — |
| 87 | D8 | — | 7 | 2 | no | — |
| 88 | M16 | — | 7 | 4 | no | — |
| 89 | M16 | — | 14 | 4 | no | — |
| 90 | C2×C8 | — | 14 | 4 | no | — |
| 91 | C2×C8 | C22×D7 | 1 | 2 | yes | — |
| 92 | C4⋊C4 | — | 4 | 2 | no | — |
| 93 | C42 | C22×D7 | 1 | 1 | yes | characteristic |
| 94 | C2×C7 | — | 2 | 1 | no | — |
| 95 | C2×C7 | D42 | 1 | 1 | yes | characteristic |
| 96 | D7 | — | 4 | 4 | no | — |
| 97 | Q8 | — | 2 | 8 | no | — |
| 98 | Q8 | — | 7 | 4 | no | — |
| 99 | Q8 | — | 28 | 4 | no | — |
| 100 | D4 | — | 7 | 4 | no | — |
| 101 | D4 | — | 14 | 4 | no | — |
| 102 | D4 | — | 28 | 4 | no | — |
| 103 | C2×C4 | — | 2 | 1 | no | — |
| 104 | C2×C4 | — | 2 | 4 | no | — |
| 105 | C2×C4 | — | 14 | 4 | no | — |
| 106 | C2×C4 | — | 28 | 8 | no | — |
| 107 | C2×C4 | C23×D7 | 1 | 1 | yes | characteristic |
| 108 | C8 | — | 14 | 4 | no | — |
| 109 | C8 | D4×D7 | 1 | 2 | yes | — |
| 110 | C8 | D4×D7 | 1 | 2 | yes | — |
| 111 | C7 | #3293 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 112 | C22 | — | 14 | 4 | no | — |
| 113 | C22 | #6548 | 1 | 1 | yes | characteristic |
| 114 | C4 | — | 2 | 2 | no | — |
| 115 | C4 | — | 4 | 4 | no | — |
| 116 | C4 | — | 14 | 4 | no | — |
| 117 | C4 | C2×D4×D7 | 1 | 2 | yes | — |
| 118 | C2 | — | 2 | 1 | no | — |
| 119 | C2 | — | 28 | 4 | no | — |
| 120 | C2 | #90408 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #90408 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#494 | C2 |
| socle | C2×C7 | D42 |
| hypercenter | #494 | D7 |
| hypocenter | C7 | #3293 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #494 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #90457 | C2 | semidirect |
| 2. C7×#494 | C2 | semidirect |
| 3. #89964 | C2 | semidirect |
| 4. #89867 | C2 | semidirect |
| 5. #89472 | C2 | semidirect |
| 39. #494 | D7 | semidirect |
| 111. C7 | #3293 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #90457 | 2 |
| C7×#494 | 2 |
| #89964 | 2 |
| #89867 | 2 |
| #89472 | 2 |
| #3293 | 7 |