Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
218 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 17 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 | C5×Q8⋊SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10602735 | — | 3 | 1 | no | maximal, Hall |
| 3 | #91953 | — | 4 | 3 | no | maximal |
| 4 | #85554 | — | 5 | 1 | no | maximal, Hall |
| 5 | C5×Q82 | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | #65384 | — | 3 | 3 | no | — |
| 7 | #64768 | — | 3 | 3 | no | — |
| 8 | C2×C5×SL(2,𝔽3) | — | 4 | 3 | no | — |
| 9 | C2×C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 10 | C5⋊GL(2,𝔽3) | — | 4 | 3 | no | — |
| 11 | C5⋊#226 | — | 4 | 3 | no | — |
| 12 | Q8⋊SL(2,𝔽3) | D5 | 1 | 1 | yes | characteristic, complemented |
| 13 | #4147 | — | 3 | 1 | no | — |
| 14 | C5×C4⋊Q8 | — | 3 | 3 | no | — |
| 15 | C4×C5×Q8 | — | 6 | 1 | no | — |
| 16 | #4088 | — | 6 | 3 | no | — |
| 17 | #3968 | — | 3 | 3 | no | — |
| 18 | #3937 | — | 6 | 3 | no | — |
| 19 | #2205 | — | 15 | 1 | no | Sylow |
| 20 | #1220 | — | 16 | 1 | no | — |
| 21 | C5×SL(2,𝔽3) | — | 4 | 3 | no | — |
| 22 | C5×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 23 | #983 | — | 20 | 3 | no | — |
| 24 | C2×C5×Q8 | — | 2 | 1 | no | — |
| 25 | C2×C5×Q8 | — | 3 | 1 | no | — |
| 26 | C2×C5×Q8 | S4 | 1 | 3 | yes | — |
| 27 | #713 | — | 12 | 1 | no | — |
| 28 | Q8×D5 | — | 12 | 1 | no | — |
| 29 | #709 | — | 6 | 3 | no | — |
| 30 | C5×C4⋊C4 | — | 6 | 3 | no | — |
| 31 | C42×C5 | — | 3 | 3 | no | — |
| 32 | #689 | — | 12 | 3 | no | — |
| 33 | #688 | — | 12 | 3 | no | — |
| 34 | #681 | — | 6 | 3 | no | — |
| 35 | Q82 | D15 | 1 | 1 | yes | characteristic, complemented |
| 36 | #456 | — | 15 | 3 | no | — |
| 37 | #356 | — | 15 | 3 | no | — |
| 38 | C22×C3×C5 | — | 16 | 1 | no | — |
| 39 | C2×D15 | — | 16 | 1 | no | — |
| 40 | Dic15 | — | 16 | 1 | no | — |
| 41 | C2×SL(2,𝔽3) | — | 4 | 3 | no | — |
| 42 | C2×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 43 | GL(2,𝔽3) | — | 20 | 3 | no | — |
| 44 | #226 | — | 20 | 3 | no | — |
| 45 | C5×Q8 | — | 2 | 1 | no | — |
| 46 | C5×Q8 | — | 4 | 3 | no | — |
| 47 | C5×Q8 | — | 6 | 1 | no | — |
| 48 | C5×Q8 | — | 12 | 1 | no | — |
| 49 | C2×C4×C5 | — | 3 | 3 | no | — |
| 50 | C2×C4×C5 | — | 6 | 1 | no | — |
| 51 | C22⋊D5 | — | 12 | 1 | no | — |
| 52 | D20 | — | 6 | 3 | no | — |
| 53 | C4×D5 | — | 12 | 3 | no | — |
| 54 | Dic10 | — | 6 | 3 | no | — |
| 55 | #168 | — | 12 | 3 | no | — |
| 56 | 2−1+4 | — | 15 | 1 | no | — |
| 57 | #137 | — | 30 | 3 | no | — |
| 58 | C4⋊Q8 | — | 3 | 3 | no | — |
| 59 | C4×Q8 | — | 6 | 1 | no | — |
| 60 | #104 | — | 30 | 3 | no | — |
| 61 | #101 | — | 15 | 3 | no | — |
| 62 | C2×C3×C5 | — | 16 | 1 | no | — |
| 63 | C2×C3×C5 | — | 32 | 1 | no | — |
| 64 | D15 | — | 32 | 1 | no | — |
| 65 | #67 | — | 80 | 1 | no | — |
| 66 | SL(2,𝔽3) | — | 4 | 3 | no | — |
| 67 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 68 | C22×C5 | #1020 | 1 | 1 | yes | characteristic |
| 69 | C2×D5 | — | 12 | 1 | no | — |
| 70 | C4×C5 | — | 6 | 1 | no | — |
| 71 | C4×C5 | — | 6 | 1 | no | — |
| 72 | C4×C5 | — | 6 | 3 | no | — |
| 73 | Dic5 | — | 12 | 1 | no | — |
| 74 | #41 | — | 30 | 3 | no | — |
| 75 | #41 | — | 60 | 1 | no | — |
| 76 | C2×Q8 | — | 2 | 1 | no | — |
| 77 | C2×Q8 | — | 3 | 1 | no | — |
| 78 | C2×Q8 | — | 60 | 1 | no | — |
| 79 | C2×Q8 | C5⋊S4 | 1 | 3 | yes | — |
| 80 | Q16 | — | 60 | 3 | no | — |
| 81 | SD16 | — | 60 | 3 | no | — |
| 82 | M16 | — | 30 | 3 | no | — |
| 83 | C4⋊C4 | — | 6 | 3 | no | — |
| 84 | C42 | — | 3 | 3 | no | — |
| 85 | C3×C5 | — | 16 | 1 | no | Hall |
| 86 | C22×C3 | — | 16 | 1 | no | — |
| 87 | C2×D3 | — | 80 | 1 | no | — |
| 88 | Dic3 | — | 80 | 1 | no | — |
| 89 | C2×C5 | — | 2 | 1 | no | — |
| 90 | C2×C5 | #5934 | 1 | 1 | yes | characteristic |
| 91 | D5 | — | 24 | 1 | no | — |
| 92 | Q8 | — | 2 | 1 | no | — |
| 93 | Q8 | — | 4 | 3 | no | — |
| 94 | Q8 | — | 6 | 1 | no | — |
| 95 | Q8 | — | 12 | 1 | no | — |
| 96 | Q8 | — | 30 | 3 | no | — |
| 97 | D4 | — | 30 | 3 | no | — |
| 98 | D4 | — | 60 | 1 | no | — |
| 99 | C2×C4 | — | 3 | 3 | no | — |
| 100 | C2×C4 | — | 6 | 1 | no | — |
| 101 | C2×C4 | — | 60 | 3 | no | — |
| 102 | C8 | — | 60 | 3 | no | — |
| 103 | C2×C3 | — | 16 | 1 | no | — |
| 104 | C2×C3 | — | 32 | 1 | no | — |
| 105 | D3 | — | 160 | 1 | no | — |
| 106 | C5 | #85554 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 107 | C22 | — | 60 | 1 | no | — |
| 108 | C22 | C5⋊#1020 | 1 | 1 | yes | characteristic |
| 109 | C4 | — | 6 | 1 | no | — |
| 110 | C4 | — | 6 | 1 | no | — |
| 111 | C4 | — | 6 | 3 | no | — |
| 112 | C4 | — | 60 | 1 | no | — |
| 113 | C3 | — | 16 | 1 | no | Sylow |
| 114 | C2 | — | 2 | 1 | no | — |
| 115 | C2 | — | 120 | 1 | no | — |
| 116 | C2 | C5⋊#5934 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C5⋊#5934 |
| commutator, cosocle, hypocenter | C5×Q8⋊SL(2,𝔽3) | C2 |
| Frattini, hypercenter | C22 | C5⋊#1020 |
| Fitting | C5×Q82 | D3 |
| socle | C2×C5 | #5934 |
| LCS | G ▷ C5×Q8⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C5×Q8⋊SL(2,𝔽3) | C2 | semidirect |
| 5. C5×Q82 | D3 | semidirect |
| 12. Q8⋊SL(2,𝔽3) | D5 | semidirect |
| 35. Q82 | D15 | semidirect |
| 106. C5 | #85554 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C5×Q8⋊SL(2,𝔽3) | 2 |
| #10602735 | 3 |
| #91953 | 4 |
| #85554 | 5 |