Finite Group Databasebeta

← C89×#63040 #36440888

Subgroups — C89×#63040 #36440888

Order 22784.

8508 conjugacy classes of subgroups; only the 5718 normal subgroups are shown.

normal subgroups only

This group has 8508 conjugacy classes of subgroups — too many to draw. Only its 5718 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3549C2127yesmaximal, complemented
2C89×#3498C2127yesmaximal, complemented
3C89×#3483C219yesmaximal, complemented
4C2×H(5,𝔽2)×C89C22118yescomplemented
5H(3,𝔽4)×C89C22136yescomplemented
6Q82×C89C2213yescomplemented
7#555×C89C22181yescomplemented
8#550×C89C22154yescomplemented
9D42×C89C221162yescomplemented
10#535×C89C221162yescomplemented
11#532×C89C22181yescomplemented
12C2×#127×C89C22127yescomplemented
13C4×#41×C89C22127yescomplemented
14H(5,𝔽2)×C89—472no—
15C2×#41×C89C23127yes—
16C2×#41×C89C23154yes—
17C22×D4×C89C231162yes—
18C4⋊Q8×C89C23127yes—
19#127×C89C23181yes—
20#127×C89C23154yes—
21#124×C89C231162yes—
22#121×C89C231324yes—
23#120×C89C231216yes—
24C4×Q8×C89C23118yes—
25C4×D4×C89C231162yes—
26#117×C89C23181yes—
27C2×C42×C89C23127yes—
28C24×C89C24136yes—
29#41×C89—2108no—
30#41×C89—4108no—
31C2×Q8×C89C24118yes—
32C2×Q8×C89C2413yes—
33C2×D4×C89—4324no—
34C2×D4×C89C24181yes—
35C2×D4×C89C241162yes—
36C22×C4×C89C24181yes—
37C4⋊C4×C89C24154yes—
38C22⋊C4×C89C241162yes—
39C42×C89C24127yes—
40C42×C89C24127yes—
41C23×C89—4108no—
42C23×C89C25127yes—
43Q8×C89—418no—
44Q8×C89H(5,𝔽2)112yescomplemented
45D4×C89—2162no—
46D4×C89—4162no—
47C2×C4×C89—2162no—
48C2×C4×C89C25127yes—
49C2×C4×C89C2519yes—
50C22×C89—281no—
51C22×C89—1636no—
52C22×C89C2611yescharacteristic
53C4×C89—227no—
54C4×C89C2×H(5,𝔽2)118yes—
55#63040C8911yesmaximal, characteristic, direct product factor, Sylow
56C2×C89—427no—
57C2×C89C22×H(5,𝔽2)13yes—
58#3549C2×C89127yescomplemented
59#3498C2×C89127yescomplemented
60#3483C2×C8919yescomplemented
61C89#6304011yescharacteristic, direct product factor, Sylow
62C2×H(5,𝔽2)C22×C89118yescomplemented
63H(3,𝔽4)C22×C89136yescomplemented
64Q82C22×C8913yescomplemented
65#555C22×C89181yescomplemented
66#550C22×C89154yescomplemented
67D42C22×C891162yescomplemented
68#535C22×C891162yescomplemented
69#532C22×C89181yescomplemented
70C2×#127C22×C89127yescomplemented
71C4×#41C22×C89127yescomplemented
72H(5,𝔽2)—472no—
73C2×#41C23×C89127yes—
74C2×#41C23×C89154yes—
75C22×D4C23×C891162yes—
76C4⋊Q8C23×C89127yes—
77#127C23×C89181yes—
78#127C23×C89154yes—
79#124C23×C891162yes—
80#121C23×C891324yes—
81#120C23×C891216yes—
82C4×Q8C23×C89118yes—
83C4×D4C23×C891162yes—
84#117C23×C89181yes—
85C2×C42C23×C89127yes—
86C24C24×C89136yes—
87#41—2108no—
88#41—4108no—
89C2×Q8C24×C89118yes—
90C2×Q8C24×C8913yes—
91C2×D4—4324no—
92C2×D4C24×C89181yes—
93C2×D4C24×C891162yes—
94C22×C4C24×C89181yes—
95C4⋊C4C24×C89154yes—
96C22⋊C4C24×C891162yes—
97C42C24×C89127yes—
98C42C24×C89127yes—
99C23—4108no—
100C23C25×C89127yes—
101Q8—418no—
102Q8H(5,𝔽2)×C89112yescomplemented
103D4—2162no—
104D4—4162no—
105C2×C4—2162no—
106C2×C4C25×C89127yes—
107C2×C4C25×C8919yes—
108C22—281no—
109C22—1636no—
110C22C26×C8911yescharacteristic
111C4—227no—
112C4C2×H(5,𝔽2)×C89118yes—
113C2—427no—
114C2C22×H(5,𝔽2)×C8913yes—
subgroupG/H
center, socleC22×C89C26
commutator, Frattini, cosocleC22C26×C89
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C89 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C89×#3549C2semidirect
2. C89×#3498C2semidirect
3. C89×#3483C2semidirect
4. C2×H(5,𝔽2)×C89C22semidirect
5. H(3,𝔽4)×C89C22semidirect
6. Q82×C89C22semidirect
7. #555×C89C22semidirect
8. #550×C89C22semidirect
9. D42×C89C22semidirect
10. #535×C89C22semidirect
11. #532×C89C22semidirect
12. C2×#127×C89C22semidirect
13. C4×#41×C89C22semidirect
44. Q8×C89H(5,𝔽2)semidirect
55. #63040C89direct (N × H)
58. #3549C2×C89semidirect
59. #3498C2×C89semidirect
60. #3483C2×C89semidirect
61. C89#63040direct (N × H)
62. C2×H(5,𝔽2)C22×C89semidirect
63. H(3,𝔽4)C22×C89semidirect
64. Q82C22×C89semidirect
65. #555C22×C89semidirect
66. #550C22×C89semidirect
67. D42C22×C89semidirect
68. #535C22×C89semidirect
69. #532C22×C89semidirect
70. C2×#127C22×C89semidirect
71. C4×#41C22×C89semidirect
102. Q8H(5,𝔽2)×C89semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C89×#35492
C89×#34982
C89×#34832
#6304089