Finite Group Databasebeta

← C23×H(5,𝔽2)×C89 #36440946

Subgroups — C23×H(5,𝔽2)×C89 #36440946

Order 22784.

65726 conjugacy classes of subgroups; only the 58526 normal subgroups are shown.

normal subgroups only

This group has 65726 conjugacy classes of subgroups — too many to draw. Only its 58526 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
1C22×H(5,𝔽2)×C89C21112yesmaximal, direct product factor
2C23×#41×C89C216yesmaximal, complemented
3C24×D4×C89C219yesmaximal, complemented
4C26×C89C2216yescomplemented
5C2×H(5,𝔽2)×C89C2211792yesdirect product factor
6C22×#41×C89C221336yescomplemented
7C23×Q8×C89C2212yescomplemented
8C23×D4×C89C22118yescomplemented
9C23×D4×C89C221504yescomplemented
10C24×C4×C89C2219yescomplemented
11C25×C89—448no—
12C25×C89C2319yes—
13C25×C89C231168yescomplemented
14H(5,𝔽2)×C89C2314096yesdirect product factor
15C2×#41×C89C2312688yescomplemented
16C22×Q8×C89C23156yescomplemented
17C22×D4×C89C231504yescomplemented
18C22×D4×C89C2314032yescomplemented
19C23×C4×C89C2316yes—
20C23×C4×C89C231252yescomplemented
21C24×C89—272no—
22C24×C89—4672no—
23C24×C89C241126yes—
24C24×C89C2411yescharacteristic
25C24×C89C241672yescomplemented
26#41×C89C2413072yescomplemented
27C2×Q8×C89C241224yescomplemented
28C2×D4×C89C2412016yescomplemented
29C2×D4×C89C2414608yescomplemented
30C22×C4×C89C24184yes—
31C22×C4×C89C2411008yescomplemented
32C23×C89—2504no—
33C23×C89—41344no—
34C23×C89H(5,𝔽2)18yesdirect product factor
35C23×C89C251252yes—
36C23×C89C2517yes—
37C23×C89C251384yescomplemented
38Q8×C89C251128yescomplemented
39D4×C89C2511152yescomplemented
40C2×C4×C89C251168yes—
41C2×C4×C89C251576yescomplemented
42C22×C89—2504no—
43C22×C89—4384no—
44C22×C89C2×H(5,𝔽2)128yesdirect product factor
45C22×C89C26172yes—
46C22×C89C2617yes—
47C4×C89C26148yes—
48C23×H(5,𝔽2)C8911yesmaximal, characteristic, direct product factor, Sylow
49C2×C89—272no—
50C2×C89C22×H(5,𝔽2)114yesdirect product factor
51C2×C89C2711yescharacteristic
52C22×H(5,𝔽2)C2×C891112yesdirect product factor
53C23×#41C2×C8916yescomplemented
54C24×D4C2×C8919yescomplemented
55C89C23×H(5,𝔽2)11yescharacteristic, direct product factor, Sylow
56C26C22×C8916yescomplemented
57C2×H(5,𝔽2)C22×C8911792yesdirect product factor
58C22×#41C22×C891336yescomplemented
59C23×Q8C22×C8912yescomplemented
60C23×D4C22×C89118yescomplemented
61C23×D4C22×C891504yescomplemented
62C24×C4C22×C8919yescomplemented
63C25—448no—
64C25C23×C8919yes—
65C25C23×C891168yescomplemented
66H(5,𝔽2)C23×C8914096yesdirect product factor
67C2×#41C23×C8912688yescomplemented
68C22×Q8C23×C89156yescomplemented
69C22×D4C23×C891504yescomplemented
70C22×D4C23×C8914032yescomplemented
71C23×C4C23×C8916yes—
72C23×C4C23×C891252yescomplemented
73C24—272no—
74C24—4672no—
75C24C24×C891126yes—
76C24C24×C8911yescharacteristic
77C24C24×C891672yescomplemented
78#41C24×C8913072yescomplemented
79C2×Q8C24×C891224yescomplemented
80C2×D4C24×C8912016yescomplemented
81C2×D4C24×C8914608yescomplemented
82C22×C4C24×C89184yes—
83C22×C4C24×C8911008yescomplemented
84C23—2504no—
85C23—41344no—
86C23H(5,𝔽2)×C8918yesdirect product factor
87C23C25×C891252yes—
88C23C25×C8917yes—
89C23C25×C891384yescomplemented
90Q8C25×C891128yescomplemented
91D4C25×C8911152yescomplemented
92C2×C4C25×C891168yes—
93C2×C4C25×C891576yescomplemented
94C22—2504no—
95C22—4384no—
96C22C2×H(5,𝔽2)×C89128yesdirect product factor
97C22C26×C89172yes—
98C22C26×C8917yes—
99C4C26×C89148yes—
100C2—272no—
101C2C22×H(5,𝔽2)×C89114yesdirect product factor
102C2C27×C8911yescharacteristic
subgroupG/H
center, socleC24×C89C24
commutator, Frattini, cosocleC2C27×C89
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C24×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C22×H(5,𝔽2)×C89C2direct (N × H)
2. C23×#41×C89C2semidirect
3. C24×D4×C89C2semidirect
4. C26×C89C22semidirect
5. C2×H(5,𝔽2)×C89C22direct (N × H)
6. C22×#41×C89C22semidirect
7. C23×Q8×C89C22semidirect
8. C23×D4×C89C22semidirect
9. C23×D4×C89C22semidirect
10. C24×C4×C89C22semidirect
13. C25×C89C23semidirect
14. H(5,𝔽2)×C89C23direct (N × H)
15. C2×#41×C89C23semidirect
16. C22×Q8×C89C23semidirect
17. C22×D4×C89C23semidirect
18. C22×D4×C89C23semidirect
20. C23×C4×C89C23semidirect
25. C24×C89C24semidirect
26. #41×C89C24semidirect
27. C2×Q8×C89C24semidirect
28. C2×D4×C89C24semidirect
29. C2×D4×C89C24semidirect
31. C22×C4×C89C24semidirect
34. C23×C89H(5,𝔽2)direct (N × H)
37. C23×C89C25semidirect
38. Q8×C89C25semidirect
39. D4×C89C25semidirect
41. C2×C4×C89C25semidirect
44. C22×C89C2×H(5,𝔽2)direct (N × H)
48. C23×H(5,𝔽2)C89direct (N × H)
50. C2×C89C22×H(5,𝔽2)direct (N × H)
52. C22×H(5,𝔽2)C2×C89direct (N × H)
53. C23×#41C2×C89semidirect
54. C24×D4C2×C89semidirect
55. C89C23×H(5,𝔽2)direct (N × H)
56. C26C22×C89semidirect
57. C2×H(5,𝔽2)C22×C89direct (N × H)
58. C22×#41C22×C89semidirect
59. C23×Q8C22×C89semidirect
60. C23×D4C22×C89semidirect
61. C23×D4C22×C89semidirect
62. C24×C4C22×C89semidirect
65. C25C23×C89semidirect
66. H(5,𝔽2)C23×C89direct (N × H)
67. C2×#41C23×C89semidirect
68. C22×Q8C23×C89semidirect
69. C22×D4C23×C89semidirect
70. C22×D4C23×C89semidirect
72. C23×C4C23×C89semidirect
77. C24C24×C89semidirect
78. #41C24×C89semidirect
79. C2×Q8C24×C89semidirect
80. C2×D4C24×C89semidirect
81. C2×D4C24×C89semidirect
83. C22×C4C24×C89semidirect
86. C23H(5,𝔽2)×C89direct (N × H)
89. C23C25×C89semidirect
90. Q8C25×C89semidirect
91. D4C25×C89semidirect
93. C2×C4C25×C89semidirect
96. C22C2×H(5,𝔽2)×C89direct (N × H)
101. C2C22×H(5,𝔽2)×C89direct (N × H)

Maximal subgroups

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

subgroupindex
C22×H(5,𝔽2)×C892
C23×#41×C892
C24×D4×C892
C23×H(5,𝔽2)89