Finite Group Databasebeta

← C22×#585×C89 #36440948

Subgroups — C22×#585×C89 #36440948

Order 22784.

60366 conjugacy classes of subgroups.

The subgroup lattice has 126 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#585×C89C2196yesmaximal, direct product factor
2C22×2−1+4×C89C216yesmaximal, complemented
3C22×H(5,𝔽2)×C89C2110yesmaximal, complemented
4C23×#41×C89C2115yesmaximal, complemented
5#585×C89C2211024yesdirect product factor
6C2×2−1+4×C89C221288yescomplemented
7C2×H(5,𝔽2)×C89C221480yescomplemented
8C22×#41×C89C22120yescomplemented
9C22×#41×C89C22160yescomplemented
10C22×#41×C89C221720yescomplemented
11C23×Q8×C89C22115yescomplemented
12C23×D4×C89C22145yescomplemented
13C24×C4×C89C22115yescomplemented
14C25×C89C23115yes—
152−1+4×C89C2311536yescomplemented
16H(5,𝔽2)×C89C2312560yescomplemented
17C2×#41×C89C231480yescomplemented
18C2×#41×C89C2311440yescomplemented
19C2×#41×C89C2313840yescomplemented
20C22×Q8×C89C23120yes—
21C22×Q8×C89C231360yescomplemented
22C22×D4×C89C23160yes—
23C22×D4×C89C2311080yescomplemented
24C23×C4×C89C23145yes—
25C23×C4×C89C23115yes—
26C23×C4×C89C231360yescomplemented
27C24×C89—460no—
28C24×C89C24115yes—
29C24×C89C241180yes—
30#41×C89C2411280yescomplemented
31#41×C89C2413840yescomplemented
32C2×Q8×C89C241240yes—
33C2×Q8×C89C241960yescomplemented
34C2×D4×C89C241720yes—
35C2×D4×C89C2412880yescomplemented
36C22×C4×C89C24115yes—
37C22×C4×C89C241180yes—
38C22×C4×C89C241540yes—
39C22×C4×C89C2411yescharacteristic
40C22×C4×C89C241960yescomplemented
41C23×C89—260no—
42C23×C89—4360no—
43C23×C89C25190yes—
44C23×C89C251240yes—
45C23×C89C2511yescharacteristic
46Q8×C89C251320yes—
47D4×C89C251960yes—
48C2×C4×C89C2516yes—
49C2×C4×C89C25190yes—
50C2×C4×C89C251240yes—
51C2×C4×C89C251720yes—
52C22×C89—2180no—
53C22×C89—4240no—
54C22×C89#58514yesdirect product factor
55C22×C89C26160yes—
56C22×C89C2613yes—
57C4×C89C2614yes—
58C4×C89C26160yes—
59C22×#585C8911yesmaximal, characteristic, direct product factor, Sylow
60C2×C89—260no—
61C2×C89C2×#58516yesdirect product factor
62C2×C89C2711yescharacteristic
63C2×#585C2×C89196yesdirect product factor
64C22×2−1+4C2×C8916yescomplemented
65C22×H(5,𝔽2)C2×C89110yescomplemented
66C23×#41C2×C89115yescomplemented
67C89C22×#58511yescharacteristic, direct product factor, Sylow
68#585C22×C8911024yesdirect product factor
69C2×2−1+4C22×C891288yescomplemented
70C2×H(5,𝔽2)C22×C891480yescomplemented
71C22×#41C22×C89120yescomplemented
72C22×#41C22×C89160yescomplemented
73C22×#41C22×C891720yescomplemented
74C23×Q8C22×C89115yescomplemented
75C23×D4C22×C89145yescomplemented
76C24×C4C22×C89115yescomplemented
77C25C23×C89115yes—
782−1+4C23×C8911536yescomplemented
79H(5,𝔽2)C23×C8912560yescomplemented
80C2×#41C23×C891480yescomplemented
81C2×#41C23×C8911440yescomplemented
82C2×#41C23×C8913840yescomplemented
83C22×Q8C23×C89120yes—
84C22×Q8C23×C891360yescomplemented
85C22×D4C23×C89160yes—
86C22×D4C23×C8911080yescomplemented
87C23×C4C23×C89115yes—
88C23×C4C23×C89145yes—
89C23×C4C23×C891360yescomplemented
90C24—460no—
91C24C24×C89115yes—
92C24C24×C891180yes—
93#41C24×C8911280yescomplemented
94#41C24×C8913840yescomplemented
95C2×Q8C24×C891240yes—
96C2×Q8C24×C891960yescomplemented
97C2×D4C24×C891720yes—
98C2×D4C24×C8912880yescomplemented
99C22×C4C24×C8911yescharacteristic
100C22×C4C24×C89115yes—
101C22×C4C24×C891180yes—
102C22×C4C24×C891540yes—
103C22×C4C24×C891960yescomplemented
104C23—260no—
105C23—4360no—
106C23C25×C89190yes—
107C23C25×C891240yes—
108C23C25×C8911yescharacteristic
109Q8C25×C891320yes—
110D4C25×C891960yes—
111C2×C4C25×C8916yes—
112C2×C4C25×C89190yes—
113C2×C4C25×C891240yes—
114C2×C4C25×C891720yes—
115C22—2180no—
116C22—4240no—
117C22#585×C8914yesdirect product factor
118C22C26×C89160yes—
119C22C26×C8913yes—
120C4C26×C8914yes—
121C4C26×C89160yes—
122C2—260no—
123C2C2×#585×C8916yesdirect product factor
124C2C27×C8911yescharacteristic
subgroupG/H
centerC22×C4×C89C24
commutator, Frattini, cosocleC2C27×C89
socleC23×C89C25
LCSG ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#585×C89C2direct (N × H)
2. C22×2−1+4×C89C2semidirect
3. C22×H(5,𝔽2)×C89C2semidirect
4. C23×#41×C89C2semidirect
5. #585×C89C22direct (N × H)
6. C2×2−1+4×C89C22semidirect
7. C2×H(5,𝔽2)×C89C22semidirect
8. C22×#41×C89C22semidirect
9. C22×#41×C89C22semidirect
10. C22×#41×C89C22semidirect
11. C23×Q8×C89C22semidirect
12. C23×D4×C89C22semidirect
13. C24×C4×C89C22semidirect
15. 2−1+4×C89C23semidirect
16. H(5,𝔽2)×C89C23semidirect
17. C2×#41×C89C23semidirect
18. C2×#41×C89C23semidirect
19. C2×#41×C89C23semidirect
21. C22×Q8×C89C23semidirect
23. C22×D4×C89C23semidirect
26. C23×C4×C89C23semidirect
30. #41×C89C24semidirect
31. #41×C89C24semidirect
33. C2×Q8×C89C24semidirect
35. C2×D4×C89C24semidirect
40. C22×C4×C89C24semidirect
54. C22×C89#585direct (N × H)
59. C22×#585C89direct (N × H)
61. C2×C89C2×#585direct (N × H)
63. C2×#585C2×C89direct (N × H)
64. C22×2−1+4C2×C89semidirect
65. C22×H(5,𝔽2)C2×C89semidirect
66. C23×#41C2×C89semidirect
67. C89C22×#585direct (N × H)
68. #585C22×C89direct (N × H)
69. C2×2−1+4C22×C89semidirect
70. C2×H(5,𝔽2)C22×C89semidirect
71. C22×#41C22×C89semidirect
72. C22×#41C22×C89semidirect
73. C22×#41C22×C89semidirect
74. C23×Q8C22×C89semidirect
75. C23×D4C22×C89semidirect
76. C24×C4C22×C89semidirect
78. 2−1+4C23×C89semidirect
79. H(5,𝔽2)C23×C89semidirect
80. C2×#41C23×C89semidirect
81. C2×#41C23×C89semidirect
82. C2×#41C23×C89semidirect
84. C22×Q8C23×C89semidirect
86. C22×D4C23×C89semidirect
89. C23×C4C23×C89semidirect
93. #41C24×C89semidirect
94. #41C24×C89semidirect
96. C2×Q8C24×C89semidirect
98. C2×D4C24×C89semidirect
103. C22×C4C24×C89semidirect
117. C22#585×C89direct (N × H)
123. C2C2×#585×C89direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#585×C892
C22×2−1+4×C892
C22×H(5,𝔽2)×C892
C23×#41×C892
C22×#58589