Finite Group Databasebeta

← C22×#15668057 #26035053

Subgroups — C22×#15668057 #26035053

Order 11392.

6560 conjugacy classes of subgroups; only the 3221 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3221 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
1C2×#15668057C2148yesmaximal, direct product factor
2C22×#13083574C212yesmaximal, complemented
3C22×#13083573C214yesmaximal, complemented
4C23×C22⋊D89C214yesmaximal, complemented
5C22×D4×D89C214yesmaximal, complemented
6C23×D4×C89C211yesmaximal, characteristic, complemented
7C24×D89C2214yescomplemented
8#15668057C221256yesdirect product factor
9C2×#13083574C22148yescomplemented
10C2×#13083573C22196yescomplemented
11C22×Dic178C2212yescomplemented
12C22×D356C2212yescomplemented
13C22×C22⋊D89C2218yescomplemented
14C22×C22⋊D89C2214yescomplemented
15C22×C22⋊D89C22196yescomplemented
16C2×D4×D89C22196yescomplemented
17C23×Dic89C2214yescomplemented
18C22×C4×D89C2214yescomplemented
19C25×C89C2212yescomplemented
20C22×D4×C89C2214yescomplemented
21C22×D4×C89C22124yescomplemented
22C23×C4×C89C2211yescharacteristic, complemented
23C23×D89—416no—
24C23×D89C2314yes—
25C23×D89C23148yescomplemented
26#13083574C231128yescomplemented
27#13083573C231256yescomplemented
28C2×Dic178C23124yescomplemented
29C2×D356C23124yescomplemented
30C2×C22⋊D89C23196yescomplemented
31C2×C22⋊D89C23148yescomplemented
32C2×C22⋊D89C231256yescomplemented
33D4×D89C231256yescomplemented
34C22×Dic89C2314yes—
35C22×Dic89C23148yescomplemented
36C2×C4×D89C23148yescomplemented
37C24×C89—48no—
38C24×C89C2314yes—
39C24×C89C2311yescharacteristic
40C24×C89C23124yescomplemented
41C2×D4×C89C23148yescomplemented
42C2×D4×C89C23164yescomplemented
43C22×C4×C89C2312yes—
44C22×C4×C89C23112yescomplemented
45C23×C89—24no—
46C23×C89—216no—
47C23×C89—448no—
48C23×C89C2416yes—
49C23×C89C24124yes—
50C23×C89C2411yescharacteristic
51C23×C89C24132yescomplemented
52C22×D89—216no—
53C22×D89—496no—
54C22×D89C24124yes—
55C22×D89C24164yescomplemented
56D4×C89C24164yescomplemented
57C2×C4×C89C24112yes—
58C2×C4×C89C24116yescomplemented
59C22⋊D89C241128yescomplemented
60C22⋊D89C24164yescomplemented
61C2×Dic89C24124yes—
62C2×Dic89C24164yescomplemented
63D356C24132yescomplemented
64C4×D89C24164yescomplemented
65Dic178C24132yescomplemented
66C22×C89—248no—
67C22×C89—212no—
68C22×C89—432no—
69C22×C89H(5,𝔽2)14yescomplemented
70C22×C89C25116yes—
71C22×C89C2514yes—
72C22×C89C2513yes—
73C2×D89—248no—
74C2×D89—464no—
75C2×D89C25116yes—
76C4×C89C2518yes—
77Dic89C25116yes—
78C2×C89—216no—
79C2×C89—24no—
80C2×C89C2×H(5,𝔽2)16yescomplemented
81C2×C89C2611yescharacteristic
82D89—216no—
83C22×H(5,𝔽2)—891nomaximal, Sylow
84C89C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
85C2×H(5,𝔽2)—8948no—
86C22×#41—894no—
87C22×#41—892no—
88C23×D4—894no—
89C23×D4—894no—
90C23×D4D8911yescharacteristic, complemented
91C25—894no—
92C25C2×D8912yescomplemented
93H(5,𝔽2)—89256no—
94C2×#41—8996no—
95C2×#41—8948no—
96C22×Q8—892no—
97C22×D4—898no—
98C22×D4—894no—
99C22×D4—8996no—
100C22×D4—8996no—
101C22×D4—892no—
102C22×D4C2×D8914yescomplemented
103C22×D4C2×D89124yescomplemented
104C23×C4—894no—
105C23×C4—894no—
106C23×C4C2×D8911yescharacteristic, complemented
107C24—48no—
108C24—8948no—
109C24—894no—
110C24—35616no—
111C24C22×D8914yes—
112C24C22×D8911yescharacteristic
113C24C22×D89124yescomplemented
114#41—89256no—
115#41—89128no—
116C2×Q8—8924no—
117C2×D4—8996no—
118C2×D4—8948no—
119C2×D4—89256no—
120C2×D4—89256no—
121C2×D4—8924no—
122C2×D4C22×D89148yescomplemented
123C2×D4C22×D89164yescomplemented
124C22×C4—8948no—
125C22×C4—894no—
126C22×C4—8948no—
127C22×C4C22×D8912yes—
128C22×C4C22×D89112yescomplemented
129C23—24no—
130C23—216no—
131C23—448no—
132C23—8964no—
133C23—8924no—
134C23—35696no—
135C23—17816no—
136C23C23×D89124yes—
137C23C23×D8916yes—
138C23C23×D8911yescharacteristic
139C23C23×D89132yescomplemented
140Q8—8932no—
141D4—89128no—
142D4—8964no—
143D4—8932no—
144D4C23×D89164yescomplemented
145C2×C4—8924no—
146C2×C4—8964no—
147C2×C4—8964no—
148C2×C4C23×D89112yes—
149C2×C4C23×D89116yescomplemented
150C22—212no—
151C22—248no—
152C22—432no—
153C22—8916no—
154C22—35664no—
155C22—17848no—
156C22#1566805714yesdirect product factor
157C22C24×D89116yes—
158C22C24×D8914yes—
159C22C24×D8913yes—
160C4—8916no—
161C4C24×D8918yes—
162C2—24no—
163C2—216no—
164C2—17816no—
165C2C2×#1566805716yesdirect product factor
166C2C25×D8911yescharacteristic
subgroupG/H
centerC23C23×D89
commutator, cosocleC2×C89C26
FrattiniC2C25×D89
FittingC23×D4×C89C2
socleC23×C89C24
hypercenterC23×D4D89
hypocenterC89C22×H(5,𝔽2)
LCSG ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15668057C2direct (N × H)
2. C22×#13083574C2semidirect
3. C22×#13083573C2semidirect
4. C23×C22⋊D89C2semidirect
5. C22×D4×D89C2semidirect
6. C23×D4×C89C2semidirect
7. C24×D89C22semidirect
8. #15668057C22direct (N × H)
9. C2×#13083574C22semidirect
10. C2×#13083573C22semidirect
11. C22×Dic178C22semidirect
12. C22×D356C22semidirect
13. C22×C22⋊D89C22semidirect
14. C22×C22⋊D89C22semidirect
15. C22×C22⋊D89C22semidirect
16. C2×D4×D89C22semidirect
17. C23×Dic89C22semidirect
18. C22×C4×D89C22semidirect
19. C25×C89C22semidirect
20. C22×D4×C89C22semidirect
21. C22×D4×C89C22semidirect
22. C23×C4×C89C22semidirect
25. C23×D89C23semidirect
26. #13083574C23semidirect
27. #13083573C23semidirect
28. C2×Dic178C23semidirect
29. C2×D356C23semidirect
30. C2×C22⋊D89C23semidirect
31. C2×C22⋊D89C23semidirect
32. C2×C22⋊D89C23semidirect
33. D4×D89C23semidirect
35. C22×Dic89C23semidirect
36. C2×C4×D89C23semidirect
40. C24×C89C23semidirect
41. C2×D4×C89C23semidirect
42. C2×D4×C89C23semidirect
44. C22×C4×C89C23semidirect
51. C23×C89C24semidirect
55. C22×D89C24semidirect
56. D4×C89C24semidirect
58. C2×C4×C89C24semidirect
59. C22⋊D89C24semidirect
60. C22⋊D89C24semidirect
62. C2×Dic89C24semidirect
63. D356C24semidirect
64. C4×D89C24semidirect
65. Dic178C24semidirect
69. C22×C89H(5,𝔽2)semidirect
80. C2×C89C2×H(5,𝔽2)semidirect
84. C89C22×H(5,𝔽2)semidirect
90. C23×D4D89semidirect
92. C25C2×D89semidirect
102. C22×D4C2×D89semidirect
103. C22×D4C2×D89semidirect
106. C23×C4C2×D89semidirect
113. C24C22×D89semidirect
122. C2×D4C22×D89semidirect
123. C2×D4C22×D89semidirect
128. C22×C4C22×D89semidirect
139. C23C23×D89semidirect
144. D4C23×D89semidirect
149. C2×C4C23×D89semidirect
156. C22#15668057direct (N × H)
165. C2C2×#15668057direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156680572
C22×#130835742
C22×#130835732
C23×C22⋊D892
C22×D4×D892
C23×D4×C892
C22×H(5,𝔽2)89