Finite Group Databasebeta

← C2×H(5,𝔽2)×D163 #35229336

Subgroups — C2×H(5,𝔽2)×D163 #35229336

Order 20864.

6560 conjugacy classes of subgroups; only the 3213 normal subgroups are shown.

normal subgroups only

This group has 6560 conjugacy classes of subgroups — too many to draw. Only its 3213 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×#19104781C216yesmaximal, complemented
2C2×#19104780C219yesmaximal, complemented
3H(5,𝔽2)×D163C2132yesmaximal, direct product factor
4C2×#41×D163C216yesmaximal, complemented
5C22×D4×D163C219yesmaximal, complemented
6C2×H(5,𝔽2)×C163C211yesmaximal, characteristic, complemented
7C24×D163C2216yescomplemented
8#19104781C22196yescomplemented
9#19104780C221144yescomplemented
10C2×#14579700C2216yescomplemented
11C2×#14579699C22118yescomplemented
12C2×#14579698C22118yescomplemented
13#41×D163C22196yescomplemented
14C2×Q8×D163C2212yescomplemented
15C22×D652C2219yescomplemented
16C22×C22⋊D163C22118yescomplemented
17C2×D4×D163C22118yescomplemented
18C2×D4×D163C221144yescomplemented
19C2×D4×D163C22136yescomplemented
20C22×C4×D163C2219yescomplemented
21H(5,𝔽2)×C163C22116yescomplemented
22C2×#41×C163C2216yescomplemented
23C22×D4×C163C2219yescomplemented
24C23×D163—424no—
25C23×D163C2319yes—
26C23×D163C23118yescomplemented
27C23×D163C23148yescomplemented
28#14579700C23148yescomplemented
29#14579699C231144yescomplemented
30#14579698C231144yescomplemented
31C2×Dic326C2316yescomplemented
32Q8×D163C23116yescomplemented
33C2×D652C23118yescomplemented
34C2×D652C23172yescomplemented
35C2×C22⋊D163C23136yescomplemented
36C2×C22⋊D163C231144yescomplemented
37D4×D163C231144yescomplemented
38D4×D163C231288yescomplemented
39C22×Dic163C2319yescomplemented
40C2×C4×D163C2316yes—
41C2×C4×D163C23118yescomplemented
42C2×C4×D163C23172yescomplemented
43C24×C163C2316yescomplemented
44#41×C163C23148yescomplemented
45C2×Q8×C163C2312yescomplemented
46C2×D4×C163C23118yescomplemented
47C2×D4×C163C23172yescomplemented
48C22×C4×C163C2319yescomplemented
49C22×D163—236no—
50C22×D163—496no—
51C22×D163—436no—
52C22×D163C24136yes—
53C22×D163C2419yes—
54C22×D163C2411yescharacteristic
55C22×D163C24172yescomplemented
56Dic326C24124yescomplemented
57D652C24172yescomplemented
58C22⋊D163C241144yescomplemented
59C2×Dic163C2416yes—
60C2×Dic163C24136yescomplemented
61C4×D163C24124yes—
62C4×D163C24172yescomplemented
63C23×C163—412no—
64C23×C163C2419yes—
65C23×C163C24124yescomplemented
66Q8×C163C2418yescomplemented
67D4×C163C24172yescomplemented
68C2×C4×C163C2416yes—
69C2×C4×C163C24136yescomplemented
70C22×C163—218no—
71C22×C163—424no—
72C22×C163C25118yes—
73C22×C163C2511yescharacteristic
74C2×D163—218no—
75C2×D163—272no—
76C2×D163—472no—
77C2×D163H(5,𝔽2)14yesdirect product factor
78C2×D163C25118yes—
79C2×D163C2512yes—
80C4×C163C25112yes—
81Dic163C25112yes—
82C2×C163—218no—
83C2×C163C2×H(5,𝔽2)12yescomplemented
84C2×C163C2611yescharacteristic
85D163—218no—
86D163C2×H(5,𝔽2)14yesdirect product factor
87C163C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
88C22×H(5,𝔽2)—1631nomaximal, Sylow
89C2×H(5,𝔽2)—16332no—
90C2×H(5,𝔽2)—1636no—
91C2×H(5,𝔽2)—1639no—
92C2×H(5,𝔽2)D16311yescharacteristic, direct product factor
93C22×#41—1636no—
94C23×D4—1639no—
95C25—1636no—
96H(5,𝔽2)—16396no—
97H(5,𝔽2)—163144no—
98H(5,𝔽2)C2×D163116yesdirect product factor
99C2×#41—16396no—
100C2×#41—1636no—
101C2×#41—16318no—
102C2×#41—16318no—
103C2×#41C2×D16316yescomplemented
104C22×Q8—1632no—
105C22×D4—16318no—
106C22×D4—16318no—
107C22×D4—16336no—
108C22×D4—163144no—
109C22×D4—1639no—
110C22×D4C2×D16319yescomplemented
111C23×C4—1639no—
112C24—65224no—
113C24—1639no—
114C24—16318no—
115C24—16348no—
116C24C22×D16316yescomplemented
117#41—16348no—
118#41—163144no—
119#41—163144no—
120#41C22×D163148yescomplemented
121C2×Q8—16316no—
122C2×Q8—1636no—
123C2×Q8C22×D16312yescomplemented
124C2×D4—16336no—
125C2×D4—163144no—
126C2×D4—163144no—
127C2×D4—163288no—
128C2×D4—16318no—
129C2×D4—16372no—
130C2×D4C22×D163118yescomplemented
131C2×D4C22×D163172yescomplemented
132C22×C4—1639no—
133C22×C4—1636no—
134C22×C4—16318no—
135C22×C4—16372no—
136C22×C4C22×D16319yescomplemented
137C23—412no—
138C23—32636no—
139C23—65236no—
140C23—65296no—
141C23—16336no—
142C23—1639no—
143C23—16372no—
144C23—1631no—
145C23C23×D16319yes—
146C23C23×D163124yescomplemented
147Q8—16324no—
148Q8C23×D16318yescomplemented
149D4—163144no—
150D4—16372no—
151D4C23×D163172yescomplemented
152C2×C4—16336no—
153C2×C4—1636no—
154C2×C4—16324no—
155C2×C4—16372no—
156C2×C4C23×D16316yes—
157C2×C4C23×D163136yescomplemented
158C22—218no—
159C22—424no—
160C22—32672no—
161C22—32618no—
162C22—65272no—
163C22—16318no—
164C22—1634no—
165C22—1632no—
166C22C24×D163118yes—
167C22C24×D16311yescharacteristic
168C4—16312no—
169C4C24×D163112yes—
170C2—218no—
171C2—32618no—
172C2—1634no—
173C2H(5,𝔽2)×D16312yesdirect product factor
174C2C25×D16311yescharacteristic
subgroupG/H
centerC22C24×D163
commutator, cosocleC2×C163C26
FrattiniC2C25×D163
FittingC2×H(5,𝔽2)×C163C2
socleC22×C163C25
hypercenterC2×H(5,𝔽2)D163
hypocenterC163C22×H(5,𝔽2)
LCSG ▷ C2×C163 ▷ C163
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#19104781C2semidirect
2. C2×#19104780C2semidirect
3. H(5,𝔽2)×D163C2direct (N × H)
4. C2×#41×D163C2semidirect
5. C22×D4×D163C2semidirect
6. C2×H(5,𝔽2)×C163C2semidirect
7. C24×D163C22semidirect
8. #19104781C22semidirect
9. #19104780C22semidirect
10. C2×#14579700C22semidirect
11. C2×#14579699C22semidirect
12. C2×#14579698C22semidirect
13. #41×D163C22semidirect
14. C2×Q8×D163C22semidirect
15. C22×D652C22semidirect
16. C22×C22⋊D163C22semidirect
17. C2×D4×D163C22semidirect
18. C2×D4×D163C22semidirect
19. C2×D4×D163C22semidirect
20. C22×C4×D163C22semidirect
21. H(5,𝔽2)×C163C22semidirect
22. C2×#41×C163C22semidirect
23. C22×D4×C163C22semidirect
26. C23×D163C23semidirect
27. C23×D163C23semidirect
28. #14579700C23semidirect
29. #14579699C23semidirect
30. #14579698C23semidirect
31. C2×Dic326C23semidirect
32. Q8×D163C23semidirect
33. C2×D652C23semidirect
34. C2×D652C23semidirect
35. C2×C22⋊D163C23semidirect
36. C2×C22⋊D163C23semidirect
37. D4×D163C23semidirect
38. D4×D163C23semidirect
39. C22×Dic163C23semidirect
41. C2×C4×D163C23semidirect
42. C2×C4×D163C23semidirect
43. C24×C163C23semidirect
44. #41×C163C23semidirect
45. C2×Q8×C163C23semidirect
46. C2×D4×C163C23semidirect
47. C2×D4×C163C23semidirect
48. C22×C4×C163C23semidirect
55. C22×D163C24semidirect
56. Dic326C24semidirect
57. D652C24semidirect
58. C22⋊D163C24semidirect
60. C2×Dic163C24semidirect
62. C4×D163C24semidirect
65. C23×C163C24semidirect
66. Q8×C163C24semidirect
67. D4×C163C24semidirect
69. C2×C4×C163C24semidirect
77. C2×D163H(5,𝔽2)direct (N × H)
83. C2×C163C2×H(5,𝔽2)semidirect
86. D163C2×H(5,𝔽2)direct (N × H)
87. C163C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D163direct (N × H)
98. H(5,𝔽2)C2×D163direct (N × H)
103. C2×#41C2×D163semidirect
110. C22×D4C2×D163semidirect
116. C24C22×D163semidirect
120. #41C22×D163semidirect
123. C2×Q8C22×D163semidirect
130. C2×D4C22×D163semidirect
131. C2×D4C22×D163semidirect
136. C22×C4C22×D163semidirect
146. C23C23×D163semidirect
148. Q8C23×D163semidirect
151. D4C23×D163semidirect
157. C2×C4C23×D163semidirect
173. C2H(5,𝔽2)×D163direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191047812
C2×#191047802
H(5,𝔽2)×D1632
C2×#41×D1632
C22×D4×D1632
C2×H(5,𝔽2)×C1632
C22×H(5,𝔽2)163