Finite Group Databasebeta

← C2×H(5,𝔽2)×D181 #37548521

Subgroups — C2×H(5,𝔽2)×D181 #37548521

Order 23168.

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×#19153862C216yesmaximal, complemented
2C2×#19153861C219yesmaximal, complemented
3H(5,𝔽2)×D181C2132yesmaximal, direct product factor
4C2×#41×D181C216yesmaximal, complemented
5C22×D4×D181C219yesmaximal, complemented
6C2×H(5,𝔽2)×C181C211yesmaximal, characteristic, complemented
7C24×D181C2216yescomplemented
8#19153862C22196yescomplemented
9#19153861C221144yescomplemented
10C2×#15668450C2216yescomplemented
11C2×#15668449C22118yescomplemented
12C2×#15668448C22118yescomplemented
13#41×D181C22196yescomplemented
14C2×Q8×D181C2212yescomplemented
15C22×D724C2219yescomplemented
16C22×C22⋊D181C22118yescomplemented
17C2×D4×D181C22118yescomplemented
18C2×D4×D181C221144yescomplemented
19C2×D4×D181C22136yescomplemented
20C22×C4×D181C2219yescomplemented
21H(5,𝔽2)×C181C22116yescomplemented
22C2×#41×C181C2216yescomplemented
23C22×D4×C181C2219yescomplemented
24C23×D181—424no—
25C23×D181C2319yes—
26C23×D181C23118yescomplemented
27C23×D181C23148yescomplemented
28#15668450C23148yescomplemented
29#15668449C231144yescomplemented
30#15668448C231144yescomplemented
31C2×Dic362C2316yescomplemented
32Q8×D181C23116yescomplemented
33C2×D724C23118yescomplemented
34C2×D724C23172yescomplemented
35C2×C22⋊D181C23136yescomplemented
36C2×C22⋊D181C231144yescomplemented
37D4×D181C231144yescomplemented
38D4×D181C231288yescomplemented
39C22×Dic181C2319yescomplemented
40C2×C4×D181C2316yes—
41C2×C4×D181C23118yescomplemented
42C2×C4×D181C23172yescomplemented
43C24×C181C2316yescomplemented
44#41×C181C23148yescomplemented
45C2×Q8×C181C2312yescomplemented
46C2×D4×C181C23118yescomplemented
47C2×D4×C181C23172yescomplemented
48C22×C4×C181C2319yescomplemented
49C22×D181—236no—
50C22×D181—496no—
51C22×D181—436no—
52C22×D181C24136yes—
53C22×D181C2419yes—
54C22×D181C2411yescharacteristic
55C22×D181C24172yescomplemented
56Dic362C24124yescomplemented
57D724C24172yescomplemented
58C22⋊D181C241144yescomplemented
59C2×Dic181C2416yes—
60C2×Dic181C24136yescomplemented
61C4×D181C24124yes—
62C4×D181C24172yescomplemented
63C23×C181—412no—
64C23×C181C2419yes—
65C23×C181C24124yescomplemented
66Q8×C181C2418yescomplemented
67D4×C181C24172yescomplemented
68C2×C4×C181C2416yes—
69C2×C4×C181C24136yescomplemented
70C22×C181—218no—
71C22×C181—424no—
72C22×C181C25118yes—
73C22×C181C2511yescharacteristic
74C2×D181—218no—
75C2×D181—272no—
76C2×D181—472no—
77C2×D181H(5,𝔽2)14yesdirect product factor
78C2×D181C25118yes—
79C2×D181C2512yes—
80C4×C181C25112yes—
81Dic181C25112yes—
82C2×C181—218no—
83C2×C181C2×H(5,𝔽2)12yescomplemented
84C2×C181C2611yescharacteristic
85D181—218no—
86D181C2×H(5,𝔽2)14yesdirect product factor
87C181C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
88C22×H(5,𝔽2)—1811nomaximal, Sylow
89C2×H(5,𝔽2)—18132no—
90C2×H(5,𝔽2)—1816no—
91C2×H(5,𝔽2)—1819no—
92C2×H(5,𝔽2)D18111yescharacteristic, direct product factor
93C22×#41—1816no—
94C23×D4—1819no—
95C25—1816no—
96H(5,𝔽2)—18196no—
97H(5,𝔽2)—181144no—
98H(5,𝔽2)C2×D181116yesdirect product factor
99C2×#41—18196no—
100C2×#41—1816no—
101C2×#41—18118no—
102C2×#41—18118no—
103C2×#41C2×D18116yescomplemented
104C22×Q8—1812no—
105C22×D4—18118no—
106C22×D4—18118no—
107C22×D4—18136no—
108C22×D4—181144no—
109C22×D4—1819no—
110C22×D4C2×D18119yescomplemented
111C23×C4—1819no—
112C24—1819no—
113C24—18118no—
114C24—18148no—
115C24—72424no—
116C24C22×D18116yescomplemented
117#41—18148no—
118#41—181144no—
119#41—181144no—
120#41C22×D181148yescomplemented
121C2×Q8—18116no—
122C2×Q8—1816no—
123C2×Q8C22×D18112yescomplemented
124C2×D4—18136no—
125C2×D4—181144no—
126C2×D4—181144no—
127C2×D4—181288no—
128C2×D4—18118no—
129C2×D4—18172no—
130C2×D4C22×D181118yescomplemented
131C2×D4C22×D181172yescomplemented
132C22×C4—1819no—
133C22×C4—1816no—
134C22×C4—18118no—
135C22×C4—18172no—
136C22×C4C22×D18119yescomplemented
137C23—412no—
138C23—36236no—
139C23—18136no—
140C23—1819no—
141C23—1811no—
142C23—18172no—
143C23—72436no—
144C23—72496no—
145C23C23×D18119yes—
146C23C23×D181124yescomplemented
147Q8—18124no—
148Q8C23×D18118yescomplemented
149D4—181144no—
150D4—18172no—
151D4C23×D181172yescomplemented
152C2×C4—18136no—
153C2×C4—1816no—
154C2×C4—18124no—
155C2×C4—18172no—
156C2×C4C23×D18116yes—
157C2×C4C23×D181136yescomplemented
158C22—218no—
159C22—424no—
160C22—36272no—
161C22—36218no—
162C22—18118no—
163C22—1812no—
164C22—1814no—
165C22—72472no—
166C22C24×D181118yes—
167C22C24×D18111yescharacteristic
168C4—18112no—
169C4C24×D181112yes—
170C2—218no—
171C2—36218no—
172C2—1814no—
173C2H(5,𝔽2)×D18112yesdirect product factor
174C2C25×D18111yescharacteristic
subgroupG/H
centerC22C24×D181
commutator, cosocleC2×C181C26
FrattiniC2C25×D181
FittingC2×H(5,𝔽2)×C181C2
socleC22×C181C25
hypercenterC2×H(5,𝔽2)D181
hypocenterC181C22×H(5,𝔽2)
LCSG ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#19153862C2semidirect
2. C2×#19153861C2semidirect
3. H(5,𝔽2)×D181C2direct (N × H)
4. C2×#41×D181C2semidirect
5. C22×D4×D181C2semidirect
6. C2×H(5,𝔽2)×C181C2semidirect
7. C24×D181C22semidirect
8. #19153862C22semidirect
9. #19153861C22semidirect
10. C2×#15668450C22semidirect
11. C2×#15668449C22semidirect
12. C2×#15668448C22semidirect
13. #41×D181C22semidirect
14. C2×Q8×D181C22semidirect
15. C22×D724C22semidirect
16. C22×C22⋊D181C22semidirect
17. C2×D4×D181C22semidirect
18. C2×D4×D181C22semidirect
19. C2×D4×D181C22semidirect
20. C22×C4×D181C22semidirect
21. H(5,𝔽2)×C181C22semidirect
22. C2×#41×C181C22semidirect
23. C22×D4×C181C22semidirect
26. C23×D181C23semidirect
27. C23×D181C23semidirect
28. #15668450C23semidirect
29. #15668449C23semidirect
30. #15668448C23semidirect
31. C2×Dic362C23semidirect
32. Q8×D181C23semidirect
33. C2×D724C23semidirect
34. C2×D724C23semidirect
35. C2×C22⋊D181C23semidirect
36. C2×C22⋊D181C23semidirect
37. D4×D181C23semidirect
38. D4×D181C23semidirect
39. C22×Dic181C23semidirect
41. C2×C4×D181C23semidirect
42. C2×C4×D181C23semidirect
43. C24×C181C23semidirect
44. #41×C181C23semidirect
45. C2×Q8×C181C23semidirect
46. C2×D4×C181C23semidirect
47. C2×D4×C181C23semidirect
48. C22×C4×C181C23semidirect
55. C22×D181C24semidirect
56. Dic362C24semidirect
57. D724C24semidirect
58. C22⋊D181C24semidirect
60. C2×Dic181C24semidirect
62. C4×D181C24semidirect
65. C23×C181C24semidirect
66. Q8×C181C24semidirect
67. D4×C181C24semidirect
69. C2×C4×C181C24semidirect
77. C2×D181H(5,𝔽2)direct (N × H)
83. C2×C181C2×H(5,𝔽2)semidirect
86. D181C2×H(5,𝔽2)direct (N × H)
87. C181C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D181direct (N × H)
98. H(5,𝔽2)C2×D181direct (N × H)
103. C2×#41C2×D181semidirect
110. C22×D4C2×D181semidirect
116. C24C22×D181semidirect
120. #41C22×D181semidirect
123. C2×Q8C22×D181semidirect
130. C2×D4C22×D181semidirect
131. C2×D4C22×D181semidirect
136. C22×C4C22×D181semidirect
146. C23C23×D181semidirect
148. Q8C23×D181semidirect
151. D4C23×D181semidirect
157. C2×C4C23×D181semidirect
173. C2H(5,𝔽2)×D181direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191538622
C2×#191538612
H(5,𝔽2)×D1812
C2×#41×D1812
C22×D4×D1812
C2×H(5,𝔽2)×C1812
C22×H(5,𝔽2)181