Finite Group Databasebeta

← C2×H(5,𝔽2)×D167 #36338938

Subgroups — C2×H(5,𝔽2)×D167 #36338938

Order 21376.

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×#19128878C216yesmaximal, complemented
2C2×#19128877C219yesmaximal, complemented
3H(5,𝔽2)×D167C2132yesmaximal, direct product factor
4C2×#41×D167C216yesmaximal, complemented
5C22×D4×D167C219yesmaximal, complemented
6C2×H(5,𝔽2)×C167C211yesmaximal, characteristic, complemented
7C24×D167C2216yescomplemented
8#19128878C22196yescomplemented
9#19128877C221144yescomplemented
10C2×#14581951C2216yescomplemented
11C2×#14581950C22118yescomplemented
12C2×#14581949C22118yescomplemented
13#41×D167C22196yescomplemented
14C2×Q8×D167C2212yescomplemented
15C22×D668C2219yescomplemented
16C22×C22⋊D167C22118yescomplemented
17C2×D4×D167C22118yescomplemented
18C2×D4×D167C221144yescomplemented
19C2×D4×D167C22136yescomplemented
20C22×C4×D167C2219yescomplemented
21H(5,𝔽2)×C167C22116yescomplemented
22C2×#41×C167C2216yescomplemented
23C22×D4×C167C2219yescomplemented
24C23×D167—424no—
25C23×D167C2319yes—
26C23×D167C23118yescomplemented
27C23×D167C23148yescomplemented
28#14581951C23148yescomplemented
29#14581950C231144yescomplemented
30#14581949C231144yescomplemented
31C2×Dic334C2316yescomplemented
32Q8×D167C23116yescomplemented
33C2×D668C23118yescomplemented
34C2×D668C23172yescomplemented
35C2×C22⋊D167C23136yescomplemented
36C2×C22⋊D167C231144yescomplemented
37D4×D167C231144yescomplemented
38D4×D167C231288yescomplemented
39C22×Dic167C2319yescomplemented
40C2×C4×D167C2316yes—
41C2×C4×D167C23118yescomplemented
42C2×C4×D167C23172yescomplemented
43C24×C167C2316yescomplemented
44#41×C167C23148yescomplemented
45C2×Q8×C167C2312yescomplemented
46C2×D4×C167C23118yescomplemented
47C2×D4×C167C23172yescomplemented
48C22×C4×C167C2319yescomplemented
49C22×D167—236no—
50C22×D167—496no—
51C22×D167—436no—
52C22×D167C24136yes—
53C22×D167C2419yes—
54C22×D167C2411yescharacteristic
55C22×D167C24172yescomplemented
56Dic334C24124yescomplemented
57D668C24172yescomplemented
58C22⋊D167C241144yescomplemented
59C2×Dic167C2416yes—
60C2×Dic167C24136yescomplemented
61C4×D167C24124yes—
62C4×D167C24172yescomplemented
63C23×C167—412no—
64C23×C167C2419yes—
65C23×C167C24124yescomplemented
66Q8×C167C2418yescomplemented
67D4×C167C24172yescomplemented
68C2×C4×C167C2416yes—
69C2×C4×C167C24136yescomplemented
70C22×C167—218no—
71C22×C167—424no—
72C22×C167C25118yes—
73C22×C167C2511yescharacteristic
74C2×D167—218no—
75C2×D167—272no—
76C2×D167—472no—
77C2×D167H(5,𝔽2)14yesdirect product factor
78C2×D167C25118yes—
79C2×D167C2512yes—
80C4×C167C25112yes—
81Dic167C25112yes—
82C2×C167—218no—
83C2×C167C2×H(5,𝔽2)12yescomplemented
84C2×C167C2611yescharacteristic
85D167—218no—
86D167C2×H(5,𝔽2)14yesdirect product factor
87C167C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
88C22×H(5,𝔽2)—1671nomaximal, Sylow
89C2×H(5,𝔽2)—16732no—
90C2×H(5,𝔽2)—1676no—
91C2×H(5,𝔽2)—1679no—
92C2×H(5,𝔽2)D16711yescharacteristic, direct product factor
93C22×#41—1676no—
94C23×D4—1679no—
95C25—1676no—
96H(5,𝔽2)—16796no—
97H(5,𝔽2)—167144no—
98H(5,𝔽2)C2×D167116yesdirect product factor
99C2×#41—16796no—
100C2×#41—1676no—
101C2×#41—16718no—
102C2×#41—16718no—
103C2×#41C2×D16716yescomplemented
104C22×Q8—1672no—
105C22×D4—16718no—
106C22×D4—16718no—
107C22×D4—16736no—
108C22×D4—167144no—
109C22×D4—1679no—
110C22×D4C2×D16719yescomplemented
111C23×C4—1679no—
112C24—66824no—
113C24—1679no—
114C24—16718no—
115C24—16748no—
116C24C22×D16716yescomplemented
117#41—16748no—
118#41—167144no—
119#41—167144no—
120#41C22×D167148yescomplemented
121C2×Q8—16716no—
122C2×Q8—1676no—
123C2×Q8C22×D16712yescomplemented
124C2×D4—16736no—
125C2×D4—167144no—
126C2×D4—167144no—
127C2×D4—167288no—
128C2×D4—16718no—
129C2×D4—16772no—
130C2×D4C22×D167118yescomplemented
131C2×D4C22×D167172yescomplemented
132C22×C4—1679no—
133C22×C4—1676no—
134C22×C4—16718no—
135C22×C4—16772no—
136C22×C4C22×D16719yescomplemented
137C23—412no—
138C23—33436no—
139C23—66836no—
140C23—66896no—
141C23—16736no—
142C23—1679no—
143C23—16772no—
144C23—1671no—
145C23C23×D16719yes—
146C23C23×D167124yescomplemented
147Q8—16724no—
148Q8C23×D16718yescomplemented
149D4—167144no—
150D4—16772no—
151D4C23×D167172yescomplemented
152C2×C4—16736no—
153C2×C4—1676no—
154C2×C4—16724no—
155C2×C4—16772no—
156C2×C4C23×D16716yes—
157C2×C4C23×D167136yescomplemented
158C22—218no—
159C22—424no—
160C22—33472no—
161C22—33418no—
162C22—66872no—
163C22—16718no—
164C22—1674no—
165C22—1672no—
166C22C24×D167118yes—
167C22C24×D16711yescharacteristic
168C4—16712no—
169C4C24×D167112yes—
170C2—218no—
171C2—33418no—
172C2—1674no—
173C2H(5,𝔽2)×D16712yesdirect product factor
174C2C25×D16711yescharacteristic
subgroupG/H
centerC22C24×D167
commutator, cosocleC2×C167C26
FrattiniC2C25×D167
FittingC2×H(5,𝔽2)×C167C2
socleC22×C167C25
hypercenterC2×H(5,𝔽2)D167
hypocenterC167C22×H(5,𝔽2)
LCSG ▷ C2×C167 ▷ C167
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#19128878C2semidirect
2. C2×#19128877C2semidirect
3. H(5,𝔽2)×D167C2direct (N × H)
4. C2×#41×D167C2semidirect
5. C22×D4×D167C2semidirect
6. C2×H(5,𝔽2)×C167C2semidirect
7. C24×D167C22semidirect
8. #19128878C22semidirect
9. #19128877C22semidirect
10. C2×#14581951C22semidirect
11. C2×#14581950C22semidirect
12. C2×#14581949C22semidirect
13. #41×D167C22semidirect
14. C2×Q8×D167C22semidirect
15. C22×D668C22semidirect
16. C22×C22⋊D167C22semidirect
17. C2×D4×D167C22semidirect
18. C2×D4×D167C22semidirect
19. C2×D4×D167C22semidirect
20. C22×C4×D167C22semidirect
21. H(5,𝔽2)×C167C22semidirect
22. C2×#41×C167C22semidirect
23. C22×D4×C167C22semidirect
26. C23×D167C23semidirect
27. C23×D167C23semidirect
28. #14581951C23semidirect
29. #14581950C23semidirect
30. #14581949C23semidirect
31. C2×Dic334C23semidirect
32. Q8×D167C23semidirect
33. C2×D668C23semidirect
34. C2×D668C23semidirect
35. C2×C22⋊D167C23semidirect
36. C2×C22⋊D167C23semidirect
37. D4×D167C23semidirect
38. D4×D167C23semidirect
39. C22×Dic167C23semidirect
41. C2×C4×D167C23semidirect
42. C2×C4×D167C23semidirect
43. C24×C167C23semidirect
44. #41×C167C23semidirect
45. C2×Q8×C167C23semidirect
46. C2×D4×C167C23semidirect
47. C2×D4×C167C23semidirect
48. C22×C4×C167C23semidirect
55. C22×D167C24semidirect
56. Dic334C24semidirect
57. D668C24semidirect
58. C22⋊D167C24semidirect
60. C2×Dic167C24semidirect
62. C4×D167C24semidirect
65. C23×C167C24semidirect
66. Q8×C167C24semidirect
67. D4×C167C24semidirect
69. C2×C4×C167C24semidirect
77. C2×D167H(5,𝔽2)direct (N × H)
83. C2×C167C2×H(5,𝔽2)semidirect
86. D167C2×H(5,𝔽2)direct (N × H)
87. C167C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D167direct (N × H)
98. H(5,𝔽2)C2×D167direct (N × H)
103. C2×#41C2×D167semidirect
110. C22×D4C2×D167semidirect
116. C24C22×D167semidirect
120. #41C22×D167semidirect
123. C2×Q8C22×D167semidirect
130. C2×D4C22×D167semidirect
131. C2×D4C22×D167semidirect
136. C22×C4C22×D167semidirect
146. C23C23×D167semidirect
148. Q8C23×D167semidirect
151. D4C23×D167semidirect
157. C2×C4C23×D167semidirect
173. C2H(5,𝔽2)×D167direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191288782
C2×#191288772
H(5,𝔽2)×D1672
C2×#41×D1672
C22×D4×D1672
C2×H(5,𝔽2)×C1672
C22×H(5,𝔽2)167