Finite Group Databasebeta

← C2×H(5,𝔽2)×D173 #36368363

Subgroups — C2×H(5,𝔽2)×D173 #36368363

Order 22144.

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×#19149725C216yesmaximal, complemented
2C2×#19149724C219yesmaximal, complemented
3H(5,𝔽2)×D173C2132yesmaximal, direct product factor
4C2×#41×D173C216yesmaximal, complemented
5C22×D4×D173C219yesmaximal, complemented
6C2×H(5,𝔽2)×C173C211yesmaximal, characteristic, complemented
7C24×D173C2216yescomplemented
8#19149725C22196yescomplemented
9#19149724C221144yescomplemented
10C2×#14583827C2216yescomplemented
11C2×#14583826C22118yescomplemented
12C2×#14583825C22118yescomplemented
13#41×D173C22196yescomplemented
14C2×Q8×D173C2212yescomplemented
15C22×D692C2219yescomplemented
16C22×C22⋊D173C22118yescomplemented
17C2×D4×D173C22118yescomplemented
18C2×D4×D173C221144yescomplemented
19C2×D4×D173C22136yescomplemented
20C22×C4×D173C2219yescomplemented
21H(5,𝔽2)×C173C22116yescomplemented
22C2×#41×C173C2216yescomplemented
23C22×D4×C173C2219yescomplemented
24C23×D173—424no—
25C23×D173C2319yes—
26C23×D173C23118yescomplemented
27C23×D173C23148yescomplemented
28#14583827C23148yescomplemented
29#14583826C231144yescomplemented
30#14583825C231144yescomplemented
31C2×Dic346C2316yescomplemented
32Q8×D173C23116yescomplemented
33C2×D692C23118yescomplemented
34C2×D692C23172yescomplemented
35C2×C22⋊D173C23136yescomplemented
36C2×C22⋊D173C231144yescomplemented
37D4×D173C231144yescomplemented
38D4×D173C231288yescomplemented
39C22×Dic173C2319yescomplemented
40C2×C4×D173C2316yes—
41C2×C4×D173C23118yescomplemented
42C2×C4×D173C23172yescomplemented
43C24×C173C2316yescomplemented
44#41×C173C23148yescomplemented
45C2×Q8×C173C2312yescomplemented
46C2×D4×C173C23118yescomplemented
47C2×D4×C173C23172yescomplemented
48C22×C4×C173C2319yescomplemented
49C22×D173—236no—
50C22×D173—496no—
51C22×D173—436no—
52C22×D173C24136yes—
53C22×D173C2419yes—
54C22×D173C2411yescharacteristic
55C22×D173C24172yescomplemented
56Dic346C24124yescomplemented
57D692C24172yescomplemented
58C22⋊D173C241144yescomplemented
59C2×Dic173C2416yes—
60C2×Dic173C24136yescomplemented
61C4×D173C24124yes—
62C4×D173C24172yescomplemented
63C23×C173—412no—
64C23×C173C2419yes—
65C23×C173C24124yescomplemented
66Q8×C173C2418yescomplemented
67D4×C173C24172yescomplemented
68C2×C4×C173C2416yes—
69C2×C4×C173C24136yescomplemented
70C22×C173—218no—
71C22×C173—424no—
72C22×C173C25118yes—
73C22×C173C2511yescharacteristic
74C2×D173—218no—
75C2×D173—272no—
76C2×D173—472no—
77C2×D173H(5,𝔽2)14yesdirect product factor
78C2×D173C25118yes—
79C2×D173C2512yes—
80C4×C173C25112yes—
81Dic173C25112yes—
82C2×C173—218no—
83C2×C173C2×H(5,𝔽2)12yescomplemented
84C2×C173C2611yescharacteristic
85D173—218no—
86D173C2×H(5,𝔽2)14yesdirect product factor
87C173C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
88C22×H(5,𝔽2)—1731nomaximal, Sylow
89C2×H(5,𝔽2)—17332no—
90C2×H(5,𝔽2)—1736no—
91C2×H(5,𝔽2)—1739no—
92C2×H(5,𝔽2)D17311yescharacteristic, direct product factor
93C22×#41—1736no—
94C23×D4—1739no—
95C25—1736no—
96H(5,𝔽2)—17396no—
97H(5,𝔽2)—173144no—
98H(5,𝔽2)C2×D173116yesdirect product factor
99C2×#41—17396no—
100C2×#41—1736no—
101C2×#41—17318no—
102C2×#41—17318no—
103C2×#41C2×D17316yescomplemented
104C22×Q8—1732no—
105C22×D4—17318no—
106C22×D4—17318no—
107C22×D4—17336no—
108C22×D4—173144no—
109C22×D4—1739no—
110C22×D4C2×D17319yescomplemented
111C23×C4—1739no—
112C24—1739no—
113C24—17318no—
114C24—17348no—
115C24—69224no—
116C24C22×D17316yescomplemented
117#41—17348no—
118#41—173144no—
119#41—173144no—
120#41C22×D173148yescomplemented
121C2×Q8—17316no—
122C2×Q8—1736no—
123C2×Q8C22×D17312yescomplemented
124C2×D4—17336no—
125C2×D4—173144no—
126C2×D4—173144no—
127C2×D4—173288no—
128C2×D4—17318no—
129C2×D4—17372no—
130C2×D4C22×D173118yescomplemented
131C2×D4C22×D173172yescomplemented
132C22×C4—1739no—
133C22×C4—1736no—
134C22×C4—17318no—
135C22×C4—17372no—
136C22×C4C22×D17319yescomplemented
137C23—412no—
138C23—34636no—
139C23—17336no—
140C23—1739no—
141C23—1731no—
142C23—17372no—
143C23—69236no—
144C23—69296no—
145C23C23×D17319yes—
146C23C23×D173124yescomplemented
147Q8—17324no—
148Q8C23×D17318yescomplemented
149D4—173144no—
150D4—17372no—
151D4C23×D173172yescomplemented
152C2×C4—17336no—
153C2×C4—1736no—
154C2×C4—17324no—
155C2×C4—17372no—
156C2×C4C23×D17316yes—
157C2×C4C23×D173136yescomplemented
158C22—218no—
159C22—424no—
160C22—34672no—
161C22—34618no—
162C22—17318no—
163C22—1734no—
164C22—1732no—
165C22—69272no—
166C22C24×D173118yes—
167C22C24×D17311yescharacteristic
168C4—17312no—
169C4C24×D173112yes—
170C2—218no—
171C2—34618no—
172C2—1734no—
173C2H(5,𝔽2)×D17312yesdirect product factor
174C2C25×D17311yescharacteristic
subgroupG/H
centerC22C24×D173
commutator, cosocleC2×C173C26
FrattiniC2C25×D173
FittingC2×H(5,𝔽2)×C173C2
socleC22×C173C25
hypercenterC2×H(5,𝔽2)D173
hypocenterC173C22×H(5,𝔽2)
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#19149725C2semidirect
2. C2×#19149724C2semidirect
3. H(5,𝔽2)×D173C2direct (N × H)
4. C2×#41×D173C2semidirect
5. C22×D4×D173C2semidirect
6. C2×H(5,𝔽2)×C173C2semidirect
7. C24×D173C22semidirect
8. #19149725C22semidirect
9. #19149724C22semidirect
10. C2×#14583827C22semidirect
11. C2×#14583826C22semidirect
12. C2×#14583825C22semidirect
13. #41×D173C22semidirect
14. C2×Q8×D173C22semidirect
15. C22×D692C22semidirect
16. C22×C22⋊D173C22semidirect
17. C2×D4×D173C22semidirect
18. C2×D4×D173C22semidirect
19. C2×D4×D173C22semidirect
20. C22×C4×D173C22semidirect
21. H(5,𝔽2)×C173C22semidirect
22. C2×#41×C173C22semidirect
23. C22×D4×C173C22semidirect
26. C23×D173C23semidirect
27. C23×D173C23semidirect
28. #14583827C23semidirect
29. #14583826C23semidirect
30. #14583825C23semidirect
31. C2×Dic346C23semidirect
32. Q8×D173C23semidirect
33. C2×D692C23semidirect
34. C2×D692C23semidirect
35. C2×C22⋊D173C23semidirect
36. C2×C22⋊D173C23semidirect
37. D4×D173C23semidirect
38. D4×D173C23semidirect
39. C22×Dic173C23semidirect
41. C2×C4×D173C23semidirect
42. C2×C4×D173C23semidirect
43. C24×C173C23semidirect
44. #41×C173C23semidirect
45. C2×Q8×C173C23semidirect
46. C2×D4×C173C23semidirect
47. C2×D4×C173C23semidirect
48. C22×C4×C173C23semidirect
55. C22×D173C24semidirect
56. Dic346C24semidirect
57. D692C24semidirect
58. C22⋊D173C24semidirect
60. C2×Dic173C24semidirect
62. C4×D173C24semidirect
65. C23×C173C24semidirect
66. Q8×C173C24semidirect
67. D4×C173C24semidirect
69. C2×C4×C173C24semidirect
77. C2×D173H(5,𝔽2)direct (N × H)
83. C2×C173C2×H(5,𝔽2)semidirect
86. D173C2×H(5,𝔽2)direct (N × H)
87. C173C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D173direct (N × H)
98. H(5,𝔽2)C2×D173direct (N × H)
103. C2×#41C2×D173semidirect
110. C22×D4C2×D173semidirect
116. C24C22×D173semidirect
120. #41C22×D173semidirect
123. C2×Q8C22×D173semidirect
130. C2×D4C22×D173semidirect
131. C2×D4C22×D173semidirect
136. C22×C4C22×D173semidirect
146. C23C23×D173semidirect
148. Q8C23×D173semidirect
151. D4C23×D173semidirect
157. C2×C4C23×D173semidirect
173. C2H(5,𝔽2)×D173direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191497252
C2×#191497242
H(5,𝔽2)×D1732
C2×#41×D1732
C22×D4×D1732
C2×H(5,𝔽2)×C1732
C22×H(5,𝔽2)173