Finite Group Databasebeta

← C2×H(5,𝔽2)×D89 #26035052

Subgroups — C2×H(5,𝔽2)×D89 #26035052

Order 11392.

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×#15668058C216yesmaximal, complemented
2C2×#15668057C219yesmaximal, complemented
3H(5,𝔽2)×D89C2132yesmaximal, direct product factor
4C2×#41×D89C216yesmaximal, complemented
5C22×D4×D89C219yesmaximal, complemented
6C2×H(5,𝔽2)×C89C211yesmaximal, characteristic, complemented
7C24×D89C2216yescomplemented
8#15668058C22196yescomplemented
9#15668057C221144yescomplemented
10C2×#13083575C2216yescomplemented
11C2×#13083574C22118yescomplemented
12C2×#13083573C22118yescomplemented
13#41×D89C22196yescomplemented
14C2×Q8×D89C2212yescomplemented
15C22×D356C2219yescomplemented
16C22×C22⋊D89C22118yescomplemented
17C2×D4×D89C22118yescomplemented
18C2×D4×D89C221144yescomplemented
19C2×D4×D89C22136yescomplemented
20C22×C4×D89C2219yescomplemented
21H(5,𝔽2)×C89C22116yescomplemented
22C2×#41×C89C2216yescomplemented
23C22×D4×C89C2219yescomplemented
24C23×D89—424no—
25C23×D89C2319yes—
26C23×D89C23148yescomplemented
27C23×D89C23118yescomplemented
28#13083575C23148yescomplemented
29#13083574C231144yescomplemented
30#13083573C231144yescomplemented
31C2×Dic178C2316yescomplemented
32Q8×D89C23116yescomplemented
33C2×D356C23118yescomplemented
34C2×D356C23172yescomplemented
35C2×C22⋊D89C23136yescomplemented
36C2×C22⋊D89C231144yescomplemented
37D4×D89C231144yescomplemented
38D4×D89C231288yescomplemented
39C22×Dic89C2319yescomplemented
40C2×C4×D89C2316yes—
41C2×C4×D89C23172yescomplemented
42C2×C4×D89C23118yescomplemented
43C24×C89C2316yescomplemented
44#41×C89C23148yescomplemented
45C2×Q8×C89C2312yescomplemented
46C2×D4×C89C23118yescomplemented
47C2×D4×C89C23172yescomplemented
48C22×C4×C89C2319yescomplemented
49C23×C89—412no—
50C23×C89C2419yes—
51C23×C89C24124yescomplemented
52C22×D89—236no—
53C22×D89—496no—
54C22×D89—436no—
55C22×D89C24136yes—
56C22×D89C2419yes—
57C22×D89C2411yescharacteristic
58C22×D89C24172yescomplemented
59Q8×C89C2418yescomplemented
60D4×C89C24172yescomplemented
61C2×C4×C89C2416yes—
62C2×C4×C89C24136yescomplemented
63C22⋊D89C241144yescomplemented
64C2×Dic89C2416yes—
65C2×Dic89C24136yescomplemented
66D356C24172yescomplemented
67C4×D89C24124yes—
68C4×D89C24172yescomplemented
69Dic178C24124yescomplemented
70C22×C89—218no—
71C22×C89—424no—
72C22×C89C25118yes—
73C22×C89C2511yescharacteristic
74C2×D89—218no—
75C2×D89—272no—
76C2×D89—472no—
77C2×D89H(5,𝔽2)14yesdirect product factor
78C2×D89C25118yes—
79C2×D89C2512yes—
80C4×C89C25112yes—
81Dic89C25112yes—
82C2×C89—218no—
83C2×C89C2×H(5,𝔽2)12yescomplemented
84C2×C89C2611yescharacteristic
85D89—218no—
86D89C2×H(5,𝔽2)14yesdirect product factor
87C22×H(5,𝔽2)—891nomaximal, Sylow
88C89C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
89C2×H(5,𝔽2)—8932no—
90C2×H(5,𝔽2)—896no—
91C2×H(5,𝔽2)—899no—
92C2×H(5,𝔽2)D8911yescharacteristic, direct product factor
93C22×#41—896no—
94C23×D4—899no—
95C25—896no—
96H(5,𝔽2)—8996no—
97H(5,𝔽2)—89144no—
98H(5,𝔽2)C2×D89116yesdirect product factor
99C2×#41—8996no—
100C2×#41—896no—
101C2×#41—8918no—
102C2×#41—8918no—
103C2×#41C2×D8916yescomplemented
104C22×Q8—892no—
105C22×D4—8918no—
106C22×D4—8918no—
107C22×D4—8936no—
108C22×D4—89144no—
109C22×D4—899no—
110C22×D4C2×D8919yescomplemented
111C23×C4—899no—
112C24—8918no—
113C24—8948no—
114C24—899no—
115C24—35624no—
116C24C22×D8916yescomplemented
117#41—8948no—
118#41—89144no—
119#41—89144no—
120#41C22×D89148yescomplemented
121C2×Q8—8916no—
122C2×Q8—896no—
123C2×Q8C22×D8912yescomplemented
124C2×D4—8936no—
125C2×D4—89144no—
126C2×D4—89144no—
127C2×D4—89288no—
128C2×D4—8918no—
129C2×D4—8972no—
130C2×D4C22×D89118yescomplemented
131C2×D4C22×D89172yescomplemented
132C22×C4—899no—
133C22×C4—896no—
134C22×C4—8918no—
135C22×C4—8972no—
136C22×C4C22×D8919yescomplemented
137C23—412no—
138C23—891no—
139C23—8972no—
140C23—8936no—
141C23—899no—
142C23—35636no—
143C23—35696no—
144C23—17836no—
145C23C23×D8919yes—
146C23C23×D89124yescomplemented
147Q8—8924no—
148Q8C23×D8918yescomplemented
149D4—89144no—
150D4—8972no—
151D4C23×D89172yescomplemented
152C2×C4—8936no—
153C2×C4—896no—
154C2×C4—8924no—
155C2×C4—8972no—
156C2×C4C23×D8916yes—
157C2×C4C23×D89136yescomplemented
158C22—218no—
159C22—424no—
160C22—894no—
161C22—892no—
162C22—8918no—
163C22—35672no—
164C22—17872no—
165C22—17818no—
166C22C24×D89118yes—
167C22C24×D8911yescharacteristic
168C4—8912no—
169C4C24×D89112yes—
170C2—218no—
171C2—894no—
172C2—17818no—
173C2H(5,𝔽2)×D8912yesdirect product factor
174C2C25×D8911yescharacteristic
subgroupG/H
centerC22C24×D89
commutator, cosocleC2×C89C26
FrattiniC2C25×D89
FittingC2×H(5,𝔽2)×C89C2
socleC22×C89C25
hypercenterC2×H(5,𝔽2)D89
hypocenterC89C22×H(5,𝔽2)
LCSG ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#15668058C2semidirect
2. C2×#15668057C2semidirect
3. H(5,𝔽2)×D89C2direct (N × H)
4. C2×#41×D89C2semidirect
5. C22×D4×D89C2semidirect
6. C2×H(5,𝔽2)×C89C2semidirect
7. C24×D89C22semidirect
8. #15668058C22semidirect
9. #15668057C22semidirect
10. C2×#13083575C22semidirect
11. C2×#13083574C22semidirect
12. C2×#13083573C22semidirect
13. #41×D89C22semidirect
14. C2×Q8×D89C22semidirect
15. C22×D356C22semidirect
16. C22×C22⋊D89C22semidirect
17. C2×D4×D89C22semidirect
18. C2×D4×D89C22semidirect
19. C2×D4×D89C22semidirect
20. C22×C4×D89C22semidirect
21. H(5,𝔽2)×C89C22semidirect
22. C2×#41×C89C22semidirect
23. C22×D4×C89C22semidirect
26. C23×D89C23semidirect
27. C23×D89C23semidirect
28. #13083575C23semidirect
29. #13083574C23semidirect
30. #13083573C23semidirect
31. C2×Dic178C23semidirect
32. Q8×D89C23semidirect
33. C2×D356C23semidirect
34. C2×D356C23semidirect
35. C2×C22⋊D89C23semidirect
36. C2×C22⋊D89C23semidirect
37. D4×D89C23semidirect
38. D4×D89C23semidirect
39. C22×Dic89C23semidirect
41. C2×C4×D89C23semidirect
42. C2×C4×D89C23semidirect
43. C24×C89C23semidirect
44. #41×C89C23semidirect
45. C2×Q8×C89C23semidirect
46. C2×D4×C89C23semidirect
47. C2×D4×C89C23semidirect
48. C22×C4×C89C23semidirect
51. C23×C89C24semidirect
58. C22×D89C24semidirect
59. Q8×C89C24semidirect
60. D4×C89C24semidirect
62. C2×C4×C89C24semidirect
63. C22⋊D89C24semidirect
65. C2×Dic89C24semidirect
66. D356C24semidirect
68. C4×D89C24semidirect
69. Dic178C24semidirect
77. C2×D89H(5,𝔽2)direct (N × H)
83. C2×C89C2×H(5,𝔽2)semidirect
86. D89C2×H(5,𝔽2)direct (N × H)
88. C89C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D89direct (N × H)
98. H(5,𝔽2)C2×D89direct (N × H)
103. C2×#41C2×D89semidirect
110. C22×D4C2×D89semidirect
116. C24C22×D89semidirect
120. #41C22×D89semidirect
123. C2×Q8C22×D89semidirect
130. C2×D4C22×D89semidirect
131. C2×D4C22×D89semidirect
136. C22×C4C22×D89semidirect
146. C23C23×D89semidirect
148. Q8C23×D89semidirect
151. D4C23×D89semidirect
157. C2×C4C23×D89semidirect
173. C2H(5,𝔽2)×D89direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156680582
C2×#156680572
H(5,𝔽2)×D892
C2×#41×D892
C22×D4×D892
C2×H(5,𝔽2)×C892
C22×H(5,𝔽2)89