Finite Group Databasebeta

← C2×H(5,𝔽2)×D97 #27156536

Subgroups — C2×H(5,𝔽2)×D97 #27156536

Order 12416.

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×#15690856C216yesmaximal, complemented
2C2×#15690855C219yesmaximal, complemented
3H(5,𝔽2)×D97C2132yesmaximal, direct product factor
4C2×#41×D97C216yesmaximal, complemented
5C22×D4×D97C219yesmaximal, complemented
6C2×H(5,𝔽2)×C97C211yesmaximal, characteristic, complemented
7C24×D97C2216yescomplemented
8#15690856C22196yescomplemented
9#15690855C221144yescomplemented
10C2×#13095512C2216yescomplemented
11C2×#13095511C22118yescomplemented
12C2×#13095510C22118yescomplemented
13#41×D97C22196yescomplemented
14C2×Q8×D97C2212yescomplemented
15C22×D388C2219yescomplemented
16C22×C22⋊D97C22118yescomplemented
17C2×D4×D97C22118yescomplemented
18C2×D4×D97C221144yescomplemented
19C2×D4×D97C22136yescomplemented
20C22×C4×D97C2219yescomplemented
21H(5,𝔽2)×C97C22116yescomplemented
22C2×#41×C97C2216yescomplemented
23C22×D4×C97C2219yescomplemented
24C23×D97—424no—
25C23×D97C2319yes—
26C23×D97C23148yescomplemented
27C23×D97C23118yescomplemented
28#13095512C23148yescomplemented
29#13095511C231144yescomplemented
30#13095510C231144yescomplemented
31C2×Dic194C2316yescomplemented
32Q8×D97C23116yescomplemented
33C2×D388C23118yescomplemented
34C2×D388C23172yescomplemented
35C2×C22⋊D97C23136yescomplemented
36C2×C22⋊D97C231144yescomplemented
37D4×D97C231144yescomplemented
38D4×D97C231288yescomplemented
39C22×Dic97C2319yescomplemented
40C2×C4×D97C2316yes—
41C2×C4×D97C23172yescomplemented
42C2×C4×D97C23118yescomplemented
43C24×C97C2316yescomplemented
44#41×C97C23148yescomplemented
45C2×Q8×C97C2312yescomplemented
46C2×D4×C97C23118yescomplemented
47C2×D4×C97C23172yescomplemented
48C22×C4×C97C2319yescomplemented
49C23×C97—412no—
50C23×C97C2419yes—
51C23×C97C24124yescomplemented
52C22×D97—236no—
53C22×D97—496no—
54C22×D97—436no—
55C22×D97C24136yes—
56C22×D97C2419yes—
57C22×D97C2411yescharacteristic
58C22×D97C24172yescomplemented
59Q8×C97C2418yescomplemented
60D4×C97C24172yescomplemented
61C2×C4×C97C2416yes—
62C2×C4×C97C24136yescomplemented
63C22⋊D97C241144yescomplemented
64C2×Dic97C2416yes—
65C2×Dic97C24136yescomplemented
66D388C24172yescomplemented
67C4×D97C24124yes—
68C4×D97C24172yescomplemented
69Dic194C24124yescomplemented
70C22×C97—218no—
71C22×C97—424no—
72C22×C97C25118yes—
73C22×C97C2511yescharacteristic
74C2×D97—218no—
75C2×D97—272no—
76C2×D97—472no—
77C2×D97H(5,𝔽2)14yesdirect product factor
78C2×D97C25118yes—
79C2×D97C2512yes—
80C4×C97C25112yes—
81Dic97C25112yes—
82C2×C97—218no—
83C2×C97C2×H(5,𝔽2)12yescomplemented
84C2×C97C2611yescharacteristic
85D97—218no—
86D97C2×H(5,𝔽2)14yesdirect product factor
87C22×H(5,𝔽2)—971nomaximal, Sylow
88C97C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
89C2×H(5,𝔽2)—9732no—
90C2×H(5,𝔽2)—976no—
91C2×H(5,𝔽2)—979no—
92C2×H(5,𝔽2)D9711yescharacteristic, direct product factor
93C22×#41—976no—
94C23×D4—979no—
95C25—976no—
96H(5,𝔽2)—9796no—
97H(5,𝔽2)—97144no—
98H(5,𝔽2)C2×D97116yesdirect product factor
99C2×#41—9796no—
100C2×#41—976no—
101C2×#41—9718no—
102C2×#41—9718no—
103C2×#41C2×D9716yescomplemented
104C22×Q8—972no—
105C22×D4—9718no—
106C22×D4—9718no—
107C22×D4—9736no—
108C22×D4—97144no—
109C22×D4—979no—
110C22×D4C2×D9719yescomplemented
111C23×C4—979no—
112C24—9718no—
113C24—9748no—
114C24—979no—
115C24—38824no—
116C24C22×D9716yescomplemented
117#41—9748no—
118#41—97144no—
119#41—97144no—
120#41C22×D97148yescomplemented
121C2×Q8—9716no—
122C2×Q8—976no—
123C2×Q8C22×D9712yescomplemented
124C2×D4—9736no—
125C2×D4—97144no—
126C2×D4—97144no—
127C2×D4—97288no—
128C2×D4—9718no—
129C2×D4—9772no—
130C2×D4C22×D97118yescomplemented
131C2×D4C22×D97172yescomplemented
132C22×C4—979no—
133C22×C4—976no—
134C22×C4—9718no—
135C22×C4—9772no—
136C22×C4C22×D9719yescomplemented
137C23—412no—
138C23—971no—
139C23—9772no—
140C23—9736no—
141C23—979no—
142C23—38836no—
143C23—38896no—
144C23—19436no—
145C23C23×D9719yes—
146C23C23×D97124yescomplemented
147Q8—9724no—
148Q8C23×D9718yescomplemented
149D4—97144no—
150D4—9772no—
151D4C23×D97172yescomplemented
152C2×C4—9736no—
153C2×C4—976no—
154C2×C4—9724no—
155C2×C4—9772no—
156C2×C4C23×D9716yes—
157C2×C4C23×D97136yescomplemented
158C22—218no—
159C22—424no—
160C22—974no—
161C22—972no—
162C22—9718no—
163C22—38872no—
164C22—19472no—
165C22—19418no—
166C22C24×D97118yes—
167C22C24×D9711yescharacteristic
168C4—9712no—
169C4C24×D97112yes—
170C2—218no—
171C2—974no—
172C2—19418no—
173C2H(5,𝔽2)×D9712yesdirect product factor
174C2C25×D9711yescharacteristic
subgroupG/H
centerC22C24×D97
commutator, cosocleC2×C97C26
FrattiniC2C25×D97
FittingC2×H(5,𝔽2)×C97C2
socleC22×C97C25
hypercenterC2×H(5,𝔽2)D97
hypocenterC97C22×H(5,𝔽2)
LCSG ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#15690856C2semidirect
2. C2×#15690855C2semidirect
3. H(5,𝔽2)×D97C2direct (N × H)
4. C2×#41×D97C2semidirect
5. C22×D4×D97C2semidirect
6. C2×H(5,𝔽2)×C97C2semidirect
7. C24×D97C22semidirect
8. #15690856C22semidirect
9. #15690855C22semidirect
10. C2×#13095512C22semidirect
11. C2×#13095511C22semidirect
12. C2×#13095510C22semidirect
13. #41×D97C22semidirect
14. C2×Q8×D97C22semidirect
15. C22×D388C22semidirect
16. C22×C22⋊D97C22semidirect
17. C2×D4×D97C22semidirect
18. C2×D4×D97C22semidirect
19. C2×D4×D97C22semidirect
20. C22×C4×D97C22semidirect
21. H(5,𝔽2)×C97C22semidirect
22. C2×#41×C97C22semidirect
23. C22×D4×C97C22semidirect
26. C23×D97C23semidirect
27. C23×D97C23semidirect
28. #13095512C23semidirect
29. #13095511C23semidirect
30. #13095510C23semidirect
31. C2×Dic194C23semidirect
32. Q8×D97C23semidirect
33. C2×D388C23semidirect
34. C2×D388C23semidirect
35. C2×C22⋊D97C23semidirect
36. C2×C22⋊D97C23semidirect
37. D4×D97C23semidirect
38. D4×D97C23semidirect
39. C22×Dic97C23semidirect
41. C2×C4×D97C23semidirect
42. C2×C4×D97C23semidirect
43. C24×C97C23semidirect
44. #41×C97C23semidirect
45. C2×Q8×C97C23semidirect
46. C2×D4×C97C23semidirect
47. C2×D4×C97C23semidirect
48. C22×C4×C97C23semidirect
51. C23×C97C24semidirect
58. C22×D97C24semidirect
59. Q8×C97C24semidirect
60. D4×C97C24semidirect
62. C2×C4×C97C24semidirect
63. C22⋊D97C24semidirect
65. C2×Dic97C24semidirect
66. D388C24semidirect
68. C4×D97C24semidirect
69. Dic194C24semidirect
77. C2×D97H(5,𝔽2)direct (N × H)
83. C2×C97C2×H(5,𝔽2)semidirect
86. D97C2×H(5,𝔽2)direct (N × H)
88. C97C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D97direct (N × H)
98. H(5,𝔽2)C2×D97direct (N × H)
103. C2×#41C2×D97semidirect
110. C22×D4C2×D97semidirect
116. C24C22×D97semidirect
120. #41C22×D97semidirect
123. C2×Q8C22×D97semidirect
130. C2×D4C22×D97semidirect
131. C2×D4C22×D97semidirect
136. C22×C4C22×D97semidirect
146. C23C23×D97semidirect
148. Q8C23×D97semidirect
151. D4C23×D97semidirect
157. C2×C4C23×D97semidirect
173. C2H(5,𝔽2)×D97direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908562
C2×#156908552
H(5,𝔽2)×D972
C2×#41×D972
C22×D4×D972
C2×H(5,𝔽2)×C972
C22×H(5,𝔽2)97