Finite Group Databasebeta

← C2×H(5,𝔽2)×D137 #31795589

Subgroups — C2×H(5,𝔽2)×D137 #31795589

Order 17536.

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×#17984134C216yesmaximal, complemented
2C2×#17984133C219yesmaximal, complemented
3H(5,𝔽2)×D137C2132yesmaximal, direct product factor
4C2×#41×D137C216yesmaximal, complemented
5C22×D4×D137C219yesmaximal, complemented
6C2×H(5,𝔽2)×C137C211yesmaximal, characteristic, complemented
7C24×D137C2216yescomplemented
8#17984134C22196yescomplemented
9#17984133C221144yescomplemented
10C2×#14555511C2216yescomplemented
11C2×#14555510C22118yescomplemented
12C2×#14555509C22118yescomplemented
13#41×D137C22196yescomplemented
14C2×Q8×D137C2212yescomplemented
15C22×D548C2219yescomplemented
16C22×C22⋊D137C22118yescomplemented
17C2×D4×D137C22118yescomplemented
18C2×D4×D137C221144yescomplemented
19C2×D4×D137C22136yescomplemented
20C22×C4×D137C2219yescomplemented
21H(5,𝔽2)×C137C22116yescomplemented
22C2×#41×C137C2216yescomplemented
23C22×D4×C137C2219yescomplemented
24C23×D137—424no—
25C23×D137C2319yes—
26C23×D137C23118yescomplemented
27C23×D137C23148yescomplemented
28#14555511C23148yescomplemented
29#14555510C231144yescomplemented
30#14555509C231144yescomplemented
31C2×Dic274C2316yescomplemented
32Q8×D137C23116yescomplemented
33C2×D548C23118yescomplemented
34C2×D548C23172yescomplemented
35C2×C22⋊D137C23136yescomplemented
36C2×C22⋊D137C231144yescomplemented
37D4×D137C231144yescomplemented
38D4×D137C231288yescomplemented
39C22×Dic137C2319yescomplemented
40C2×C4×D137C2316yes—
41C2×C4×D137C23118yescomplemented
42C2×C4×D137C23172yescomplemented
43C24×C137C2316yescomplemented
44#41×C137C23148yescomplemented
45C2×Q8×C137C2312yescomplemented
46C2×D4×C137C23118yescomplemented
47C2×D4×C137C23172yescomplemented
48C22×C4×C137C2319yescomplemented
49C22×D137—236no—
50C22×D137—496no—
51C22×D137—436no—
52C22×D137C24136yes—
53C22×D137C2419yes—
54C22×D137C2411yescharacteristic
55C22×D137C24172yescomplemented
56Dic274C24124yescomplemented
57D548C24172yescomplemented
58C22⋊D137C241144yescomplemented
59C2×Dic137C2416yes—
60C2×Dic137C24136yescomplemented
61C4×D137C24124yes—
62C4×D137C24172yescomplemented
63C23×C137—412no—
64C23×C137C2419yes—
65C23×C137C24124yescomplemented
66Q8×C137C2418yescomplemented
67D4×C137C24172yescomplemented
68C2×C4×C137C2416yes—
69C2×C4×C137C24136yescomplemented
70C22×C137—218no—
71C22×C137—424no—
72C22×C137C25118yes—
73C22×C137C2511yescharacteristic
74C2×D137—218no—
75C2×D137—272no—
76C2×D137—472no—
77C2×D137H(5,𝔽2)14yesdirect product factor
78C2×D137C25118yes—
79C2×D137C2512yes—
80C4×C137C25112yes—
81Dic137C25112yes—
82C2×C137—218no—
83C2×C137C2×H(5,𝔽2)12yescomplemented
84C2×C137C2611yescharacteristic
85D137—218no—
86D137C2×H(5,𝔽2)14yesdirect product factor
87C137C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
88C22×H(5,𝔽2)—1371nomaximal, Sylow
89C2×H(5,𝔽2)—13732no—
90C2×H(5,𝔽2)—1376no—
91C2×H(5,𝔽2)—1379no—
92C2×H(5,𝔽2)D13711yescharacteristic, direct product factor
93C22×#41—1376no—
94C23×D4—1379no—
95C25—1376no—
96H(5,𝔽2)—13796no—
97H(5,𝔽2)—137144no—
98H(5,𝔽2)C2×D137116yesdirect product factor
99C2×#41—13796no—
100C2×#41—1376no—
101C2×#41—13718no—
102C2×#41—13718no—
103C2×#41C2×D13716yescomplemented
104C22×Q8—1372no—
105C22×D4—13718no—
106C22×D4—13718no—
107C22×D4—13736no—
108C22×D4—137144no—
109C22×D4—1379no—
110C22×D4C2×D13719yescomplemented
111C23×C4—1379no—
112C24—54824no—
113C24—1379no—
114C24—13718no—
115C24—13748no—
116C24C22×D13716yescomplemented
117#41—13748no—
118#41—137144no—
119#41—137144no—
120#41C22×D137148yescomplemented
121C2×Q8—13716no—
122C2×Q8—1376no—
123C2×Q8C22×D13712yescomplemented
124C2×D4—13736no—
125C2×D4—137144no—
126C2×D4—137144no—
127C2×D4—137288no—
128C2×D4—13718no—
129C2×D4—13772no—
130C2×D4C22×D137118yescomplemented
131C2×D4C22×D137172yescomplemented
132C22×C4—1379no—
133C22×C4—1376no—
134C22×C4—13718no—
135C22×C4—13772no—
136C22×C4C22×D13719yescomplemented
137C23—412no—
138C23—27436no—
139C23—54836no—
140C23—54896no—
141C23—13736no—
142C23—1379no—
143C23—13772no—
144C23—1371no—
145C23C23×D13719yes—
146C23C23×D137124yescomplemented
147Q8—13724no—
148Q8C23×D13718yescomplemented
149D4—137144no—
150D4—13772no—
151D4C23×D137172yescomplemented
152C2×C4—13736no—
153C2×C4—1376no—
154C2×C4—13724no—
155C2×C4—13772no—
156C2×C4C23×D13716yes—
157C2×C4C23×D137136yescomplemented
158C22—218no—
159C22—424no—
160C22—27472no—
161C22—27418no—
162C22—54872no—
163C22—13718no—
164C22—1374no—
165C22—1372no—
166C22C24×D137118yes—
167C22C24×D13711yescharacteristic
168C4—13712no—
169C4C24×D137112yes—
170C2—218no—
171C2—27418no—
172C2—1374no—
173C2H(5,𝔽2)×D13712yesdirect product factor
174C2C25×D13711yescharacteristic
subgroupG/H
centerC22C24×D137
commutator, cosocleC2×C137C26
FrattiniC2C25×D137
FittingC2×H(5,𝔽2)×C137C2
socleC22×C137C25
hypercenterC2×H(5,𝔽2)D137
hypocenterC137C22×H(5,𝔽2)
LCSG ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#17984134C2semidirect
2. C2×#17984133C2semidirect
3. H(5,𝔽2)×D137C2direct (N × H)
4. C2×#41×D137C2semidirect
5. C22×D4×D137C2semidirect
6. C2×H(5,𝔽2)×C137C2semidirect
7. C24×D137C22semidirect
8. #17984134C22semidirect
9. #17984133C22semidirect
10. C2×#14555511C22semidirect
11. C2×#14555510C22semidirect
12. C2×#14555509C22semidirect
13. #41×D137C22semidirect
14. C2×Q8×D137C22semidirect
15. C22×D548C22semidirect
16. C22×C22⋊D137C22semidirect
17. C2×D4×D137C22semidirect
18. C2×D4×D137C22semidirect
19. C2×D4×D137C22semidirect
20. C22×C4×D137C22semidirect
21. H(5,𝔽2)×C137C22semidirect
22. C2×#41×C137C22semidirect
23. C22×D4×C137C22semidirect
26. C23×D137C23semidirect
27. C23×D137C23semidirect
28. #14555511C23semidirect
29. #14555510C23semidirect
30. #14555509C23semidirect
31. C2×Dic274C23semidirect
32. Q8×D137C23semidirect
33. C2×D548C23semidirect
34. C2×D548C23semidirect
35. C2×C22⋊D137C23semidirect
36. C2×C22⋊D137C23semidirect
37. D4×D137C23semidirect
38. D4×D137C23semidirect
39. C22×Dic137C23semidirect
41. C2×C4×D137C23semidirect
42. C2×C4×D137C23semidirect
43. C24×C137C23semidirect
44. #41×C137C23semidirect
45. C2×Q8×C137C23semidirect
46. C2×D4×C137C23semidirect
47. C2×D4×C137C23semidirect
48. C22×C4×C137C23semidirect
55. C22×D137C24semidirect
56. Dic274C24semidirect
57. D548C24semidirect
58. C22⋊D137C24semidirect
60. C2×Dic137C24semidirect
62. C4×D137C24semidirect
65. C23×C137C24semidirect
66. Q8×C137C24semidirect
67. D4×C137C24semidirect
69. C2×C4×C137C24semidirect
77. C2×D137H(5,𝔽2)direct (N × H)
83. C2×C137C2×H(5,𝔽2)semidirect
86. D137C2×H(5,𝔽2)direct (N × H)
87. C137C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D137direct (N × H)
98. H(5,𝔽2)C2×D137direct (N × H)
103. C2×#41C2×D137semidirect
110. C22×D4C2×D137semidirect
116. C24C22×D137semidirect
120. #41C22×D137semidirect
123. C2×Q8C22×D137semidirect
130. C2×D4C22×D137semidirect
131. C2×D4C22×D137semidirect
136. C22×C4C22×D137semidirect
146. C23C23×D137semidirect
148. Q8C23×D137semidirect
151. D4C23×D137semidirect
157. C2×C4C23×D137semidirect
173. C2H(5,𝔽2)×D137direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179841342
C2×#179841332
H(5,𝔽2)×D1372
C2×#41×D1372
C22×D4×D1372
C2×H(5,𝔽2)×C1372
C22×H(5,𝔽2)137