Finite Group Databasebeta

← C2×H(5,𝔽2)×D131 #30680761

Subgroups — C2×H(5,𝔽2)×D131 #30680761

Order 16768.

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×#16862292C216yesmaximal, complemented
2C2×#16862291C219yesmaximal, complemented
3H(5,𝔽2)×D131C2132yesmaximal, direct product factor
4C2×#41×D131C216yesmaximal, complemented
5C22×D4×D131C219yesmaximal, complemented
6C2×H(5,𝔽2)×C131C211yesmaximal, characteristic, complemented
7C24×D131C2216yescomplemented
8#16862292C22196yescomplemented
9#16862291C221144yescomplemented
10C2×#14523514C2216yescomplemented
11C2×#14523513C22118yescomplemented
12C2×#14523512C22118yescomplemented
13#41×D131C22196yescomplemented
14C2×Q8×D131C2212yescomplemented
15C22×D524C2219yescomplemented
16C22×C22⋊D131C22118yescomplemented
17C2×D4×D131C22118yescomplemented
18C2×D4×D131C221144yescomplemented
19C2×D4×D131C22136yescomplemented
20C22×C4×D131C2219yescomplemented
21H(5,𝔽2)×C131C22116yescomplemented
22C2×#41×C131C2216yescomplemented
23C22×D4×C131C2219yescomplemented
24C23×D131—424no—
25C23×D131C2319yes—
26C23×D131C23118yescomplemented
27C23×D131C23148yescomplemented
28#14523514C23148yescomplemented
29#14523513C231144yescomplemented
30#14523512C231144yescomplemented
31C2×Dic262C2316yescomplemented
32Q8×D131C23116yescomplemented
33C2×D524C23118yescomplemented
34C2×D524C23172yescomplemented
35C2×C22⋊D131C23136yescomplemented
36C2×C22⋊D131C231144yescomplemented
37D4×D131C231144yescomplemented
38D4×D131C231288yescomplemented
39C22×Dic131C2319yescomplemented
40C2×C4×D131C2316yes—
41C2×C4×D131C23118yescomplemented
42C2×C4×D131C23172yescomplemented
43C24×C131C2316yescomplemented
44#41×C131C23148yescomplemented
45C2×Q8×C131C2312yescomplemented
46C2×D4×C131C23118yescomplemented
47C2×D4×C131C23172yescomplemented
48C22×C4×C131C2319yescomplemented
49C22×D131—236no—
50C22×D131—496no—
51C22×D131—436no—
52C22×D131C24136yes—
53C22×D131C2419yes—
54C22×D131C2411yescharacteristic
55C22×D131C24172yescomplemented
56Dic262C24124yescomplemented
57D524C24172yescomplemented
58C22⋊D131C241144yescomplemented
59C2×Dic131C2416yes—
60C2×Dic131C24136yescomplemented
61C4×D131C24124yes—
62C4×D131C24172yescomplemented
63C23×C131—412no—
64C23×C131C2419yes—
65C23×C131C24124yescomplemented
66Q8×C131C2418yescomplemented
67D4×C131C24172yescomplemented
68C2×C4×C131C2416yes—
69C2×C4×C131C24136yescomplemented
70C22×C131—218no—
71C22×C131—424no—
72C22×C131C25118yes—
73C22×C131C2511yescharacteristic
74C2×D131—218no—
75C2×D131—272no—
76C2×D131—472no—
77C2×D131H(5,𝔽2)14yesdirect product factor
78C2×D131C25118yes—
79C2×D131C2512yes—
80C4×C131C25112yes—
81Dic131C25112yes—
82C2×C131—218no—
83C2×C131C2×H(5,𝔽2)12yescomplemented
84C2×C131C2611yescharacteristic
85D131—218no—
86D131C2×H(5,𝔽2)14yesdirect product factor
87C131C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
88C22×H(5,𝔽2)—1311nomaximal, Sylow
89C2×H(5,𝔽2)—13132no—
90C2×H(5,𝔽2)—1316no—
91C2×H(5,𝔽2)—1319no—
92C2×H(5,𝔽2)D13111yescharacteristic, direct product factor
93C22×#41—1316no—
94C23×D4—1319no—
95C25—1316no—
96H(5,𝔽2)—13196no—
97H(5,𝔽2)—131144no—
98H(5,𝔽2)C2×D131116yesdirect product factor
99C2×#41—13196no—
100C2×#41—1316no—
101C2×#41—13118no—
102C2×#41—13118no—
103C2×#41C2×D13116yescomplemented
104C22×Q8—1312no—
105C22×D4—13118no—
106C22×D4—13118no—
107C22×D4—13136no—
108C22×D4—131144no—
109C22×D4—1319no—
110C22×D4C2×D13119yescomplemented
111C23×C4—1319no—
112C24—52424no—
113C24—1319no—
114C24—13118no—
115C24—13148no—
116C24C22×D13116yescomplemented
117#41—13148no—
118#41—131144no—
119#41—131144no—
120#41C22×D131148yescomplemented
121C2×Q8—13116no—
122C2×Q8—1316no—
123C2×Q8C22×D13112yescomplemented
124C2×D4—13136no—
125C2×D4—131144no—
126C2×D4—131144no—
127C2×D4—131288no—
128C2×D4—13118no—
129C2×D4—13172no—
130C2×D4C22×D131118yescomplemented
131C2×D4C22×D131172yescomplemented
132C22×C4—1319no—
133C22×C4—1316no—
134C22×C4—13118no—
135C22×C4—13172no—
136C22×C4C22×D13119yescomplemented
137C23—412no—
138C23—26236no—
139C23—52436no—
140C23—52496no—
141C23—13136no—
142C23—1319no—
143C23—13172no—
144C23—1311no—
145C23C23×D13119yes—
146C23C23×D131124yescomplemented
147Q8—13124no—
148Q8C23×D13118yescomplemented
149D4—131144no—
150D4—13172no—
151D4C23×D131172yescomplemented
152C2×C4—13136no—
153C2×C4—1316no—
154C2×C4—13124no—
155C2×C4—13172no—
156C2×C4C23×D13116yes—
157C2×C4C23×D131136yescomplemented
158C22—218no—
159C22—424no—
160C22—26272no—
161C22—26218no—
162C22—52472no—
163C22—13118no—
164C22—1314no—
165C22—1312no—
166C22C24×D131118yes—
167C22C24×D13111yescharacteristic
168C4—13112no—
169C4C24×D131112yes—
170C2—218no—
171C2—26218no—
172C2—1314no—
173C2H(5,𝔽2)×D13112yesdirect product factor
174C2C25×D13111yescharacteristic
subgroupG/H
centerC22C24×D131
commutator, cosocleC2×C131C26
FrattiniC2C25×D131
FittingC2×H(5,𝔽2)×C131C2
socleC22×C131C25
hypercenterC2×H(5,𝔽2)D131
hypocenterC131C22×H(5,𝔽2)
LCSG ▷ C2×C131 ▷ C131
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16862292C2semidirect
2. C2×#16862291C2semidirect
3. H(5,𝔽2)×D131C2direct (N × H)
4. C2×#41×D131C2semidirect
5. C22×D4×D131C2semidirect
6. C2×H(5,𝔽2)×C131C2semidirect
7. C24×D131C22semidirect
8. #16862292C22semidirect
9. #16862291C22semidirect
10. C2×#14523514C22semidirect
11. C2×#14523513C22semidirect
12. C2×#14523512C22semidirect
13. #41×D131C22semidirect
14. C2×Q8×D131C22semidirect
15. C22×D524C22semidirect
16. C22×C22⋊D131C22semidirect
17. C2×D4×D131C22semidirect
18. C2×D4×D131C22semidirect
19. C2×D4×D131C22semidirect
20. C22×C4×D131C22semidirect
21. H(5,𝔽2)×C131C22semidirect
22. C2×#41×C131C22semidirect
23. C22×D4×C131C22semidirect
26. C23×D131C23semidirect
27. C23×D131C23semidirect
28. #14523514C23semidirect
29. #14523513C23semidirect
30. #14523512C23semidirect
31. C2×Dic262C23semidirect
32. Q8×D131C23semidirect
33. C2×D524C23semidirect
34. C2×D524C23semidirect
35. C2×C22⋊D131C23semidirect
36. C2×C22⋊D131C23semidirect
37. D4×D131C23semidirect
38. D4×D131C23semidirect
39. C22×Dic131C23semidirect
41. C2×C4×D131C23semidirect
42. C2×C4×D131C23semidirect
43. C24×C131C23semidirect
44. #41×C131C23semidirect
45. C2×Q8×C131C23semidirect
46. C2×D4×C131C23semidirect
47. C2×D4×C131C23semidirect
48. C22×C4×C131C23semidirect
55. C22×D131C24semidirect
56. Dic262C24semidirect
57. D524C24semidirect
58. C22⋊D131C24semidirect
60. C2×Dic131C24semidirect
62. C4×D131C24semidirect
65. C23×C131C24semidirect
66. Q8×C131C24semidirect
67. D4×C131C24semidirect
69. C2×C4×C131C24semidirect
77. C2×D131H(5,𝔽2)direct (N × H)
83. C2×C131C2×H(5,𝔽2)semidirect
86. D131C2×H(5,𝔽2)direct (N × H)
87. C131C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D131direct (N × H)
98. H(5,𝔽2)C2×D131direct (N × H)
103. C2×#41C2×D131semidirect
110. C22×D4C2×D131semidirect
116. C24C22×D131semidirect
120. #41C22×D131semidirect
123. C2×Q8C22×D131semidirect
130. C2×D4C22×D131semidirect
131. C2×D4C22×D131semidirect
136. C22×C4C22×D131semidirect
146. C23C23×D131semidirect
148. Q8C23×D131semidirect
151. D4C23×D131semidirect
157. C2×C4C23×D131semidirect
173. C2H(5,𝔽2)×D131direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168622922
C2×#168622912
H(5,𝔽2)×D1312
C2×#41×D1312
C22×D4×D1312
C2×H(5,𝔽2)×C1312
C22×H(5,𝔽2)131