Finite Group Databasebeta

← C2×H(5,𝔽2)×D251 #44565165

Subgroups — C2×H(5,𝔽2)×D251 #44565165

Order 32128.

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×#22548761C216yesmaximal, complemented
2C2×#22548760C219yesmaximal, complemented
3H(5,𝔽2)×D251C2132yesmaximal, direct product factor
4C2×#41×D251C216yesmaximal, complemented
5C22×D4×D251C219yesmaximal, complemented
6C2×H(5,𝔽2)×C251C211yesmaximal, characteristic, complemented
7C24×D251C2216yescomplemented
8#22548761C22196yescomplemented
9#22548760C221144yescomplemented
10C2×#16860811C2216yescomplemented
11C2×#16860810C22118yescomplemented
12C2×#16860809C22118yescomplemented
13#41×D251C22196yescomplemented
14C2×Q8×D251C2212yescomplemented
15C22×D1004C2219yescomplemented
16C22×C22⋊D251C22118yescomplemented
17C2×D4×D251C22118yescomplemented
18C2×D4×D251C221144yescomplemented
19C2×D4×D251C22136yescomplemented
20C22×C4×D251C2219yescomplemented
21H(5,𝔽2)×C251C22116yescomplemented
22C2×#41×C251C2216yescomplemented
23C22×D4×C251C2219yescomplemented
24C23×D251—424no—
25C23×D251C2319yes—
26C23×D251C23118yescomplemented
27C23×D251C23148yescomplemented
28#16860811C23148yescomplemented
29#16860810C231144yescomplemented
30#16860809C231144yescomplemented
31C2×Dic502C2316yescomplemented
32Q8×D251C23116yescomplemented
33C2×D1004C23118yescomplemented
34C2×D1004C23172yescomplemented
35C2×C22⋊D251C23136yescomplemented
36C2×C22⋊D251C231144yescomplemented
37D4×D251C231144yescomplemented
38D4×D251C231288yescomplemented
39C22×Dic251C2319yescomplemented
40C2×C4×D251C2316yes—
41C2×C4×D251C23118yescomplemented
42C2×C4×D251C23172yescomplemented
43C24×C251C2316yescomplemented
44#41×C251C23148yescomplemented
45C2×Q8×C251C2312yescomplemented
46C2×D4×C251C23118yescomplemented
47C2×D4×C251C23172yescomplemented
48C22×C4×C251C2319yescomplemented
49C22×D251—236no—
50C22×D251—496no—
51C22×D251—436no—
52C22×D251C24136yes—
53C22×D251C2419yes—
54C22×D251C2411yescharacteristic
55C22×D251C24172yescomplemented
56Dic502C24124yescomplemented
57D1004C24172yescomplemented
58C22⋊D251C241144yescomplemented
59C2×Dic251C2416yes—
60C2×Dic251C24136yescomplemented
61C4×D251C24124yes—
62C4×D251C24172yescomplemented
63C23×C251—412no—
64C23×C251C2419yes—
65C23×C251C24124yescomplemented
66Q8×C251C2418yescomplemented
67D4×C251C24172yescomplemented
68C2×C4×C251C2416yes—
69C2×C4×C251C24136yescomplemented
70C22×C251—218no—
71C22×C251—424no—
72C22×C251C25118yes—
73C22×C251C2511yescharacteristic
74C2×D251—218no—
75C2×D251—272no—
76C2×D251—472no—
77C2×D251H(5,𝔽2)14yesdirect product factor
78C2×D251C25118yes—
79C2×D251C2512yes—
80C4×C251C25112yes—
81Dic251C25112yes—
82C2×C251—218no—
83C2×C251C2×H(5,𝔽2)12yescomplemented
84C2×C251C2611yescharacteristic
85D251—218no—
86D251C2×H(5,𝔽2)14yesdirect product factor
87C251C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
88C22×H(5,𝔽2)—2511nomaximal, Sylow
89C2×H(5,𝔽2)—25132no—
90C2×H(5,𝔽2)—2516no—
91C2×H(5,𝔽2)—2519no—
92C2×H(5,𝔽2)D25111yescharacteristic, direct product factor
93C22×#41—2516no—
94C23×D4—2519no—
95C25—2516no—
96H(5,𝔽2)—25196no—
97H(5,𝔽2)—251144no—
98H(5,𝔽2)C2×D251116yesdirect product factor
99C2×#41—25196no—
100C2×#41—2516no—
101C2×#41—25118no—
102C2×#41—25118no—
103C2×#41C2×D25116yescomplemented
104C22×Q8—2512no—
105C22×D4—25118no—
106C22×D4—25118no—
107C22×D4—25136no—
108C22×D4—251144no—
109C22×D4—2519no—
110C22×D4C2×D25119yescomplemented
111C23×C4—2519no—
112C24—100424no—
113C24—25118no—
114C24—25148no—
115C24—2519no—
116C24C22×D25116yescomplemented
117#41—25148no—
118#41—251144no—
119#41—251144no—
120#41C22×D251148yescomplemented
121C2×Q8—25116no—
122C2×Q8—2516no—
123C2×Q8C22×D25112yescomplemented
124C2×D4—25136no—
125C2×D4—251144no—
126C2×D4—251144no—
127C2×D4—251288no—
128C2×D4—25118no—
129C2×D4—25172no—
130C2×D4C22×D251118yescomplemented
131C2×D4C22×D251172yescomplemented
132C22×C4—2519no—
133C22×C4—2516no—
134C22×C4—25118no—
135C22×C4—25172no—
136C22×C4C22×D25119yescomplemented
137C23—412no—
138C23—100436no—
139C23—100496no—
140C23—50236no—
141C23—25172no—
142C23—25136no—
143C23—2519no—
144C23—2511no—
145C23C23×D25119yes—
146C23C23×D251124yescomplemented
147Q8—25124no—
148Q8C23×D25118yescomplemented
149D4—251144no—
150D4—25172no—
151D4C23×D251172yescomplemented
152C2×C4—25136no—
153C2×C4—2516no—
154C2×C4—25124no—
155C2×C4—25172no—
156C2×C4C23×D25116yes—
157C2×C4C23×D251136yescomplemented
158C22—218no—
159C22—424no—
160C22—100472no—
161C22—50272no—
162C22—50218no—
163C22—25118no—
164C22—2514no—
165C22—2512no—
166C22C24×D251118yes—
167C22C24×D25111yescharacteristic
168C4—25112no—
169C4C24×D251112yes—
170C2—218no—
171C2—50218no—
172C2—2514no—
173C2H(5,𝔽2)×D25112yesdirect product factor
174C2C25×D25111yescharacteristic
subgroupG/H
centerC22C24×D251
commutator, cosocleC2×C251C26
FrattiniC2C25×D251
FittingC2×H(5,𝔽2)×C251C2
socleC22×C251C25
hypercenterC2×H(5,𝔽2)D251
hypocenterC251C22×H(5,𝔽2)
LCSG ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#22548761C2semidirect
2. C2×#22548760C2semidirect
3. H(5,𝔽2)×D251C2direct (N × H)
4. C2×#41×D251C2semidirect
5. C22×D4×D251C2semidirect
6. C2×H(5,𝔽2)×C251C2semidirect
7. C24×D251C22semidirect
8. #22548761C22semidirect
9. #22548760C22semidirect
10. C2×#16860811C22semidirect
11. C2×#16860810C22semidirect
12. C2×#16860809C22semidirect
13. #41×D251C22semidirect
14. C2×Q8×D251C22semidirect
15. C22×D1004C22semidirect
16. C22×C22⋊D251C22semidirect
17. C2×D4×D251C22semidirect
18. C2×D4×D251C22semidirect
19. C2×D4×D251C22semidirect
20. C22×C4×D251C22semidirect
21. H(5,𝔽2)×C251C22semidirect
22. C2×#41×C251C22semidirect
23. C22×D4×C251C22semidirect
26. C23×D251C23semidirect
27. C23×D251C23semidirect
28. #16860811C23semidirect
29. #16860810C23semidirect
30. #16860809C23semidirect
31. C2×Dic502C23semidirect
32. Q8×D251C23semidirect
33. C2×D1004C23semidirect
34. C2×D1004C23semidirect
35. C2×C22⋊D251C23semidirect
36. C2×C22⋊D251C23semidirect
37. D4×D251C23semidirect
38. D4×D251C23semidirect
39. C22×Dic251C23semidirect
41. C2×C4×D251C23semidirect
42. C2×C4×D251C23semidirect
43. C24×C251C23semidirect
44. #41×C251C23semidirect
45. C2×Q8×C251C23semidirect
46. C2×D4×C251C23semidirect
47. C2×D4×C251C23semidirect
48. C22×C4×C251C23semidirect
55. C22×D251C24semidirect
56. Dic502C24semidirect
57. D1004C24semidirect
58. C22⋊D251C24semidirect
60. C2×Dic251C24semidirect
62. C4×D251C24semidirect
65. C23×C251C24semidirect
66. Q8×C251C24semidirect
67. D4×C251C24semidirect
69. C2×C4×C251C24semidirect
77. C2×D251H(5,𝔽2)direct (N × H)
83. C2×C251C2×H(5,𝔽2)semidirect
86. D251C2×H(5,𝔽2)direct (N × H)
87. C251C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D251direct (N × H)
98. H(5,𝔽2)C2×D251direct (N × H)
103. C2×#41C2×D251semidirect
110. C22×D4C2×D251semidirect
116. C24C22×D251semidirect
120. #41C22×D251semidirect
123. C2×Q8C22×D251semidirect
130. C2×D4C22×D251semidirect
131. C2×D4C22×D251semidirect
136. C22×C4C22×D251semidirect
146. C23C23×D251semidirect
148. Q8C23×D251semidirect
151. D4C23×D251semidirect
157. C2×C4C23×D251semidirect
173. C2H(5,𝔽2)×D251direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487612
C2×#225487602
H(5,𝔽2)×D2512
C2×#41×D2512
C22×D4×D2512
C2×H(5,𝔽2)×C2512
C22×H(5,𝔽2)251