Finite Group Databasebeta

← C2×H(5,𝔽2)×D113 #28405374

Subgroups — C2×H(5,𝔽2)×D113 #28405374

Order 14464.

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×#16813668C216yesmaximal, complemented
2C2×#16813667C219yesmaximal, complemented
3H(5,𝔽2)×D113C2132yesmaximal, direct product factor
4C2×#41×D113C216yesmaximal, complemented
5C22×D4×D113C219yesmaximal, complemented
6C2×H(5,𝔽2)×C113C211yesmaximal, characteristic, complemented
7C24×D113C2216yescomplemented
8#16813668C22196yescomplemented
9#16813667C221144yescomplemented
10C2×#14268648C2216yescomplemented
11C2×#14268647C22118yescomplemented
12C2×#14268646C22118yescomplemented
13#41×D113C22196yescomplemented
14C2×Q8×D113C2212yescomplemented
15C22×D452C2219yescomplemented
16C22×C22⋊D113C22118yescomplemented
17C2×D4×D113C22118yescomplemented
18C2×D4×D113C221144yescomplemented
19C2×D4×D113C22136yescomplemented
20C22×C4×D113C2219yescomplemented
21H(5,𝔽2)×C113C22116yescomplemented
22C2×#41×C113C2216yescomplemented
23C22×D4×C113C2219yescomplemented
24C23×D113—424no—
25C23×D113C2319yes—
26C23×D113C23148yescomplemented
27C23×D113C23118yescomplemented
28#14268648C23148yescomplemented
29#14268647C231144yescomplemented
30#14268646C231144yescomplemented
31C2×Dic226C2316yescomplemented
32Q8×D113C23116yescomplemented
33C2×D452C23118yescomplemented
34C2×D452C23172yescomplemented
35C2×C22⋊D113C23136yescomplemented
36C2×C22⋊D113C231144yescomplemented
37D4×D113C231144yescomplemented
38D4×D113C231288yescomplemented
39C22×Dic113C2319yescomplemented
40C2×C4×D113C2316yes—
41C2×C4×D113C23172yescomplemented
42C2×C4×D113C23118yescomplemented
43C24×C113C2316yescomplemented
44#41×C113C23148yescomplemented
45C2×Q8×C113C2312yescomplemented
46C2×D4×C113C23118yescomplemented
47C2×D4×C113C23172yescomplemented
48C22×C4×C113C2319yescomplemented
49C23×C113—412no—
50C23×C113C2419yes—
51C23×C113C24124yescomplemented
52C22×D113—236no—
53C22×D113—496no—
54C22×D113—436no—
55C22×D113C24136yes—
56C22×D113C2419yes—
57C22×D113C2411yescharacteristic
58C22×D113C24172yescomplemented
59Q8×C113C2418yescomplemented
60D4×C113C24172yescomplemented
61C2×C4×C113C2416yes—
62C2×C4×C113C24136yescomplemented
63C22⋊D113C241144yescomplemented
64C2×Dic113C2416yes—
65C2×Dic113C24136yescomplemented
66D452C24172yescomplemented
67C4×D113C24124yes—
68C4×D113C24172yescomplemented
69Dic226C24124yescomplemented
70C22×C113—218no—
71C22×C113—424no—
72C22×C113C25118yes—
73C22×C113C2511yescharacteristic
74C2×D113—218no—
75C2×D113—272no—
76C2×D113—472no—
77C2×D113H(5,𝔽2)14yesdirect product factor
78C2×D113C25118yes—
79C2×D113C2512yes—
80C4×C113C25112yes—
81Dic113C25112yes—
82C2×C113—218no—
83C2×C113C2×H(5,𝔽2)12yescomplemented
84C2×C113C2611yescharacteristic
85D113—218no—
86D113C2×H(5,𝔽2)14yesdirect product factor
87C22×H(5,𝔽2)—1131nomaximal, Sylow
88C113C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
89C2×H(5,𝔽2)—11332no—
90C2×H(5,𝔽2)—1136no—
91C2×H(5,𝔽2)—1139no—
92C2×H(5,𝔽2)D11311yescharacteristic, direct product factor
93C22×#41—1136no—
94C23×D4—1139no—
95C25—1136no—
96H(5,𝔽2)—11396no—
97H(5,𝔽2)—113144no—
98H(5,𝔽2)C2×D113116yesdirect product factor
99C2×#41—11396no—
100C2×#41—1136no—
101C2×#41—11318no—
102C2×#41—11318no—
103C2×#41C2×D11316yescomplemented
104C22×Q8—1132no—
105C22×D4—11318no—
106C22×D4—11318no—
107C22×D4—11336no—
108C22×D4—113144no—
109C22×D4—1139no—
110C22×D4C2×D11319yescomplemented
111C23×C4—1139no—
112C24—11318no—
113C24—11348no—
114C24—1139no—
115C24—45224no—
116C24C22×D11316yescomplemented
117#41—11348no—
118#41—113144no—
119#41—113144no—
120#41C22×D113148yescomplemented
121C2×Q8—11316no—
122C2×Q8—1136no—
123C2×Q8C22×D11312yescomplemented
124C2×D4—11336no—
125C2×D4—113144no—
126C2×D4—113144no—
127C2×D4—113288no—
128C2×D4—11318no—
129C2×D4—11372no—
130C2×D4C22×D113118yescomplemented
131C2×D4C22×D113172yescomplemented
132C22×C4—1139no—
133C22×C4—1136no—
134C22×C4—11318no—
135C22×C4—11372no—
136C22×C4C22×D11319yescomplemented
137C23—412no—
138C23—1131no—
139C23—11372no—
140C23—11336no—
141C23—1139no—
142C23—45236no—
143C23—45296no—
144C23—22636no—
145C23C23×D11319yes—
146C23C23×D113124yescomplemented
147Q8—11324no—
148Q8C23×D11318yescomplemented
149D4—113144no—
150D4—11372no—
151D4C23×D113172yescomplemented
152C2×C4—11336no—
153C2×C4—1136no—
154C2×C4—11324no—
155C2×C4—11372no—
156C2×C4C23×D11316yes—
157C2×C4C23×D113136yescomplemented
158C22—218no—
159C22—424no—
160C22—1134no—
161C22—1132no—
162C22—11318no—
163C22—45272no—
164C22—22672no—
165C22—22618no—
166C22C24×D113118yes—
167C22C24×D11311yescharacteristic
168C4—11312no—
169C4C24×D113112yes—
170C2—218no—
171C2—1134no—
172C2—22618no—
173C2H(5,𝔽2)×D11312yesdirect product factor
174C2C25×D11311yescharacteristic
subgroupG/H
centerC22C24×D113
commutator, cosocleC2×C113C26
FrattiniC2C25×D113
FittingC2×H(5,𝔽2)×C113C2
socleC22×C113C25
hypercenterC2×H(5,𝔽2)D113
hypocenterC113C22×H(5,𝔽2)
LCSG ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16813668C2semidirect
2. C2×#16813667C2semidirect
3. H(5,𝔽2)×D113C2direct (N × H)
4. C2×#41×D113C2semidirect
5. C22×D4×D113C2semidirect
6. C2×H(5,𝔽2)×C113C2semidirect
7. C24×D113C22semidirect
8. #16813668C22semidirect
9. #16813667C22semidirect
10. C2×#14268648C22semidirect
11. C2×#14268647C22semidirect
12. C2×#14268646C22semidirect
13. #41×D113C22semidirect
14. C2×Q8×D113C22semidirect
15. C22×D452C22semidirect
16. C22×C22⋊D113C22semidirect
17. C2×D4×D113C22semidirect
18. C2×D4×D113C22semidirect
19. C2×D4×D113C22semidirect
20. C22×C4×D113C22semidirect
21. H(5,𝔽2)×C113C22semidirect
22. C2×#41×C113C22semidirect
23. C22×D4×C113C22semidirect
26. C23×D113C23semidirect
27. C23×D113C23semidirect
28. #14268648C23semidirect
29. #14268647C23semidirect
30. #14268646C23semidirect
31. C2×Dic226C23semidirect
32. Q8×D113C23semidirect
33. C2×D452C23semidirect
34. C2×D452C23semidirect
35. C2×C22⋊D113C23semidirect
36. C2×C22⋊D113C23semidirect
37. D4×D113C23semidirect
38. D4×D113C23semidirect
39. C22×Dic113C23semidirect
41. C2×C4×D113C23semidirect
42. C2×C4×D113C23semidirect
43. C24×C113C23semidirect
44. #41×C113C23semidirect
45. C2×Q8×C113C23semidirect
46. C2×D4×C113C23semidirect
47. C2×D4×C113C23semidirect
48. C22×C4×C113C23semidirect
51. C23×C113C24semidirect
58. C22×D113C24semidirect
59. Q8×C113C24semidirect
60. D4×C113C24semidirect
62. C2×C4×C113C24semidirect
63. C22⋊D113C24semidirect
65. C2×Dic113C24semidirect
66. D452C24semidirect
68. C4×D113C24semidirect
69. Dic226C24semidirect
77. C2×D113H(5,𝔽2)direct (N × H)
83. C2×C113C2×H(5,𝔽2)semidirect
86. D113C2×H(5,𝔽2)direct (N × H)
88. C113C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D113direct (N × H)
98. H(5,𝔽2)C2×D113direct (N × H)
103. C2×#41C2×D113semidirect
110. C22×D4C2×D113semidirect
116. C24C22×D113semidirect
120. #41C22×D113semidirect
123. C2×Q8C22×D113semidirect
130. C2×D4C22×D113semidirect
131. C2×D4C22×D113semidirect
136. C22×C4C22×D113semidirect
146. C23C23×D113semidirect
148. Q8C23×D113semidirect
151. D4C23×D113semidirect
157. C2×C4C23×D113semidirect
173. C2H(5,𝔽2)×D113direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168136682
C2×#168136672
H(5,𝔽2)×D1132
C2×#41×D1132
C22×D4×D1132
C2×H(5,𝔽2)×C1132
C22×H(5,𝔽2)113