Finite Group Databasebeta

← C2×H(5,𝔽2)×D269 #46825685

Subgroups — C2×H(5,𝔽2)×D269 #46825685

Order 34432.

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×#22575499C216yesmaximal, complemented
2C2×#22575498C219yesmaximal, complemented
3H(5,𝔽2)×D269C2132yesmaximal, direct product factor
4C2×#41×D269C216yesmaximal, complemented
5C22×D4×D269C219yesmaximal, complemented
6C2×H(5,𝔽2)×C269C211yesmaximal, characteristic, complemented
7C24×D269C2216yescomplemented
8#22575499C22196yescomplemented
9#22575498C221144yescomplemented
10C2×#16864403C2216yescomplemented
11C2×#16864402C22118yescomplemented
12C2×#16864401C22118yescomplemented
13#41×D269C22196yescomplemented
14C2×Q8×D269C2212yescomplemented
15C22×D1076C2219yescomplemented
16C22×C22⋊D269C22118yescomplemented
17C2×D4×D269C22118yescomplemented
18C2×D4×D269C221144yescomplemented
19C2×D4×D269C22136yescomplemented
20C22×C4×D269C2219yescomplemented
21H(5,𝔽2)×C269C22116yescomplemented
22C2×#41×C269C2216yescomplemented
23C22×D4×C269C2219yescomplemented
24C23×D269—424no—
25C23×D269C2319yes—
26C23×D269C23118yescomplemented
27C23×D269C23148yescomplemented
28#16864403C23148yescomplemented
29#16864402C231144yescomplemented
30#16864401C231144yescomplemented
31C2×Dic538C2316yescomplemented
32Q8×D269C23116yescomplemented
33C2×D1076C23118yescomplemented
34C2×D1076C23172yescomplemented
35C2×C22⋊D269C23136yescomplemented
36C2×C22⋊D269C231144yescomplemented
37D4×D269C231144yescomplemented
38D4×D269C231288yescomplemented
39C22×Dic269C2319yescomplemented
40C2×C4×D269C2316yes—
41C2×C4×D269C23118yescomplemented
42C2×C4×D269C23172yescomplemented
43C24×C269C2316yescomplemented
44#41×C269C23148yescomplemented
45C2×Q8×C269C2312yescomplemented
46C2×D4×C269C23118yescomplemented
47C2×D4×C269C23172yescomplemented
48C22×C4×C269C2319yescomplemented
49C22×D269—236no—
50C22×D269—496no—
51C22×D269—436no—
52C22×D269C24136yes—
53C22×D269C2419yes—
54C22×D269C2411yescharacteristic
55C22×D269C24172yescomplemented
56Dic538C24124yescomplemented
57D1076C24172yescomplemented
58C22⋊D269C241144yescomplemented
59C2×Dic269C2416yes—
60C2×Dic269C24136yescomplemented
61C4×D269C24124yes—
62C4×D269C24172yescomplemented
63C23×C269—412no—
64C23×C269C2419yes—
65C23×C269C24124yescomplemented
66Q8×C269C2418yescomplemented
67D4×C269C24172yescomplemented
68C2×C4×C269C2416yes—
69C2×C4×C269C24136yescomplemented
70C22×C269—218no—
71C22×C269—424no—
72C22×C269C25118yes—
73C22×C269C2511yescharacteristic
74C2×D269—218no—
75C2×D269—272no—
76C2×D269—472no—
77C2×D269H(5,𝔽2)14yesdirect product factor
78C2×D269C25118yes—
79C2×D269C2512yes—
80C4×C269C25112yes—
81Dic269C25112yes—
82C2×C269—218no—
83C2×C269C2×H(5,𝔽2)12yescomplemented
84C2×C269C2611yescharacteristic
85D269—218no—
86D269C2×H(5,𝔽2)14yesdirect product factor
87C269C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
88C22×H(5,𝔽2)—2691nomaximal, Sylow
89C2×H(5,𝔽2)—26932no—
90C2×H(5,𝔽2)—2696no—
91C2×H(5,𝔽2)—2699no—
92C2×H(5,𝔽2)D26911yescharacteristic, direct product factor
93C22×#41—2696no—
94C23×D4—2699no—
95C25—2696no—
96H(5,𝔽2)—26996no—
97H(5,𝔽2)—269144no—
98H(5,𝔽2)C2×D269116yesdirect product factor
99C2×#41—26996no—
100C2×#41—2696no—
101C2×#41—26918no—
102C2×#41—26918no—
103C2×#41C2×D26916yescomplemented
104C22×Q8—2692no—
105C22×D4—26918no—
106C22×D4—26918no—
107C22×D4—26936no—
108C22×D4—269144no—
109C22×D4—2699no—
110C22×D4C2×D26919yescomplemented
111C23×C4—2699no—
112C24—2699no—
113C24—26918no—
114C24—26948no—
115C24—107624no—
116C24C22×D26916yescomplemented
117#41—26948no—
118#41—269144no—
119#41—269144no—
120#41C22×D269148yescomplemented
121C2×Q8—26916no—
122C2×Q8—2696no—
123C2×Q8C22×D26912yescomplemented
124C2×D4—26936no—
125C2×D4—269144no—
126C2×D4—269144no—
127C2×D4—269288no—
128C2×D4—26918no—
129C2×D4—26972no—
130C2×D4C22×D269118yescomplemented
131C2×D4C22×D269172yescomplemented
132C22×C4—2699no—
133C22×C4—2696no—
134C22×C4—26918no—
135C22×C4—26972no—
136C22×C4C22×D26919yescomplemented
137C23—412no—
138C23—2691no—
139C23—26936no—
140C23—2699no—
141C23—26972no—
142C23—53836no—
143C23—107636no—
144C23—107696no—
145C23C23×D26919yes—
146C23C23×D269124yescomplemented
147Q8—26924no—
148Q8C23×D26918yescomplemented
149D4—269144no—
150D4—26972no—
151D4C23×D269172yescomplemented
152C2×C4—26936no—
153C2×C4—2696no—
154C2×C4—26924no—
155C2×C4—26972no—
156C2×C4C23×D26916yes—
157C2×C4C23×D269136yescomplemented
158C22—218no—
159C22—424no—
160C22—2694no—
161C22—2692no—
162C22—26918no—
163C22—53872no—
164C22—53818no—
165C22—107672no—
166C22C24×D269118yes—
167C22C24×D26911yescharacteristic
168C4—26912no—
169C4C24×D269112yes—
170C2—218no—
171C2—2694no—
172C2—53818no—
173C2H(5,𝔽2)×D26912yesdirect product factor
174C2C25×D26911yescharacteristic
subgroupG/H
centerC22C24×D269
commutator, cosocleC2×C269C26
FrattiniC2C25×D269
FittingC2×H(5,𝔽2)×C269C2
socleC22×C269C25
hypercenterC2×H(5,𝔽2)D269
hypocenterC269C22×H(5,𝔽2)
LCSG ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#22575499C2semidirect
2. C2×#22575498C2semidirect
3. H(5,𝔽2)×D269C2direct (N × H)
4. C2×#41×D269C2semidirect
5. C22×D4×D269C2semidirect
6. C2×H(5,𝔽2)×C269C2semidirect
7. C24×D269C22semidirect
8. #22575499C22semidirect
9. #22575498C22semidirect
10. C2×#16864403C22semidirect
11. C2×#16864402C22semidirect
12. C2×#16864401C22semidirect
13. #41×D269C22semidirect
14. C2×Q8×D269C22semidirect
15. C22×D1076C22semidirect
16. C22×C22⋊D269C22semidirect
17. C2×D4×D269C22semidirect
18. C2×D4×D269C22semidirect
19. C2×D4×D269C22semidirect
20. C22×C4×D269C22semidirect
21. H(5,𝔽2)×C269C22semidirect
22. C2×#41×C269C22semidirect
23. C22×D4×C269C22semidirect
26. C23×D269C23semidirect
27. C23×D269C23semidirect
28. #16864403C23semidirect
29. #16864402C23semidirect
30. #16864401C23semidirect
31. C2×Dic538C23semidirect
32. Q8×D269C23semidirect
33. C2×D1076C23semidirect
34. C2×D1076C23semidirect
35. C2×C22⋊D269C23semidirect
36. C2×C22⋊D269C23semidirect
37. D4×D269C23semidirect
38. D4×D269C23semidirect
39. C22×Dic269C23semidirect
41. C2×C4×D269C23semidirect
42. C2×C4×D269C23semidirect
43. C24×C269C23semidirect
44. #41×C269C23semidirect
45. C2×Q8×C269C23semidirect
46. C2×D4×C269C23semidirect
47. C2×D4×C269C23semidirect
48. C22×C4×C269C23semidirect
55. C22×D269C24semidirect
56. Dic538C24semidirect
57. D1076C24semidirect
58. C22⋊D269C24semidirect
60. C2×Dic269C24semidirect
62. C4×D269C24semidirect
65. C23×C269C24semidirect
66. Q8×C269C24semidirect
67. D4×C269C24semidirect
69. C2×C4×C269C24semidirect
77. C2×D269H(5,𝔽2)direct (N × H)
83. C2×C269C2×H(5,𝔽2)semidirect
86. D269C2×H(5,𝔽2)direct (N × H)
87. C269C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D269direct (N × H)
98. H(5,𝔽2)C2×D269direct (N × H)
103. C2×#41C2×D269semidirect
110. C22×D4C2×D269semidirect
116. C24C22×D269semidirect
120. #41C22×D269semidirect
123. C2×Q8C22×D269semidirect
130. C2×D4C22×D269semidirect
131. C2×D4C22×D269semidirect
136. C22×C4C22×D269semidirect
146. C23C23×D269semidirect
148. Q8C23×D269semidirect
151. D4C23×D269semidirect
157. C2×C4C23×D269semidirect
173. C2H(5,𝔽2)×D269direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225754992
C2×#225754982
H(5,𝔽2)×D2692
C2×#41×D2692
C22×D4×D2692
C2×H(5,𝔽2)×C2692
C22×H(5,𝔽2)269