Finite Group Databasebeta

← C2×H(5,𝔽2)×D101 #27185932

Subgroups — C2×H(5,𝔽2)×D101 #27185932

Order 12928.

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×#15692301C216yesmaximal, complemented
2C2×#15692300C219yesmaximal, complemented
3H(5,𝔽2)×D101C2132yesmaximal, direct product factor
4C2×#41×D101C216yesmaximal, complemented
5C22×D4×D101C219yesmaximal, complemented
6C2×H(5,𝔽2)×C101C211yesmaximal, characteristic, complemented
7C24×D101C2216yescomplemented
8#15692301C22196yescomplemented
9#15692300C221144yescomplemented
10C2×#13108075C2216yescomplemented
11C2×#13108074C22118yescomplemented
12C2×#13108073C22118yescomplemented
13#41×D101C22196yescomplemented
14C2×Q8×D101C2212yescomplemented
15C22×D404C2219yescomplemented
16C22×C22⋊D101C22118yescomplemented
17C2×D4×D101C22118yescomplemented
18C2×D4×D101C221144yescomplemented
19C2×D4×D101C22136yescomplemented
20C22×C4×D101C2219yescomplemented
21H(5,𝔽2)×C101C22116yescomplemented
22C2×#41×C101C2216yescomplemented
23C22×D4×C101C2219yescomplemented
24C23×D101—424no—
25C23×D101C2319yes—
26C23×D101C23148yescomplemented
27C23×D101C23118yescomplemented
28#13108075C23148yescomplemented
29#13108074C231144yescomplemented
30#13108073C231144yescomplemented
31C2×Dic202C2316yescomplemented
32Q8×D101C23116yescomplemented
33C2×D404C23118yescomplemented
34C2×D404C23172yescomplemented
35C2×C22⋊D101C23136yescomplemented
36C2×C22⋊D101C231144yescomplemented
37D4×D101C231144yescomplemented
38D4×D101C231288yescomplemented
39C22×Dic101C2319yescomplemented
40C2×C4×D101C2316yes—
41C2×C4×D101C23172yescomplemented
42C2×C4×D101C23118yescomplemented
43C24×C101C2316yescomplemented
44#41×C101C23148yescomplemented
45C2×Q8×C101C2312yescomplemented
46C2×D4×C101C23118yescomplemented
47C2×D4×C101C23172yescomplemented
48C22×C4×C101C2319yescomplemented
49C23×C101—412no—
50C23×C101C2419yes—
51C23×C101C24124yescomplemented
52C22×D101—236no—
53C22×D101—496no—
54C22×D101—436no—
55C22×D101C24136yes—
56C22×D101C2419yes—
57C22×D101C2411yescharacteristic
58C22×D101C24172yescomplemented
59Q8×C101C2418yescomplemented
60D4×C101C24172yescomplemented
61C2×C4×C101C2416yes—
62C2×C4×C101C24136yescomplemented
63C22⋊D101C241144yescomplemented
64C2×Dic101C2416yes—
65C2×Dic101C24136yescomplemented
66D404C24172yescomplemented
67C4×D101C24124yes—
68C4×D101C24172yescomplemented
69Dic202C24124yescomplemented
70C22×C101—218no—
71C22×C101—424no—
72C22×C101C25118yes—
73C22×C101C2511yescharacteristic
74C2×D101—218no—
75C2×D101—272no—
76C2×D101—472no—
77C2×D101H(5,𝔽2)14yesdirect product factor
78C2×D101C25118yes—
79C2×D101C2512yes—
80C4×C101C25112yes—
81Dic101C25112yes—
82C2×C101—218no—
83C2×C101C2×H(5,𝔽2)12yescomplemented
84C2×C101C2611yescharacteristic
85D101—218no—
86D101C2×H(5,𝔽2)14yesdirect product factor
87C22×H(5,𝔽2)—1011nomaximal, Sylow
88C101C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
89C2×H(5,𝔽2)—10132no—
90C2×H(5,𝔽2)—1016no—
91C2×H(5,𝔽2)—1019no—
92C2×H(5,𝔽2)D10111yescharacteristic, direct product factor
93C22×#41—1016no—
94C23×D4—1019no—
95C25—1016no—
96H(5,𝔽2)—10196no—
97H(5,𝔽2)—101144no—
98H(5,𝔽2)C2×D101116yesdirect product factor
99C2×#41—10196no—
100C2×#41—1016no—
101C2×#41—10118no—
102C2×#41—10118no—
103C2×#41C2×D10116yescomplemented
104C22×Q8—1012no—
105C22×D4—10118no—
106C22×D4—10118no—
107C22×D4—10136no—
108C22×D4—101144no—
109C22×D4—1019no—
110C22×D4C2×D10119yescomplemented
111C23×C4—1019no—
112C24—10118no—
113C24—10148no—
114C24—1019no—
115C24—40424no—
116C24C22×D10116yescomplemented
117#41—10148no—
118#41—101144no—
119#41—101144no—
120#41C22×D101148yescomplemented
121C2×Q8—10116no—
122C2×Q8—1016no—
123C2×Q8C22×D10112yescomplemented
124C2×D4—10136no—
125C2×D4—101144no—
126C2×D4—101144no—
127C2×D4—101288no—
128C2×D4—10118no—
129C2×D4—10172no—
130C2×D4C22×D101118yescomplemented
131C2×D4C22×D101172yescomplemented
132C22×C4—1019no—
133C22×C4—1016no—
134C22×C4—10118no—
135C22×C4—10172no—
136C22×C4C22×D10119yescomplemented
137C23—412no—
138C23—1011no—
139C23—10172no—
140C23—10136no—
141C23—1019no—
142C23—40436no—
143C23—40496no—
144C23—20236no—
145C23C23×D10119yes—
146C23C23×D101124yescomplemented
147Q8—10124no—
148Q8C23×D10118yescomplemented
149D4—101144no—
150D4—10172no—
151D4C23×D101172yescomplemented
152C2×C4—10136no—
153C2×C4—1016no—
154C2×C4—10124no—
155C2×C4—10172no—
156C2×C4C23×D10116yes—
157C2×C4C23×D101136yescomplemented
158C22—218no—
159C22—424no—
160C22—1014no—
161C22—1012no—
162C22—10118no—
163C22—40472no—
164C22—20272no—
165C22—20218no—
166C22C24×D101118yes—
167C22C24×D10111yescharacteristic
168C4—10112no—
169C4C24×D101112yes—
170C2—218no—
171C2—1014no—
172C2—20218no—
173C2H(5,𝔽2)×D10112yesdirect product factor
174C2C25×D10111yescharacteristic
subgroupG/H
centerC22C24×D101
commutator, cosocleC2×C101C26
FrattiniC2C25×D101
FittingC2×H(5,𝔽2)×C101C2
socleC22×C101C25
hypercenterC2×H(5,𝔽2)D101
hypocenterC101C22×H(5,𝔽2)
LCSG ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#15692301C2semidirect
2. C2×#15692300C2semidirect
3. H(5,𝔽2)×D101C2direct (N × H)
4. C2×#41×D101C2semidirect
5. C22×D4×D101C2semidirect
6. C2×H(5,𝔽2)×C101C2semidirect
7. C24×D101C22semidirect
8. #15692301C22semidirect
9. #15692300C22semidirect
10. C2×#13108075C22semidirect
11. C2×#13108074C22semidirect
12. C2×#13108073C22semidirect
13. #41×D101C22semidirect
14. C2×Q8×D101C22semidirect
15. C22×D404C22semidirect
16. C22×C22⋊D101C22semidirect
17. C2×D4×D101C22semidirect
18. C2×D4×D101C22semidirect
19. C2×D4×D101C22semidirect
20. C22×C4×D101C22semidirect
21. H(5,𝔽2)×C101C22semidirect
22. C2×#41×C101C22semidirect
23. C22×D4×C101C22semidirect
26. C23×D101C23semidirect
27. C23×D101C23semidirect
28. #13108075C23semidirect
29. #13108074C23semidirect
30. #13108073C23semidirect
31. C2×Dic202C23semidirect
32. Q8×D101C23semidirect
33. C2×D404C23semidirect
34. C2×D404C23semidirect
35. C2×C22⋊D101C23semidirect
36. C2×C22⋊D101C23semidirect
37. D4×D101C23semidirect
38. D4×D101C23semidirect
39. C22×Dic101C23semidirect
41. C2×C4×D101C23semidirect
42. C2×C4×D101C23semidirect
43. C24×C101C23semidirect
44. #41×C101C23semidirect
45. C2×Q8×C101C23semidirect
46. C2×D4×C101C23semidirect
47. C2×D4×C101C23semidirect
48. C22×C4×C101C23semidirect
51. C23×C101C24semidirect
58. C22×D101C24semidirect
59. Q8×C101C24semidirect
60. D4×C101C24semidirect
62. C2×C4×C101C24semidirect
63. C22⋊D101C24semidirect
65. C2×Dic101C24semidirect
66. D404C24semidirect
68. C4×D101C24semidirect
69. Dic202C24semidirect
77. C2×D101H(5,𝔽2)direct (N × H)
83. C2×C101C2×H(5,𝔽2)semidirect
86. D101C2×H(5,𝔽2)direct (N × H)
88. C101C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D101direct (N × H)
98. H(5,𝔽2)C2×D101direct (N × H)
103. C2×#41C2×D101semidirect
110. C22×D4C2×D101semidirect
116. C24C22×D101semidirect
120. #41C22×D101semidirect
123. C2×Q8C22×D101semidirect
130. C2×D4C22×D101semidirect
131. C2×D4C22×D101semidirect
136. C22×C4C22×D101semidirect
146. C23C23×D101semidirect
148. Q8C23×D101semidirect
151. D4C23×D101semidirect
157. C2×C4C23×D101semidirect
173. C2H(5,𝔽2)×D101direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156923012
C2×#156923002
H(5,𝔽2)×D1012
C2×#41×D1012
C22×D4×D1012
C2×H(5,𝔽2)×C1012
C22×H(5,𝔽2)101