Finite Group Databasebeta

← C2×H(5,𝔽2)×D127 #30653530

Subgroups — C2×H(5,𝔽2)×D127 #30653530

Order 16256.

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×#16861255C216yesmaximal, complemented
2C2×#16861254C219yesmaximal, complemented
3H(5,𝔽2)×D127C2132yesmaximal, direct product factor
4C2×#41×D127C216yesmaximal, complemented
5C22×D4×D127C219yesmaximal, complemented
6C2×H(5,𝔽2)×C127C211yesmaximal, characteristic, complemented
7C24×D127C2216yescomplemented
8#16861255C22196yescomplemented
9#16861254C221144yescomplemented
10C2×#14523249C2216yescomplemented
11C2×#14523248C22118yescomplemented
12C2×#14523247C22118yescomplemented
13#41×D127C22196yescomplemented
14C2×Q8×D127C2212yescomplemented
15C22×D508C2219yescomplemented
16C22×C22⋊D127C22118yescomplemented
17C2×D4×D127C22118yescomplemented
18C2×D4×D127C221144yescomplemented
19C2×D4×D127C22136yescomplemented
20C22×C4×D127C2219yescomplemented
21H(5,𝔽2)×C127C22116yescomplemented
22C2×#41×C127C2216yescomplemented
23C22×D4×C127C2219yescomplemented
24C23×D127—424no—
25C23×D127C2319yes—
26C23×D127C23118yescomplemented
27C23×D127C23148yescomplemented
28#14523249C23148yescomplemented
29#14523248C231144yescomplemented
30#14523247C231144yescomplemented
31C2×Dic254C2316yescomplemented
32Q8×D127C23116yescomplemented
33C2×D508C23118yescomplemented
34C2×D508C23172yescomplemented
35C2×C22⋊D127C23136yescomplemented
36C2×C22⋊D127C231144yescomplemented
37D4×D127C231144yescomplemented
38D4×D127C231288yescomplemented
39C22×Dic127C2319yescomplemented
40C2×C4×D127C2316yes—
41C2×C4×D127C23118yescomplemented
42C2×C4×D127C23172yescomplemented
43C24×C127C2316yescomplemented
44#41×C127C23148yescomplemented
45C2×Q8×C127C2312yescomplemented
46C2×D4×C127C23118yescomplemented
47C2×D4×C127C23172yescomplemented
48C22×C4×C127C2319yescomplemented
49C22×D127—236no—
50C22×D127—496no—
51C22×D127—436no—
52C22×D127C24136yes—
53C22×D127C2419yes—
54C22×D127C2411yescharacteristic
55C22×D127C24172yescomplemented
56Dic254C24124yescomplemented
57D508C24172yescomplemented
58C22⋊D127C241144yescomplemented
59C2×Dic127C2416yes—
60C2×Dic127C24136yescomplemented
61C4×D127C24124yes—
62C4×D127C24172yescomplemented
63C23×C127—412no—
64C23×C127C2419yes—
65C23×C127C24124yescomplemented
66Q8×C127C2418yescomplemented
67D4×C127C24172yescomplemented
68C2×C4×C127C2416yes—
69C2×C4×C127C24136yescomplemented
70C22×C127—218no—
71C22×C127—424no—
72C22×C127C25118yes—
73C22×C127C2511yescharacteristic
74C2×D127—218no—
75C2×D127—272no—
76C2×D127—472no—
77C2×D127H(5,𝔽2)14yesdirect product factor
78C2×D127C25118yes—
79C2×D127C2512yes—
80C4×C127C25112yes—
81Dic127C25112yes—
82C2×C127—218no—
83C2×C127C2×H(5,𝔽2)12yescomplemented
84C2×C127C2611yescharacteristic
85D127—218no—
86D127C2×H(5,𝔽2)14yesdirect product factor
87C22×H(5,𝔽2)—1271nomaximal, Sylow
88C127C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
89C2×H(5,𝔽2)—12732no—
90C2×H(5,𝔽2)—1276no—
91C2×H(5,𝔽2)—1279no—
92C2×H(5,𝔽2)D12711yescharacteristic, direct product factor
93C22×#41—1276no—
94C23×D4—1279no—
95C25—1276no—
96H(5,𝔽2)—12796no—
97H(5,𝔽2)—127144no—
98H(5,𝔽2)C2×D127116yesdirect product factor
99C2×#41—12796no—
100C2×#41—1276no—
101C2×#41—12718no—
102C2×#41—12718no—
103C2×#41C2×D12716yescomplemented
104C22×Q8—1272no—
105C22×D4—12718no—
106C22×D4—12718no—
107C22×D4—12736no—
108C22×D4—127144no—
109C22×D4—1279no—
110C22×D4C2×D12719yescomplemented
111C23×C4—1279no—
112C24—12718no—
113C24—12748no—
114C24—1279no—
115C24—50824no—
116C24C22×D12716yescomplemented
117#41—12748no—
118#41—127144no—
119#41—127144no—
120#41C22×D127148yescomplemented
121C2×Q8—12716no—
122C2×Q8—1276no—
123C2×Q8C22×D12712yescomplemented
124C2×D4—12736no—
125C2×D4—127144no—
126C2×D4—127144no—
127C2×D4—127288no—
128C2×D4—12718no—
129C2×D4—12772no—
130C2×D4C22×D127118yescomplemented
131C2×D4C22×D127172yescomplemented
132C22×C4—1279no—
133C22×C4—1276no—
134C22×C4—12718no—
135C22×C4—12772no—
136C22×C4C22×D12719yescomplemented
137C23—412no—
138C23—1271no—
139C23—12772no—
140C23—12736no—
141C23—1279no—
142C23—50836no—
143C23—50896no—
144C23—25436no—
145C23C23×D12719yes—
146C23C23×D127124yescomplemented
147Q8—12724no—
148Q8C23×D12718yescomplemented
149D4—127144no—
150D4—12772no—
151D4C23×D127172yescomplemented
152C2×C4—12736no—
153C2×C4—1276no—
154C2×C4—12724no—
155C2×C4—12772no—
156C2×C4C23×D12716yes—
157C2×C4C23×D127136yescomplemented
158C22—218no—
159C22—424no—
160C22—1274no—
161C22—1272no—
162C22—12718no—
163C22—50872no—
164C22—25472no—
165C22—25418no—
166C22C24×D127118yes—
167C22C24×D12711yescharacteristic
168C4—12712no—
169C4C24×D127112yes—
170C2—218no—
171C2—1274no—
172C2—25418no—
173C2H(5,𝔽2)×D12712yesdirect product factor
174C2C25×D12711yescharacteristic
subgroupG/H
centerC22C24×D127
commutator, cosocleC2×C127C26
FrattiniC2C25×D127
FittingC2×H(5,𝔽2)×C127C2
socleC22×C127C25
hypercenterC2×H(5,𝔽2)D127
hypocenterC127C22×H(5,𝔽2)
LCSG ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#16861255C2semidirect
2. C2×#16861254C2semidirect
3. H(5,𝔽2)×D127C2direct (N × H)
4. C2×#41×D127C2semidirect
5. C22×D4×D127C2semidirect
6. C2×H(5,𝔽2)×C127C2semidirect
7. C24×D127C22semidirect
8. #16861255C22semidirect
9. #16861254C22semidirect
10. C2×#14523249C22semidirect
11. C2×#14523248C22semidirect
12. C2×#14523247C22semidirect
13. #41×D127C22semidirect
14. C2×Q8×D127C22semidirect
15. C22×D508C22semidirect
16. C22×C22⋊D127C22semidirect
17. C2×D4×D127C22semidirect
18. C2×D4×D127C22semidirect
19. C2×D4×D127C22semidirect
20. C22×C4×D127C22semidirect
21. H(5,𝔽2)×C127C22semidirect
22. C2×#41×C127C22semidirect
23. C22×D4×C127C22semidirect
26. C23×D127C23semidirect
27. C23×D127C23semidirect
28. #14523249C23semidirect
29. #14523248C23semidirect
30. #14523247C23semidirect
31. C2×Dic254C23semidirect
32. Q8×D127C23semidirect
33. C2×D508C23semidirect
34. C2×D508C23semidirect
35. C2×C22⋊D127C23semidirect
36. C2×C22⋊D127C23semidirect
37. D4×D127C23semidirect
38. D4×D127C23semidirect
39. C22×Dic127C23semidirect
41. C2×C4×D127C23semidirect
42. C2×C4×D127C23semidirect
43. C24×C127C23semidirect
44. #41×C127C23semidirect
45. C2×Q8×C127C23semidirect
46. C2×D4×C127C23semidirect
47. C2×D4×C127C23semidirect
48. C22×C4×C127C23semidirect
55. C22×D127C24semidirect
56. Dic254C24semidirect
57. D508C24semidirect
58. C22⋊D127C24semidirect
60. C2×Dic127C24semidirect
62. C4×D127C24semidirect
65. C23×C127C24semidirect
66. Q8×C127C24semidirect
67. D4×C127C24semidirect
69. C2×C4×C127C24semidirect
77. C2×D127H(5,𝔽2)direct (N × H)
83. C2×C127C2×H(5,𝔽2)semidirect
86. D127C2×H(5,𝔽2)direct (N × H)
88. C127C22×H(5,𝔽2)semidirect
92. C2×H(5,𝔽2)D127direct (N × H)
98. H(5,𝔽2)C2×D127direct (N × H)
103. C2×#41C2×D127semidirect
110. C22×D4C2×D127semidirect
116. C24C22×D127semidirect
120. #41C22×D127semidirect
123. C2×Q8C22×D127semidirect
130. C2×D4C22×D127semidirect
131. C2×D4C22×D127semidirect
136. C22×C4C22×D127semidirect
146. C23C23×D127semidirect
148. Q8C23×D127semidirect
151. D4C23×D127semidirect
157. C2×C4C23×D127semidirect
173. C2H(5,𝔽2)×D127direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168612552
C2×#168612542
H(5,𝔽2)×D1272
C2×#41×D1272
C22×D4×D1272
C2×H(5,𝔽2)×C1272
C22×H(5,𝔽2)127