Finite Group Databasebeta

← C2×H(5,𝔽2)×D31 #16860511

Subgroups — C2×H(5,𝔽2)×D31 #16860511

Order 3968.

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×#11758566C216yesmaximal, complemented
2C2×#11758561C219yesmaximal, complemented
3H(5,𝔽2)×D31C2132yesmaximal, direct product factor
4C2×#41×D31C216yesmaximal, complemented
5C22×D4×D31C219yesmaximal, complemented
6C2×C31×H(5,𝔽2)C211yesmaximal, characteristic, complemented
7C24×D31C2216yescomplemented
8C31×H(5,𝔽2)C22116yescomplemented
9C2×#41×C31C2216yescomplemented
10C22×D4×C31C2219yescomplemented
11C22×C22⋊D31C22118yescomplemented
12#11758566C22196yescomplemented
13#41×D31C22196yescomplemented
14C2×#92499C2216yescomplemented
15C2×Q8×D31C2212yescomplemented
16#11758561C221144yescomplemented
17C2×#92497C22118yescomplemented
18C2×D4×D31C22118yescomplemented
19C2×D4×D31C221144yescomplemented
20C2×D4×D31C22136yescomplemented
21C2×#92495C22118yescomplemented
22C22×D124C2219yescomplemented
23C22×C4×D31C2219yescomplemented
24C24×C31C2316yescomplemented
25C23×D31—424no—
26C23×D31C2319yes—
27C23×D31C23148yescomplemented
28C23×D31C23118yescomplemented
29#41×C31C23148yescomplemented
30C2×Q8×C31C2312yescomplemented
31C2×D4×C31C23118yescomplemented
32C2×D4×C31C23172yescomplemented
33C22×C4×C31C2319yescomplemented
34C2×C22⋊D31C23136yescomplemented
35C2×C22⋊D31C231144yescomplemented
36C22×Dic31C2319yescomplemented
37#92499C23148yescomplemented
38Q8×D31C23116yescomplemented
39#92497C231144yescomplemented
40D4×D31C231144yescomplemented
41D4×D31C231288yescomplemented
42#92495C231144yescomplemented
43C2×D124C23118yescomplemented
44C2×D124C23172yescomplemented
45C2×C4×D31C2316yes—
46C2×C4×D31C23172yescomplemented
47C2×C4×D31C23118yescomplemented
48C2×Dic62C2316yescomplemented
49C23×C31—412no—
50C23×C31C2419yes—
51C23×C31C24124yescomplemented
52C22×D31—236no—
53C22×D31—496no—
54C22×D31—436no—
55C22×D31C2419yes—
56C22×D31C24136yes—
57C22×D31C2411yescharacteristic
58C22×D31C24172yescomplemented
59Q8×C31C2418yescomplemented
60D4×C31C24172yescomplemented
61C2×C4×C31C2416yes—
62C2×C4×C31C24136yescomplemented
63C22⋊D31C241144yescomplemented
64C2×Dic31C2416yes—
65C2×Dic31C24136yescomplemented
66D124C24172yescomplemented
67C4×D31C24124yes—
68C4×D31C24172yescomplemented
69Dic62C24124yescomplemented
70C22×H(5,𝔽2)—311nomaximal, Sylow
71C22×C31—218no—
72C22×C31—424no—
73C22×C31C25118yes—
74C22×C31C2511yescharacteristic
75C2×D31—218no—
76C2×D31—272no—
77C2×D31—472no—
78C2×D31H(5,𝔽2)14yesdirect product factor
79C2×D31C25118yes—
80C2×D31C2512yes—
81C4×C31C25112yes—
82Dic31C25112yes—
83C2×H(5,𝔽2)—3132no—
84C2×H(5,𝔽2)—316no—
85C2×H(5,𝔽2)—319no—
86C2×H(5,𝔽2)D3111yescharacteristic, direct product factor
87C22×#41—316no—
88C23×D4—319no—
89C2×C31—218no—
90C2×C31C2×H(5,𝔽2)12yescomplemented
91C2×C31C2611yescharacteristic
92D31—218no—
93D31C2×H(5,𝔽2)14yesdirect product factor
94C25—316no—
95H(5,𝔽2)—3196no—
96H(5,𝔽2)—31144no—
97H(5,𝔽2)C2×D31116yesdirect product factor
98C2×#41—3196no—
99C2×#41—316no—
100C2×#41—3118no—
101C2×#41—3118no—
102C2×#41C2×D3116yescomplemented
103C22×Q8—312no—
104C22×D4—3118no—
105C22×D4—3118no—
106C22×D4—3136no—
107C22×D4—31144no—
108C22×D4—319no—
109C22×D4C2×D3119yescomplemented
110C23×C4—319no—
111C31C22×H(5,𝔽2)11yescharacteristic, complemented, Sylow
112C24—319no—
113C24—3118no—
114C24—3148no—
115C24—12424no—
116C24C22×D3116yescomplemented
117#41—3148no—
118#41—31144no—
119#41—31144no—
120#41C22×D31148yescomplemented
121C2×Q8—3116no—
122C2×Q8—316no—
123C2×Q8C22×D3112yescomplemented
124C2×D4—3136no—
125C2×D4—31144no—
126C2×D4—31144no—
127C2×D4—31288no—
128C2×D4—3118no—
129C2×D4—3172no—
130C2×D4C22×D31118yescomplemented
131C2×D4C22×D31172yescomplemented
132C22×C4—319no—
133C22×C4—316no—
134C22×C4—3118no—
135C22×C4—3172no—
136C22×C4C22×D3119yescomplemented
137C23—412no—
138C23—311no—
139C23—3136no—
140C23—319no—
141C23—3172no—
142C23—6236no—
143C23—12436no—
144C23—12496no—
145C23C23×D3119yes—
146C23C23×D31124yescomplemented
147Q8—3124no—
148Q8C23×D3118yescomplemented
149D4—31144no—
150D4—3172no—
151D4C23×D31172yescomplemented
152C2×C4—3136no—
153C2×C4—316no—
154C2×C4—3124no—
155C2×C4—3172no—
156C2×C4C23×D3116yes—
157C2×C4C23×D31136yescomplemented
158C22—218no—
159C22—424no—
160C22—314no—
161C22—312no—
162C22—3118no—
163C22—6272no—
164C22—6218no—
165C22—12472no—
166C22C24×D31118yes—
167C22C24×D3111yescharacteristic
168C4—3112no—
169C4C24×D31112yes—
170C2—218no—
171C2—314no—
172C2—6218no—
173C2H(5,𝔽2)×D3112yesdirect product factor
174C2C25×D3111yescharacteristic
subgroupG/H
centerC22C24×D31
commutator, cosocleC2×C31C26
FrattiniC2C25×D31
FittingC2×C31×H(5,𝔽2)C2
socleC22×C31C25
hypercenterC2×H(5,𝔽2)D31
hypocenterC31C22×H(5,𝔽2)
LCSG ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×#11758566C2semidirect
2. C2×#11758561C2semidirect
3. H(5,𝔽2)×D31C2direct (N × H)
4. C2×#41×D31C2semidirect
5. C22×D4×D31C2semidirect
6. C2×C31×H(5,𝔽2)C2semidirect
7. C24×D31C22semidirect
8. C31×H(5,𝔽2)C22semidirect
9. C2×#41×C31C22semidirect
10. C22×D4×C31C22semidirect
11. C22×C22⋊D31C22semidirect
12. #11758566C22semidirect
13. #41×D31C22semidirect
14. C2×#92499C22semidirect
15. C2×Q8×D31C22semidirect
16. #11758561C22semidirect
17. C2×#92497C22semidirect
18. C2×D4×D31C22semidirect
19. C2×D4×D31C22semidirect
20. C2×D4×D31C22semidirect
21. C2×#92495C22semidirect
22. C22×D124C22semidirect
23. C22×C4×D31C22semidirect
24. C24×C31C23semidirect
27. C23×D31C23semidirect
28. C23×D31C23semidirect
29. #41×C31C23semidirect
30. C2×Q8×C31C23semidirect
31. C2×D4×C31C23semidirect
32. C2×D4×C31C23semidirect
33. C22×C4×C31C23semidirect
34. C2×C22⋊D31C23semidirect
35. C2×C22⋊D31C23semidirect
36. C22×Dic31C23semidirect
37. #92499C23semidirect
38. Q8×D31C23semidirect
39. #92497C23semidirect
40. D4×D31C23semidirect
41. D4×D31C23semidirect
42. #92495C23semidirect
43. C2×D124C23semidirect
44. C2×D124C23semidirect
46. C2×C4×D31C23semidirect
47. C2×C4×D31C23semidirect
48. C2×Dic62C23semidirect
51. C23×C31C24semidirect
58. C22×D31C24semidirect
59. Q8×C31C24semidirect
60. D4×C31C24semidirect
62. C2×C4×C31C24semidirect
63. C22⋊D31C24semidirect
65. C2×Dic31C24semidirect
66. D124C24semidirect
68. C4×D31C24semidirect
69. Dic62C24semidirect
78. C2×D31H(5,𝔽2)direct (N × H)
86. C2×H(5,𝔽2)D31direct (N × H)
90. C2×C31C2×H(5,𝔽2)semidirect
93. D31C2×H(5,𝔽2)direct (N × H)
97. H(5,𝔽2)C2×D31direct (N × H)
102. C2×#41C2×D31semidirect
109. C22×D4C2×D31semidirect
111. C31C22×H(5,𝔽2)semidirect
116. C24C22×D31semidirect
120. #41C22×D31semidirect
123. C2×Q8C22×D31semidirect
130. C2×D4C22×D31semidirect
131. C2×D4C22×D31semidirect
136. C22×C4C22×D31semidirect
146. C23C23×D31semidirect
148. Q8C23×D31semidirect
151. D4C23×D31semidirect
157. C2×C4C23×D31semidirect
173. C2H(5,𝔽2)×D31direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117585662
C2×#117585612
H(5,𝔽2)×D312
C2×#41×D312
C22×D4×D312
C2×C31×H(5,𝔽2)2
C22×H(5,𝔽2)31