Finite Group Databasebeta

← #34111936

Subgroups — #34111936

Order 20096.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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
1#23758078C212yesmaximal, complemented
2#23758075C212yesmaximal, complemented
3#23757650C211yesmaximal, characteristic, complemented
4#23757544C212yesmaximal, complemented
5#23757543C212yesmaximal, complemented
6#23757442C214yesmaximal, complemented
7#23757391C211yesmaximal, characteristic, complemented
8#443×C157C211yesmaximal, characteristic, complemented
9#19097817C2214yescomplemented
10#19097799—24no—
11#19097799C2214yescomplemented
12#19097798—24no—
13#19097796C2214yescomplemented
14#19097794—22no—
15#19097793—22no—
16#19097793C2211yescharacteristic
17C2×D1256—22no—
18C2×D1256C2212yescomplemented
19C2×D1256C2212yescomplemented
20D8×D157—22no—
21#19097777C2212yescomplemented
22#19097768C2212yescomplemented
23#19097707C2211yescharacteristic, complemented
24#19097699C2212yescomplemented
25#19097687C2214yescomplemented
26#135×C157C2211yescharacteristic
27#131×C157C2212yescomplemented
28#108×C157C2211yescharacteristic, complemented
29#104×C157C2212yescomplemented
30C4×C8×C157C2211yescharacteristic, complemented
31#14578955—24no—
32#14578954—24no—
33#14578954C2312yes—
34C2×D628—24no—
35C2×D628—42no—
36C2×D628C2314yes—
37D4×D157—24no—
38D4×D157—44no—
39Dic628D411yescharacteristic, complemented
40#14578943—24no—
41#14578943D412yescomplemented
42#14578941—24no—
43D1256—24no—
44D1256—22no—
45D1256—22no—
46D1256—22no—
47D1256—22no—
48D1256D411yescharacteristic, complemented
49#14578939—24no—
50#14578938C2312yes—
51#14578937—22no—
52C8×D157—22no—
53#41×C157C2312yes—
54Q16×C157D411yescharacteristic, complemented
55SD16×C157D412yescomplemented
56D8×C157D411yescharacteristic, complemented
57M16×C157—22no—
58M16×C157C2312yes—
59C2×C8×C157—22no—
60C2×C8×C157C2311yescharacteristic
61C2×C8×C157C2311yescharacteristic
62C42×C157C2311yescharacteristic
63C22×D157—24no—
64C22×D157—44no—
65Dic314C2×D412yes—
66D628—24no—
67D628—24no—
68D628—44no—
69D628—84no—
70D628C2×D412yes—
71C22⋊D157—22no—
72C22⋊D157—44no—
73C4×D157—22no—
74C4×D157—44no—
75#11929434—22no—
76Q8×C157C2×D412yes—
77D4×C157—22no—
78D4×C157C2×D412yes—
79C2×C4×C157—21no—
80C2×C4×C157—22no—
81C2×C4×C157C2411yescharacteristic
82C8×C157—22no—
83C8×C157C2×D411yescharacteristic
84C8×C157C2×D411yescharacteristic
85C8×C157C2×D411yescharacteristic
86C8×C157C2×D411yescharacteristic
87C22×C157—22no—
88C22×C157H(5,𝔽2)11yescharacteristic
89C2×D157—22no—
90C2×D157—44no—
91C2×D157—84no—
92C2×D157—164no—
93C4×C157—22no—
94C4×C157—22no—
95C4×C157C22×D411yescharacteristic
96C4×C157C22×D411yescharacteristic
97Dic157—22no—
98C2×C157—21no—
99C2×C157—42no—
100C2×C157D4211yescharacteristic
101D157—42no—
102D157—84no—
103C157#329211yescharacteristic, complemented, Sylow
104#3292—1571nomaximal, Sylow
105#576—1572no—
106#576—1572no—
107#493—1571no—
108#472—1572no—
109#472—1572no—
110#453—1574no—
111#443—1571no—
112#443D15711yescharacteristic, complemented
113H(5,𝔽2)—1574no—
114AGL(1,ℤ/8)—3144no—
115AGL(1,ℤ/8)—3144no—
116AGL(1,ℤ/8)—1574no—
117AGL(1,ℤ/8)—1574no—
118#135—3142no—
119#135—3142no—
120#135—1571no—
121#135C2×D15711yescharacteristic
122C2×D8—3142no—
123C2×D8—3142no—
124C2×D8—1572no—
125C2×D8—1572no—
126#131—1572no—
127#131C2×D15712yescomplemented
128#127—1572no—
129#108—1571no—
130#108C2×D15711yescharacteristic, complemented
131#104—1572no—
132#104C2×D15712yescomplemented
133#100—1574no—
134C4×C8C2×D15711yescharacteristic, complemented
135#41—3144no—
136#41—3144no—
137#41—1572no—
138#41C22×D15712yes—
139C2×D4—3144no—
140C2×D4—3144no—
141C2×D4—6282no—
142C2×D4—6284no—
143C2×D4—1574no—
144Q16—1571no—
145Q16D62811yescharacteristic, complemented
146SD16—3144no—
147SD16—3144no—
148SD16—1572no—
149SD16D62812yescomplemented
150D8—3144no—
151D8—3144no—
152D8—3142no—
153D8—3142no—
154D8—3142no—
155D8—3142no—
156D8—1571no—
157D8D62811yescharacteristic, complemented
158M16—22no—
159M16—3142no—
160M16—1572no—
161M16C22×D15712yes—
162C2×C8—22no—
163C2×C8—3142no—
164C2×C8C22×D15711yescharacteristic
165C2×C8C22×D15711yescharacteristic
166C42C22×D15711yescharacteristic
167C23—3144no—
168C23—6284no—
169Q8—1572no—
170Q8C2×D62812yes—
171D4—22no—
172D4—3142no—
173D4—3144no—
174D4—3144no—
175D4—6284no—
176D4—6284no—
177D4—1572no—
178D4—12564no—
179D4C2×D62812yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—3142no—
183C2×C4—6284no—
184C2×C4C23×D15711yescharacteristic
185C8—22no—
186C8—3142no—
187C8D4×D15711yescharacteristic
188C8D4×D15711yescharacteristic
189C8C2×D62811yescharacteristic
190C8C2×D62811yescharacteristic
191C22—22no—
192C22—3142no—
193C22—6284no—
194C22—25124no—
195C22—12564no—
196C22#1909781711yescharacteristic
197C4—22no—
198C4—22no—
199C4—3142no—
200C4C2×D4×D15711yescharacteristic
201C4C22×D62811yescharacteristic
202C2—21no—
203C2—42no—
204C2—6282no—
205C2—12564no—
206C2D4×D62811yescharacteristic
subgroupG/H
centerC2D4×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#443×C157C2
socleC2×C157D42
hypercenter#443D157
hypocenterC157#3292
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #23758078C2semidirect
2. #23758075C2semidirect
3. #23757650C2semidirect
4. #23757544C2semidirect
5. #23757543C2semidirect
6. #23757442C2semidirect
7. #23757391C2semidirect
8. #443×C157C2semidirect
9. #19097817C22semidirect
11. #19097799C22semidirect
13. #19097796C22semidirect
18. C2×D1256C22semidirect
19. C2×D1256C22semidirect
21. #19097777C22semidirect
22. #19097768C22semidirect
23. #19097707C22semidirect
24. #19097699C22semidirect
25. #19097687C22semidirect
27. #131×C157C22semidirect
28. #108×C157C22semidirect
29. #104×C157C22semidirect
30. C4×C8×C157C22semidirect
39. Dic628D4semidirect
41. #14578943D4semidirect
48. D1256D4semidirect
54. Q16×C157D4semidirect
55. SD16×C157D4semidirect
56. D8×C157D4semidirect
103. C157#3292semidirect
112. #443D157semidirect
127. #131C2×D157semidirect
130. #108C2×D157semidirect
132. #104C2×D157semidirect
134. C4×C8C2×D157semidirect
145. Q16D628semidirect
149. SD16D628semidirect
157. D8D628semidirect

Maximal subgroups

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

subgroupindex
#237580782
#237580752
#237576502
#237575442
#237575432
#237574422
#237573912
#443×C1572
#3292157