Finite Group Databasebeta

← #34110509

Subgroups — #34110509

Order 20096.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#23757829C211yesmaximal, characteristic, complemented
2#127×D157C211yesmaximal, characteristic
3#23757639C211yesmaximal, characteristic, complemented
4#23757638C211yesmaximal, characteristic, complemented
5#23757637C211yesmaximal, characteristic, complemented
6#23757618C214yesmaximal, complemented
7#23757276C214yesmaximal, complemented
8#23757225C211yesmaximal, characteristic, complemented
9#490×C157C211yesmaximal, characteristic, complemented
10C2×D4×D157—22no—
11C2×D4×D157C2212yes—
12C2×#14578937C2212yes—
13#19097770C2212yescomplemented
14#19097768C2211yescharacteristic
15#19097767—22no—
16#19097767C2212yes—
17#19097763C2211yescharacteristic, complemented
18#19097749—22no—
19#19097736—22no—
20#19097729C2212yes—
21C42×D157C2211yescharacteristic
22#19097693C2214yescomplemented
23#19097692C2214yescomplemented
24#19097691C2214yescomplemented
25#19097682C2211yescharacteristic, complemented
26C8×Dic157C2212yescomplemented
27#127×C157C2211yescharacteristic
28#125×C157C2211yescharacteristic, complemented
29#102×C157C2214yescomplemented
30#97×C157C2211yescharacteristic, complemented
31C23×D157—22no—
32C2×Dic314—22no—
33C2×D628—22no—
34C2×D628C2312yes—
35C2×C22⋊D157—22no—
36C2×C22⋊D157—24no—
37D4×D157—28no—
38D4×D157—48no—
39C2×C4×D157D412yes—
40C2×C4×D157C2311yescharacteristic
41#14578937—24no—
42C2×#11929434C2312yes—
43C4⋊Dic157C2312yes—
44#14578933—22no—
45#14578933—22no—
46#14578931—22no—
47C4×Dic157—22no—
48C4×Dic157D412yes—
49C4×Dic157C2311yescharacteristic
50C2×D4×C157—22no—
51C2×D4×C157C2312yes—
52C2×C8×C157C2312yes—
53C4⋊C4×C157—22no—
54C4⋊C4×C157C2312yes—
55C42×C157C2311yescharacteristic
56C22×D157—22no—
57C22×D157—48no—
58C22×D157—44no—
59C22×D157C2×D411yescharacteristic
60Dic314—44no—
61D628—24no—
62D628—44no—
63C22⋊D157—44no—
64C22⋊D157—48no—
65C2×Dic157—22no—
66C2×Dic157C2×D411yescharacteristic
67C2×Dic157C2×D412yes—
68C4×D157—24no—
69C4×D157#4114yes—
70#11929434—22no—
71C23×C157—22no—
72D4×C157—24no—
73D4×C157—44no—
74C2×C4×C157—22no—
75C2×C4×C157C2×D412yes—
76C2×C4×C157C2411yescharacteristic
77C8×C157—22no—
78C22×C157—42no—
79C22×C157—44no—
80C22×C157C22×D411yescharacteristic
81C2×D157—22no—
82C2×D157—42no—
83C2×D157—44no—
84C2×D157—88no—
85C2×D157#12412yes—
86C4×C157—22no—
87C4×C157—42no—
88C4×C157C2×#4112yes—
89Dic157—42no—
90Dic157#12412yes—
91Dic157AGL(1,ℤ/8)14yes—
92C2×C157—82no—
93C2×C157C2×#12411yescharacteristic
94C2×C157C2×AGL(1,ℤ/8)12yes—
95D157—22no—
96D157—82no—
97C157#313811yescharacteristic, complemented, Sylow
98#3138—1571nomaximal, Sylow
99#533—1571no—
100C2×#127—1571no—
101#490—1571no—
102#490—1571no—
103#490—1571no—
104#490D15711yescharacteristic, complemented
105#486—1574no—
106#418—1574no—
107C4×M16—1571no—
108C22×D4—3142no—
109C22×D4—1572no—
110C2×M16—1572no—
111C4⋊Q8—1572no—
112#127—3142no—
113#127—1572no—
114#127—1571no—
115#127C2×D15711yescharacteristic
116#125—1571no—
117#125C2×D15711yescharacteristic, complemented
118C22⋊Q8—3142no—
119C4×Q8—3142no—
120#117—1572no—
121C2×C42—1571no—
122#102—1574no—
123#102—1574no—
124#102—1574no—
125#102C2×D15714yescomplemented
126#97—1571no—
127#97C2×D15711yescharacteristic, complemented
128C4×C8—1572no—
129C24—3142no—
130C2×Q8—3142no—
131C2×D4—22no—
132C2×D4—3148no—
133C2×D4—3142no—
134C2×D4—3144no—
135C2×D4—3142no—
136C2×D4—6288no—
137C2×D4—1572no—
138C2×D4C22×D15712yes—
139C22×C4—1572no—
140C22×C4—1571no—
141M16—3144no—
142C2×C8—1572no—
143C2×C8C22×D15712yes—
144C4⋊C4—22no—
145C4⋊C4—3142no—
146C4⋊C4—3142no—
147C4⋊C4—1572no—
148C4⋊C4C22×D15712yes—
149C22⋊C4—3142no—
150C42—3142no—
151C42—1572no—
152C42—1571no—
153C42C22×D15711yescharacteristic
154C23—22no—
155C23—3142no—
156C23—6288no—
157C23—6284no—
158C23—1571no—
159Q8—6284no—
160D4—24no—
161D4—44no—
162D4—3144no—
163D4—6284no—
164D4—6284no—
165D4—6288no—
166C2×C4—22no—
167C2×C4—3144no—
168C2×C4—3142no—
169C2×C4—1571no—
170C2×C4—1572no—
171C2×C4—1574no—
172C2×C4D4×D15712yes—
173C2×C4C23×D15711yescharacteristic
174C8—22no—
175C8—3142no—
176C22—42no—
177C22—44no—
178C22—3142no—
179C22—6282no—
180C22—6284no—
181C22—1572no—
182C22—12568no—
183C22C2×D4×D15711yescharacteristic
184C4—22no—
185C4—42no—
186C4—6282no—
187C4—1572no—
188C4—1574no—
189C4#41×D15712yes—
190C2—82no—
191C2—3142no—
192C2—12562no—
193C2#124×D15711yescharacteristic
194C2AGL(1,ℤ/8)×D15712yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#490×C157C2
socleC22×C157C22×D4
prime radical#127×D157C2
hypercenter#490D157
hypocenterC157#3138
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #23757829C2semidirect
3. #23757639C2semidirect
4. #23757638C2semidirect
5. #23757637C2semidirect
6. #23757618C2semidirect
7. #23757276C2semidirect
8. #23757225C2semidirect
9. #490×C157C2semidirect
13. #19097770C22semidirect
17. #19097763C22semidirect
22. #19097693C22semidirect
23. #19097692C22semidirect
24. #19097691C22semidirect
25. #19097682C22semidirect
26. C8×Dic157C22semidirect
28. #125×C157C22semidirect
29. #102×C157C22semidirect
30. #97×C157C22semidirect
97. C157#3138semidirect
104. #490D157semidirect
117. #125C2×D157semidirect
125. #102C2×D157semidirect
127. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578292
#127×D1572
#237576392
#237576382
#237576372
#237576182
#237572762
#237572252
#490×C1572
#3138157