Finite Group Databasebeta

← #34111902

Subgroups — #34111902

Order 20096.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#23758079C212yesmaximal, complemented
2#23758074C212yesmaximal, complemented
3#23757540C212yesmaximal, complemented
4#23757533C212yesmaximal, complemented
5#23757450C211yesmaximal, characteristic, complemented
6#23757445C211yesmaximal, characteristic, complemented
7#23757440C212yesmaximal, complemented
8#23757396C212yesmaximal, complemented
9#496×C157C211yesmaximal, characteristic, complemented
10#19097819C2211yescharacteristic, complemented
11#19097816C2211yescharacteristic
12#19097816C2212yescomplemented
13#19097802C2212yescomplemented
14#19097800—22no—
15#19097798—22no—
16#19097797C2212yes—
17#19097797C2212yescomplemented
18#19097797C2212yescomplemented
19#19097795—22no—
20#19097795—22no—
21#19097793C2212yes—
22#19097792—22no—
23#19097790—22no—
24SD16×D157—22no—
25D8×D157—22no—
26#19097777C2212yescomplemented
27#19097777C2212yescomplemented
28#19097707C2212yescomplemented
29#19097699C2212yescomplemented
30#19097699C2212yescomplemented
31#19097690C2211yescharacteristic, complemented
32#19097688C2212yescomplemented
33#19097688C2211yescharacteristic, complemented
34C2×SD16×C157C2212yescomplemented
35C2×D8×C157C2212yescomplemented
36#127×C157C2211yescharacteristic, complemented
37#124×C157C2211yescharacteristic, complemented
38#97×C157C2211yescharacteristic, complemented
39#14578955—22no—
40#14578954—41no—
41#14578954C2312yes—
42#14578954C2312yes—
43#14578953—22no—
44#14578953—22no—
45#14578953—22no—
46Q8×D157—22no—
47C2×C22⋊D157—41no—
48C2×C22⋊D157—42no—
49D4×D157—22no—
50D4×D157—22no—
51D4×D157—22no—
52Dic628D412yescomplemented
53#14578944—22no—
54#14578943D412yescomplemented
55#14578943D412yescomplemented
56#14578942—22no—
57#14578942—22no—
58#14578942—22no—
59#14578941—22no—
60D1256D412yescomplemented
61#14578939—22no—
62#14578939—22no—
63#14578939—22no—
64#14578938C2312yes—
65#14578938C2312yes—
66#14578937—22no—
67#14578937—22no—
68C8×D157—22no—
69C8×D157—22no—
70C2×Q8×C157C2311yescharacteristic
71C2×D4×C157—41no—
72C2×D4×C157C2311yescharacteristic
73C2×D4×C157C2312yes—
74SD16×C157—22no—
75SD16×C157—22no—
76D8×C157—22no—
77D8×C157—22no—
78C2×C8×C157C2312yes—
79C22⋊C4×C157—41no—
80C42×C157C2311yescharacteristic
81C22×D157—42no—
82C22×D157—41no—
83Dic314—41no—
84Dic314C2×D412yes—
85Dic314C2×D412yes—
86D628—41no—
87D628C2×D412yes—
88D628C2×D412yes—
89C22⋊D157—22no—
90C22⋊D157—22no—
91C22⋊D157—42no—
92C22⋊D157—42no—
93C22⋊D157—42no—
94C2×Dic157—42no—
95C2×Dic157—41no—
96C4×D157—22no—
97C4×D157—22no—
98C4×D157—42no—
99#11929434—22no—
100#11929434—22no—
101C23×C157—21no—
102C23×C157—22no—
103Q8×C157—22no—
104D4×C157—22no—
105D4×C157—22no—
106D4×C157—22no—
107D4×C157—82no—
108C2×C4×C157—21no—
109C2×C4×C157—21no—
110C2×C4×C157C2411yescharacteristic
111C8×C157C2×D412yes—
112C8×C157C2×D412yes—
113C22×C157—41no—
114C22×C157—42no—
115C22×C157—82no—
116C22×C157—81no—
117C22×C157H(5,𝔽2)11yescharacteristic
118C2×D157—22no—
119C2×D157—22no—
120C2×D157—162no—
121C2×D157—161no—
122C4×C157—41no—
123C4×C157—41no—
124C4×C157C22×D412yes—
125Dic157—22no—
126Dic157—22no—
127C2×C157—21no—
128C2×C157—82no—
129C2×C157—81no—
130C2×C157D4211yescharacteristic
131D157—42no—
132D157—42no—
133C157#328811yescharacteristic, complemented, Sylow
134#3288—1571nomaximal, Sylow
135#577—1572no—
136#576—1572no—
137#496D15711yescharacteristic, complemented
138#472—1572no—
139#471—1572no—
140#455—1571no—
141#454—1571no—
142#453—1572no—
143#444—1572no—
1442−1+4—1571no—
145H(5,𝔽2)—1571no—
146H(5,𝔽2)—1572no—
147#137—3142no—
148#137—1572no—
149AGL(1,ℤ/8)—3142no—
150AGL(1,ℤ/8)—3142no—
151AGL(1,ℤ/8)—3142no—
152AGL(1,ℤ/8)—1572no—
153AGL(1,ℤ/8)—1572no—
154AGL(1,ℤ/8)—1572no—
155#135—3142no—
156#135—3142no—
157#135—1572no—
158C2×SD16—3142no—
159C2×SD16C2×D15712yescomplemented
160C2×D8—3142no—
161C2×D8C2×D15712yescomplemented
162#131—1572no—
163#131—1572no—
164#127C2×D15711yescharacteristic, complemented
165#124C2×D15711yescharacteristic, complemented
166#108—1572no—
167#104—1572no—
168#104—1572no—
169#101—1571no—
170#100—1572no—
171#100—1571no—
172#97C2×D15711yescharacteristic, complemented
173#41—3142no—
174#41—3142no—
175#41—3142no—
176#41—3142no—
177#41—6281no—
178#41—1572no—
179#41—1572no—
180C2×Q8—3142no—
181C2×Q8C22×D15711yescharacteristic
182C2×D4—41no—
183C2×D4—3142no—
184C2×D4—3142no—
185C2×D4—3142no—
186C2×D4—6281no—
187C2×D4—6282no—
188C2×D4C22×D15711yescharacteristic
189C2×D4C22×D15712yes—
190Q16—3142no—
191Q16—1572no—
192SD16—22no—
193SD16—22no—
194SD16—3142no—
195SD16—3142no—
196SD16—3142no—
197SD16—3142no—
198SD16—1572no—
199SD16—1572no—
200D8—22no—
201D8—22no—
202D8—3142no—
203D8—3142no—
204D8—3142no—
205D8—1572no—
206M16—3142no—
207M16—3142no—
208M16—1572no—
209M16—1572no—
210C2×C8—3142no—
211C2×C8—3142no—
212C2×C8C22×D15712yes—
213C22⋊C4—41no—
214C42C22×D15711yescharacteristic
215C23—21no—
216C23—22no—
217C23—6282no—
218C23—6281no—
219Q8—22no—
220Q8—6281no—
221Q8—1572no—
222Q8—1572no—
223D4—22no—
224D4—22no—
225D4—22no—
226D4—82no—
227D4—3142no—
228D4—3142no—
229D4—6281no—
230D4—6282no—
231D4—6282no—
232D4—6282no—
233D4—1572no—
234D4—1572no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—3142no—
238C2×C4—3142no—
239C2×C4—6282no—
240C2×C4—6281no—
241C2×C4—6282no—
242C2×C4C23×D15711yescharacteristic
243C8—3142no—
244C8—3142no—
245C8D4×D15712yes—
246C8D4×D15712yes—
247C22—41no—
248C22—42no—
249C22—82no—
250C22—81no—
251C22—3142no—
252C22—3142no—
253C22—25121no—
254C22—25122no—
255C22#1909781611yescharacteristic
256C4—41no—
257C4—41no—
258C4—3142no—
259C4—3142no—
260C4C2×D4×D15712yes—
261C2—21no—
262C2—82no—
263C2—81no—
264C2—6282no—
265C2—6282no—
266C2#2375790811yescharacteristic
subgroupG/H
centerC2#23757908
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#496×C157C2
socleC2×C157D42
hypercenter#496D157
hypocenterC157#3288
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #23758079C2semidirect
2. #23758074C2semidirect
3. #23757540C2semidirect
4. #23757533C2semidirect
5. #23757450C2semidirect
6. #23757445C2semidirect
7. #23757440C2semidirect
8. #23757396C2semidirect
9. #496×C157C2semidirect
10. #19097819C22semidirect
12. #19097816C22semidirect
13. #19097802C22semidirect
17. #19097797C22semidirect
18. #19097797C22semidirect
26. #19097777C22semidirect
27. #19097777C22semidirect
28. #19097707C22semidirect
29. #19097699C22semidirect
30. #19097699C22semidirect
31. #19097690C22semidirect
32. #19097688C22semidirect
33. #19097688C22semidirect
34. C2×SD16×C157C22semidirect
35. C2×D8×C157C22semidirect
36. #127×C157C22semidirect
37. #124×C157C22semidirect
38. #97×C157C22semidirect
52. Dic628D4semidirect
54. #14578943D4semidirect
55. #14578943D4semidirect
60. D1256D4semidirect
133. C157#3288semidirect
137. #496D157semidirect
159. C2×SD16C2×D157semidirect
161. C2×D8C2×D157semidirect
164. #127C2×D157semidirect
165. #124C2×D157semidirect
172. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580792
#237580742
#237575402
#237575332
#237574502
#237574452
#237574402
#237573962
#496×C1572
#3288157