Finite Group Databasebeta

← #34111921

Subgroups — #34111921

Order 20096.

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

normal subgroups only

This group has 454 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#23758086C211yesmaximal, characteristic, complemented
2#23758079C212yesmaximal, complemented
3#23758074C211yesmaximal, characteristic, complemented
4#23757545C211yesmaximal, characteristic, complemented
5#23757540C211yesmaximal, characteristic, complemented
6#23757533C212yesmaximal, complemented
7#23757450C212yesmaximal, complemented
8#23757445C212yesmaximal, complemented
9#23757391C211yesmaximal, characteristic, complemented
10#23757391C211yesmaximal, characteristic, complemented
11#495×C157C211yesmaximal, characteristic, complemented
12#19097819C2212yescomplemented
13#19097816C2212yes—
14#19097803—22no—
15#19097802C2212yes—
16#19097802C2212yescomplemented
17#19097800—22no—
18#19097798—22no—
19#19097797C2212yes—
20#19097797C2212yescomplemented
21#19097795—22no—
22#19097794—21no—
23#19097793C2211yescharacteristic
24#19097793C2211yescharacteristic
25#19097792—22no—
26#19097790—21no—
27Q16×D157—21no—
28SD16×D157—22no—
29D8×D157—21no—
30#19097777C2211yescharacteristic
31#19097777C2211yescharacteristic, complemented
32#19097777C2212yescomplemented
33#19097707C2211yescharacteristic, complemented
34#19097707C2211yescharacteristic, complemented
35#19097699C2211yescharacteristic
36#19097699C2211yescharacteristic, complemented
37#19097699C2212yescomplemented
38#19097690C2212yescomplemented
39#19097688C2212yescomplemented
40C2×Q16×C157C2211yescharacteristic, complemented
41C2×SD16×C157C2212yescomplemented
42C2×D8×C157C2211yescharacteristic
43#124×C157C2212yescomplemented
44C4×C8×C157C2211yescharacteristic, complemented
45#14578955—22no—
46#14578955—22no—
47#14578954—42no—
48#14578954C2311yescharacteristic
49#14578954C2312yes—
50#14578954C2311yescharacteristic
51#14578953—22no—
52#14578953—22no—
53Q8×D157—22no—
54Q8×D157—22no—
55C2×C22⋊D157—42no—
56D4×D157—22no—
57D4×D157—22no—
58Dic628D411yescharacteristic
59Dic628D411yescharacteristic
60#14578944—22no—
61#14578944—22no—
62#14578943D411yescharacteristic
63#14578943D411yescharacteristic
64#14578943D412yes—
65#14578942—22no—
66#14578942—22no—
67#14578941—22no—
68#14578941—22no—
69D1256D411yescharacteristic
70D1256D411yescharacteristic
71#14578939—22no—
72#14578939—22no—
73#14578938C2311yescharacteristic
74#14578938C2312yes—
75#14578938C2311yescharacteristic
76#14578937—21no—
77#14578937—22no—
78#14578937—21no—
79C8×D157—21no—
80C8×D157—22no—
81C8×D157—21no—
82C2×Q8×C157C2312yes—
83C2×D4×C157C2312yes—
84Q16×C157—21no—
85Q16×C157—21no—
86SD16×C157—22no—
87SD16×C157—22no—
88D8×C157—21no—
89D8×C157—21no—
90C2×C8×C157C2311yescharacteristic
91C2×C8×C157C2311yescharacteristic
92C22⋊C4×C157—42no—
93C42×C157C2311yescharacteristic
94C22×D157—42no—
95Dic314—42no—
96Dic314C2×D411yescharacteristic
97Dic314C2×D412yes—
98Dic314C2×D411yescharacteristic
99D628—42no—
100D628C2×D411yescharacteristic
101D628C2×D412yes—
102D628C2×D411yescharacteristic
103C22⋊D157—21no—
104C22⋊D157—22no—
105C22⋊D157—21no—
106C22⋊D157—42no—
107C22⋊D157—42no—
108C2×Dic157—42no—
109C4×D157—21no—
110C4×D157—22no—
111C4×D157—21no—
112C4×D157—42no—
113C4×D157—42no—
114#11929434—21no—
115#11929434—22no—
116#11929434—21no—
117C23×C157—22no—
118Q8×C157—22no—
119Q8×C157—22no—
120D4×C157—22no—
121D4×C157—22no—
122C2×C4×C157—22no—
123C2×C4×C157—21no—
124C2×C4×C157C2411yescharacteristic
125C8×C157C2×D411yescharacteristic
126C8×C157C2×D411yescharacteristic
127C8×C157C2×D411yescharacteristic
128C8×C157C2×D411yescharacteristic
129C22×C157—42no—
130C22×C157—82no—
131C22×C157H(5,𝔽2)11yescharacteristic
132C2×D157—21no—
133C2×D157—22no—
134C2×D157—21no—
135C2×D157—162no—
136C4×C157—41no—
137C4×C157—42no—
138C4×C157C22×D411yescharacteristic
139C4×C157C22×D411yescharacteristic
140Dic157—21no—
141Dic157—22no—
142Dic157—21no—
143C2×C157—21no—
144C2×C157—82no—
145C2×C157D4211yescharacteristic
146D157—41no—
147D157—42no—
148D157—41no—
149C157#329011yescharacteristic, complemented, Sylow
150#3290—1571nomaximal, Sylow
151#578—1571no—
152#577—1572no—
153#576—1571no—
154#495D15711yescharacteristic, complemented
155#473—1571no—
156#472—1571no—
157#471—1572no—
158#455—1572no—
159#454—1572no—
160#443—1571no—
161#443—1571no—
1622−1+4—1572no—
163H(5,𝔽2)—1572no—
164#137—3142no—
165#137—3142no—
166#137—1572no—
167#137—1572no—
168AGL(1,ℤ/8)—3142no—
169AGL(1,ℤ/8)—3142no—
170AGL(1,ℤ/8)—1572no—
171AGL(1,ℤ/8)—1572no—
172#135—3141no—
173#135—3142no—
174#135—3141no—
175#135—1571no—
176#135—1571no—
177C2×Q16—3141no—
178C2×Q16C2×D15711yescharacteristic, complemented
179C2×SD16—3142no—
180C2×SD16C2×D15712yescomplemented
181C2×D8—3141no—
182C2×D8C2×D15711yescharacteristic
183#131—1571no—
184#131—1572no—
185#131—1571no—
186#124C2×D15712yescomplemented
187#108—1571no—
188#108—1571no—
189#104—1571no—
190#104—1571no—
191#104—1572no—
192#101—1572no—
193#100—1572no—
194C4×C8C2×D15711yescharacteristic, complemented
195#41—3142no—
196#41—3142no—
197#41—3142no—
198#41—3142no—
199#41—6282no—
200#41—1571no—
201#41—1572no—
202#41—1571no—
203C2×Q8—3142no—
204C2×Q8—3142no—
205C2×Q8C22×D15712yes—
206C2×D4—3142no—
207C2×D4—3142no—
208C2×D4—6282no—
209C2×D4C22×D15712yes—
210Q16—21no—
211Q16—21no—
212Q16—3142no—
213Q16—3142no—
214Q16—1571no—
215Q16—1571no—
216SD16—22no—
217SD16—22no—
218SD16—3142no—
219SD16—3142no—
220SD16—3142no—
221SD16—3142no—
222SD16—1571no—
223SD16—1571no—
224SD16—1572no—
225D8—21no—
226D8—21no—
227D8—3142no—
228D8—3142no—
229D8—1571no—
230D8—1571no—
231M16—3141no—
232M16—3142no—
233M16—3141no—
234M16—1571no—
235M16—1572no—
236M16—1571no—
237C2×C8—3141no—
238C2×C8—3142no—
239C2×C8—3141no—
240C2×C8C22×D15711yescharacteristic
241C2×C8C22×D15711yescharacteristic
242C22⋊C4—42no—
243C42C22×D15711yescharacteristic
244C23—22no—
245C23—6282no—
246Q8—22no—
247Q8—22no—
248Q8—6282no—
249Q8—1571no—
250Q8—1572no—
251Q8—1571no—
252D4—22no—
253D4—22no—
254D4—3141no—
255D4—3142no—
256D4—3141no—
257D4—6282no—
258D4—6282no—
259D4—6282no—
260D4—1571no—
261D4—1572no—
262D4—1571no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—3141no—
266C2×C4—3142no—
267C2×C4—3141no—
268C2×C4—6282no—
269C2×C4—6282no—
270C2×C4—6282no—
271C2×C4C23×D15711yescharacteristic
272C8—3141no—
273C8—3142no—
274C8—3141no—
275C8D4×D15711yescharacteristic
276C8D4×D15711yescharacteristic
277C8D4×D15711yescharacteristic
278C8D4×D15711yescharacteristic
279C22—42no—
280C22—82no—
281C22—3141no—
282C22—3142no—
283C22—3141no—
284C22—25122no—
285C22#1909781611yescharacteristic
286C4—41no—
287C4—42no—
288C4—3141no—
289C4—3142no—
290C4—3141no—
291C4C2×D4×D15711yescharacteristic
292C4C2×D4×D15711yescharacteristic
293C2—21no—
294C2—82no—
295C2—6281no—
296C2—6282no—
297C2—6281no—
298C2#2375790811yescharacteristic
subgroupG/H
centerC2#23757908
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#495×C157C2
socleC2×C157D42
hypercenter#495D157
hypocenterC157#3290
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #23758086C2semidirect
2. #23758079C2semidirect
3. #23758074C2semidirect
4. #23757545C2semidirect
5. #23757540C2semidirect
6. #23757533C2semidirect
7. #23757450C2semidirect
8. #23757445C2semidirect
9. #23757391C2semidirect
10. #23757391C2semidirect
11. #495×C157C2semidirect
12. #19097819C22semidirect
16. #19097802C22semidirect
20. #19097797C22semidirect
31. #19097777C22semidirect
32. #19097777C22semidirect
33. #19097707C22semidirect
34. #19097707C22semidirect
36. #19097699C22semidirect
37. #19097699C22semidirect
38. #19097690C22semidirect
39. #19097688C22semidirect
40. C2×Q16×C157C22semidirect
41. C2×SD16×C157C22semidirect
43. #124×C157C22semidirect
44. C4×C8×C157C22semidirect
149. C157#3290semidirect
154. #495D157semidirect
178. C2×Q16C2×D157semidirect
180. C2×SD16C2×D157semidirect
186. #124C2×D157semidirect
194. C4×C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580862
#237580792
#237580742
#237575452
#237575402
#237575332
#237574502
#237574452
#237573912
#495×C1572
#3290157