Finite Group Databasebeta

← #34111934

Subgroups — #34111934

Order 20096.

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

normal subgroups only

This group has 476 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#23758078C211yesmaximal, characteristic, complemented
2#23758077C212yesmaximal, complemented
3#23758074C211yesmaximal, characteristic, complemented
4#23757648C211yesmaximal, characteristic, complemented
5#23757544C211yesmaximal, characteristic, complemented
6#23757542C212yesmaximal, complemented
7#23757443C212yesmaximal, complemented
8#23757439C212yesmaximal, complemented
9#23757389C211yesmaximal, characteristic, complemented
10#23757388C211yesmaximal, characteristic, complemented
11#472×C157C211yesmaximal, characteristic, complemented
12#19097817C2212yescomplemented
13#19097816C2212yescomplemented
14#19097799—22no—
15#19097799C2212yescomplemented
16#19097799C2212yescomplemented
17#19097798—22no—
18#19097797C2212yescomplemented
19#19097795—22no—
20#19097795—22no—
21#19097793—21no—
22#19097793C2211yescharacteristic
23#19097792—22no—
24#19097791C2211yescharacteristic
25#19097790—21no—
26C2×D1256—21no—
27C2×D1256C2211yescharacteristic, complemented
28C2×#14578939C2212yescomplemented
29D8×D157—21no—
30D8×D157—22no—
31#19097779C2212yescomplemented
32#19097777C2211yescharacteristic, complemented
33#19097767C2212yescomplemented
34#19097706C2211yescharacteristic, complemented
35#19097698C2211yescharacteristic, complemented
36#19097697C2211yescharacteristic, complemented
37#19097697C2212yescomplemented
38#19097687C2212yescomplemented
39C8×Dic157C2211yescharacteristic, complemented
40AGL(1,ℤ/8)×C157C2212yescomplemented
41C2×D8×C157C2211yescharacteristic, complemented
42#131×C157C2211yescharacteristic, complemented
43#108×C157C2211yescharacteristic, complemented
44#100×C157C2212yescomplemented
45#14578955—22no—
46#14578954—22no—
47#14578954C2311yescharacteristic
48#14578954C2312yes—
49#14578953—22no—
50#14578953—22no—
51C2×D628—22no—
52C2×D628C2312yes—
53C2×C22⋊D157—42no—
54C2×C22⋊D157—42no—
55D4×D157—22no—
56D4×D157—22no—
57D4×D157—22no—
58D4×D157—42no—
59Dic628D411yescharacteristic, complemented
60#14578944D411yescharacteristic, complemented
61#14578943—22no—
62#14578943—22no—
63#14578943D412yescomplemented
64#14578942—22no—
65#14578942D411yescharacteristic, complemented
66#14578941D411yescharacteristic, complemented
67D1256—22no—
68D1256—22no—
69D1256—21no—
70D1256—21no—
71D1256D411yescharacteristic, complemented
72#14578939—22no—
73#14578939—22no—
74#14578939—22no—
75#14578939D411yescharacteristic, complemented
76#14578938C2311yescharacteristic
77#14578937—22no—
78#14578937—21no—
79C2×#11929434C2311yescharacteristic
80C8×D157—22no—
81C8×D157—21no—
82C4×Dic157C2311yescharacteristic
83#41×C157C2311yescharacteristic
84C2×D4×C157C2312yes—
85SD16×C157—22no—
86D8×C157—22no—
87D8×C157—21no—
88D8×C157—21no—
89M16×C157—21no—
90M16×C157C2311yescharacteristic
91M16×C157C2312yes—
92C2×C8×C157—21no—
93C2×C8×C157C2311yescharacteristic
94C22×D157—22no—
95C22×D157—42no—
96C22×D157—42no—
97Dic314C2×D411yescharacteristic
98Dic314C2×D412yes—
99D628—22no—
100D628—22no—
101D628—42no—
102D628C2×D411yescharacteristic
103D628C2×D412yes—
104C22⋊D157—22no—
105C22⋊D157—21no—
106C22⋊D157—42no—
107C22⋊D157—42no—
108C22⋊D157—42no—
109C22⋊D157—84no—
110C2×Dic157—21no—
111C2×Dic157—42no—
112C4×D157—21no—
113C4×D157—22no—
114C4×D157—42no—
115#11929434—21no—
116#11929434C2×D411yescharacteristic
117#11929434C2×D411yescharacteristic
118C23×C157—22no—
119Q8×C157C2×D411yescharacteristic
120D4×C157—21no—
121D4×C157—22no—
122D4×C157—22no—
123D4×C157C2×D411yescharacteristic
124C2×C4×C157—21no—
125C2×C4×C157C2411yescharacteristic
126C8×C157—21no—
127C8×C157—22no—
128C8×C157C2×D411yescharacteristic
129C8×C157C2×D411yescharacteristic
130C22×C157—21no—
131C22×C157—42no—
132C22×C157—82no—
133C22×C157H(5,𝔽2)11yescharacteristic
134C2×D157—21no—
135C2×D157—22no—
136C2×D157—42no—
137C2×D157—82no—
138C2×D157—162no—
139C2×D157—162no—
140C4×C157—21no—
141C4×C157C22×D411yescharacteristic
142C4×C157C22×D411yescharacteristic
143Dic157—22no—
144Dic157—22no—
145Dic157—21no—
146C2×C157—21no—
147C2×C157—41no—
148C2×C157—82no—
149C2×C157D4211yescharacteristic
150D157—41no—
151D157—42no—
152D157—82no—
153C157#329211yescharacteristic, complemented, Sylow
154#3292—1571nomaximal, Sylow
155#576—1572no—
156#576—1571no—
157#576—1571no—
158#493—1571no—
159#472—1571no—
160#472—1572no—
161#472D15711yescharacteristic, complemented
162#453—1572no—
163#453—1572no—
164#443—1571no—
165#443—1571no—
166H(5,𝔽2)—1572no—
167H(5,𝔽2)—1572no—
168AGL(1,ℤ/8)—3142no—
169AGL(1,ℤ/8)—3142no—
170AGL(1,ℤ/8)—3142no—
171AGL(1,ℤ/8)—3142no—
172AGL(1,ℤ/8)—1572no—
173AGL(1,ℤ/8)—1572no—
174AGL(1,ℤ/8)—1572no—
175AGL(1,ℤ/8)C2×D15712yescomplemented
176#135—3141no—
177#135—3141no—
178#135—3142no—
179#135—1571no—
180#135—1571no—
181C2×D8—3141no—
182C2×D8—3142no—
183C2×D8—3141no—
184C2×D8—1572no—
185C2×D8—1571no—
186C2×D8C2×D15711yescharacteristic, complemented
187#131—1572no—
188#131—1571no—
189#131C2×D15711yescharacteristic, complemented
190#127—1572no—
191#108—1571no—
192#108C2×D15711yescharacteristic, complemented
193#104—1571no—
194#104—1572no—
195#104—1571no—
196#100—1572no—
197#100C2×D15712yescomplemented
198C4×C8—1571no—
199#41—3142no—
200#41—3142no—
201#41—3142no—
202#41—3142no—
203#41—1571no—
204#41—1572no—
205#41C22×D15711yescharacteristic
206C2×D4—3142no—
207C2×D4—3142no—
208C2×D4—3142no—
209C2×D4—3142no—
210C2×D4—6282no—
211C2×D4—6282no—
212C2×D4—6282no—
213C2×D4—1572no—
214C2×D4C22×D15712yes—
215Q16—1571no—
216Q16—1571no—
217SD16—22no—
218SD16—3142no—
219SD16—3142no—
220SD16—3142no—
221SD16—1571no—
222SD16—1571no—
223SD16—1572no—
224D8—22no—
225D8—21no—
226D8—21no—
227D8—3142no—
228D8—3142no—
229D8—3142no—
230D8—3142no—
231D8—3142no—
232D8—3141no—
233D8—3141no—
234D8—1571no—
235D8—1571no—
236M16—21no—
237M16—3142no—
238M16—3141no—
239M16—1571no—
240M16C22×D15712yes—
241M16C22×D15711yescharacteristic
242C2×C8—21no—
243C2×C8—3142no—
244C2×C8—3141no—
245C2×C8—1571no—
246C2×C8C22×D15711yescharacteristic
247C42—1571no—
248C23—22no—
249C23—3142no—
250C23—6282no—
251C23—6282no—
252Q8—1571no—
253Q8—1572no—
254Q8D4×D15711yescharacteristic
255D4—21no—
256D4—22no—
257D4—22no—
258D4—3142no—
259D4—3141no—
260D4—3142no—
261D4—3142no—
262D4—6282no—
263D4—6282no—
264D4—6282no—
265D4—6282no—
266D4—1571no—
267D4—1572no—
268D4—12564no—
269D4D4×D15711yescharacteristic
270C2×C4—21no—
271C2×C4—3141no—
272C2×C4—3142no—
273C2×C4—3141no—
274C2×C4—6282no—
275C2×C4—6282no—
276C2×C4C23×D15711yescharacteristic
277C8—21no—
278C8—22no—
279C8—3141no—
280C8—1571no—
281C8—1571no—
282C8D4×D15711yescharacteristic
283C8D4×D15711yescharacteristic
284C22—21no—
285C22—42no—
286C22—82no—
287C22—3142no—
288C22—3141no—
289C22—6282no—
290C22—25122no—
291C22—25122no—
292C22—12562no—
293C22#1909781711yescharacteristic
294C4—21no—
295C4—3142no—
296C4—3142no—
297C4—3141no—
298C4C2×D4×D15711yescharacteristic
299C4C2×D4×D15711yescharacteristic
300C2—21no—
301C2—41no—
302C2—82no—
303C2—6282no—
304C2—6281no—
305C2—12562no—
306C2#2375790711yescharacteristic
subgroupG/H
centerC2#23757907
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#472×C157C2
socleC2×C157D42
hypercenter#472D157
hypocenterC157#3292
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #472

Complements

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

NH ≅ G/Ntype
1. #23758078C2semidirect
2. #23758077C2semidirect
3. #23758074C2semidirect
4. #23757648C2semidirect
5. #23757544C2semidirect
6. #23757542C2semidirect
7. #23757443C2semidirect
8. #23757439C2semidirect
9. #23757389C2semidirect
10. #23757388C2semidirect
11. #472×C157C2semidirect
12. #19097817C22semidirect
13. #19097816C22semidirect
15. #19097799C22semidirect
16. #19097799C22semidirect
18. #19097797C22semidirect
27. C2×D1256C22semidirect
28. C2×#14578939C22semidirect
31. #19097779C22semidirect
32. #19097777C22semidirect
33. #19097767C22semidirect
34. #19097706C22semidirect
35. #19097698C22semidirect
36. #19097697C22semidirect
37. #19097697C22semidirect
38. #19097687C22semidirect
39. C8×Dic157C22semidirect
40. AGL(1,ℤ/8)×C157C22semidirect
41. C2×D8×C157C22semidirect
42. #131×C157C22semidirect
43. #108×C157C22semidirect
44. #100×C157C22semidirect
59. Dic628D4semidirect
60. #14578944D4semidirect
63. #14578943D4semidirect
65. #14578942D4semidirect
66. #14578941D4semidirect
71. D1256D4semidirect
75. #14578939D4semidirect
153. C157#3292semidirect
161. #472D157semidirect
175. AGL(1,ℤ/8)C2×D157semidirect
186. C2×D8C2×D157semidirect
189. #131C2×D157semidirect
192. #108C2×D157semidirect
197. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580782
#237580772
#237580742
#237576482
#237575442
#237575422
#237574432
#237574392
#237573892
#237573882
#472×C1572
#3292157