Finite Group Databasebeta

← #34111933

Subgroups — #34111933

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#23758076C212yesmaximal, complemented
2#23758075C211yesmaximal, characteristic, complemented
3#23757648C211yesmaximal, characteristic, complemented
4#23757543C211yesmaximal, characteristic, complemented
5#23757541C212yesmaximal, complemented
6#23757540C211yesmaximal, characteristic, complemented
7#23757441C212yesmaximal, complemented
8#23757439C212yesmaximal, complemented
9#23757390C211yesmaximal, characteristic, complemented
10#23757388C211yesmaximal, characteristic, complemented
11#576×C157C211yesmaximal, characteristic, complemented
12#19097817C2212yescomplemented
13#19097798—22no—
14#19097797—22no—
15#19097797C2212yescomplemented
16#19097796—22no—
17#19097796C2212yescomplemented
18#19097796C2212yescomplemented
19#19097794—21no—
20#19097791—22no—
21#19097791C2211yescharacteristic
22C2×D1256C2211yescharacteristic, complemented
23C2×#14578939—22no—
24C2×#14578939C2212yescomplemented
25D8×D157—21no—
26#19097778C2212yescomplemented
27#19097777C2211yescharacteristic, complemented
28#19097767C2212yescomplemented
29#19097707C2211yescharacteristic, complemented
30#19097706C2211yescharacteristic, complemented
31#19097698C2211yescharacteristic, complemented
32#19097698C2212yescomplemented
33#19097697C2211yescharacteristic, complemented
34#19097688C2212yescomplemented
35#19097687C2212yescomplemented
36C8×Dic157C2211yescharacteristic, complemented
37H(5,𝔽2)×C157C2212yescomplemented
38AGL(1,ℤ/8)×C157—22no—
39AGL(1,ℤ/8)×C157C2212yescomplemented
40#135×C157—21no—
41#135×C157C2211yescharacteristic
42C2×D8×C157—21no—
43C2×D8×C157C2211yescharacteristic, complemented
44#131×C157C2211yescharacteristic, complemented
45#14578955—22no—
46#14578954—22no—
47#14578954C2311yescharacteristic
48C2×D628—22no—
49C2×D628C2312yes—
50C2×C22⋊D157—42no—
51D4×D157—22no—
52D4×D157—42no—
53#14578944D411yescharacteristic, complemented
54#14578942—22no—
55#14578942D411yescharacteristic, complemented
56#14578941—22no—
57#14578941—22no—
58#14578941D411yescharacteristic, complemented
59D1256—21no—
60D1256—21no—
61#14578939—22no—
62#14578939—22no—
63#14578939—22no—
64#14578939—22no—
65#14578939—22no—
66#14578939D411yescharacteristic, complemented
67#14578938—22no—
68#14578938C2312yes—
69#14578938C2311yescharacteristic
70#14578937—21no—
71C2×#11929434—22no—
72C2×#11929434C2311yescharacteristic
73C8×D157—21no—
74C4×Dic157C2311yescharacteristic
75#41×C157—22no—
76#41×C157—22no—
77#41×C157C2311yescharacteristic
78#41×C157C2312yes—
79C2×D4×C157—22no—
80C2×D4×C157—22no—
81C2×D4×C157—42no—
82C2×D4×C157C2312yes—
83Q16×C157D411yescharacteristic, complemented
84SD16×C157—22no—
85SD16×C157D412yescomplemented
86D8×C157—22no—
87D8×C157—21no—
88D8×C157—21no—
89D8×C157D411yescharacteristic, complemented
90M16×C157—21no—
91M16×C157C2311yescharacteristic
92C2×C8×C157—21no—
93C2×C8×C157C2311yescharacteristic
94C22×D157—22no—
95C22×D157—42no—
96Dic314C2×D411yescharacteristic
97D628—22no—
98D628—22no—
99D628—42no—
100D628C2×D411yescharacteristic
101C22⋊D157—21no—
102C22⋊D157—42no—
103C22⋊D157—84no—
104C2×Dic157—21no—
105C4×D157—21no—
106C4×D157—42no—
107#11929434—21no—
108#11929434—22no—
109#11929434C2×D411yescharacteristic
110#11929434C2×D411yescharacteristic
111C23×C157—22no—
112C23×C157—42no—
113Q8×C157C2×D411yescharacteristic
114Q8×C157C2×D412yes—
115D4×C157—22no—
116D4×C157—21no—
117D4×C157—22no—
118D4×C157—22no—
119D4×C157—42no—
120D4×C157—42no—
121D4×C157C2×D411yescharacteristic
122D4×C157C2×D412yes—
123C2×C4×C157—22no—
124C2×C4×C157—21no—
125C2×C4×C157—42no—
126C2×C4×C157C2411yescharacteristic
127C8×C157—21no—
128C8×C157C2×D411yescharacteristic
129C8×C157C2×D411yescharacteristic
130C22×C157—21no—
131C22×C157—22no—
132C22×C157—42no—
133C22×C157—82no—
134C22×C157—162no—
135C22×C157H(5,𝔽2)11yescharacteristic
136C2×D157—21no—
137C2×D157—42no—
138C2×D157—82no—
139C2×D157—162no—
140C4×C157—21no—
141C4×C157—22no—
142C4×C157C22×D411yescharacteristic
143C4×C157C22×D411yescharacteristic
144Dic157—22no—
145Dic157—21no—
146C2×C157—21no—
147C2×C157—42no—
148C2×C157—41no—
149C2×C157—82no—
150C2×C157D4211yescharacteristic
151D157—41no—
152D157—82no—
153C157#329211yescharacteristic, complemented, Sylow
154#3292—1571nomaximal, Sylow
155#576—1572no—
156#576—1571no—
157#576D15711yescharacteristic, complemented
158#493—1571no—
159#472—1572no—
160#472—1571no—
161#472—1571no—
162#453—1572no—
163#453—1572no—
164#443—1571no—
165#443—1571no—
166H(5,𝔽2)—1572no—
167H(5,𝔽2)C2×D15712yescomplemented
168AGL(1,ℤ/8)—22no—
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—21no—
177#135—3142no—
178#135—3141no—
179#135—1571no—
180#135C2×D15711yescharacteristic
181C2×D8—21no—
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#108—1571no—
193#104—1571no—
194#104—1572no—
195#104—1571no—
196#100—1572no—
197#100—1572no—
198C4×C8—1571no—
199#41—22no—
200#41—22no—
201#41—3142no—
202#41—3142no—
203#41—1571no—
204#41C22×D15711yescharacteristic
205#41C22×D15712yes—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—42no—
209C2×D4—3142no—
210C2×D4—3142no—
211C2×D4—6282no—
212C2×D4—6282no—
213C2×D4—1572no—
214C2×D4C22×D15712yes—
215Q16—1571no—
216Q16C22⋊D15711yescharacteristic, complemented
217SD16—22no—
218SD16—3142no—
219SD16—3142no—
220SD16—3142no—
221SD16—1571no—
222SD16—1571no—
223SD16C22⋊D15712yescomplemented
224D8—22no—
225D8—21no—
226D8—21no—
227D8—3142no—
228D8—3142no—
229D8—3142no—
230D8—3142no—
231D8—3142no—
232D8—3141no—
233D8—3141no—
234D8—1571no—
235D8C22⋊D15711yescharacteristic, complemented
236M16—21no—
237M16—3142no—
238M16—3141no—
239M16—1572no—
240M16—1571no—
241M16C22×D15711yescharacteristic
242C2×C8—21no—
243C2×C8—3142no—
244C2×C8—3141no—
245C2×C8—1571no—
246C2×C8C22×D15711yescharacteristic
247C42—1571no—
248C23—22no—
249C23—42no—
250C23—3142no—
251C23—6282no—
252Q8—1571no—
253Q8D4×D15711yescharacteristic
254Q8C2×C22⋊D15712yes—
255D4—21no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—42no—
260D4—42no—
261D4—3141no—
262D4—3142no—
263D4—3142no—
264D4—6282no—
265D4—6282no—
266D4—1571no—
267D4—12564no—
268D4D4×D15711yescharacteristic
269D4C2×C22⋊D15712yes—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—42no—
273C2×C4—3141no—
274C2×C4—3141no—
275C2×C4—6282no—
276C2×C4C23×D15711yescharacteristic
277C8—21no—
278C8—3142no—
279C8—3141no—
280C8—1571no—
281C8—1571no—
282C8C2×C22⋊D15711yescharacteristic
283C8C2×C22⋊D15711yescharacteristic
284C22—21no—
285C22—22no—
286C22—42no—
287C22—82no—
288C22—162no—
289C22—3141no—
290C22—6282no—
291C22—25122no—
292C22—12562no—
293C22#1909781611yescharacteristic
294C4—21no—
295C4—22no—
296C4—3142no—
297C4—3141no—
298C4C2×D4×D15711yescharacteristic
299C4C22×C22⋊D15711yescharacteristic
300C2—21no—
301C2—41no—
302C2—42no—
303C2—82no—
304C2—6281no—
305C2—12562no—
306C2D4×C22⋊D15711yescharacteristic
subgroupG/H
centerC2D4×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#576×C157C2
socleC2×C157D42
hypercenter#576D157
hypocenterC157#3292
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #576

Complements

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

NH ≅ G/Ntype
1. #23758076C2semidirect
2. #23758075C2semidirect
3. #23757648C2semidirect
4. #23757543C2semidirect
5. #23757541C2semidirect
6. #23757540C2semidirect
7. #23757441C2semidirect
8. #23757439C2semidirect
9. #23757390C2semidirect
10. #23757388C2semidirect
11. #576×C157C2semidirect
12. #19097817C22semidirect
15. #19097797C22semidirect
17. #19097796C22semidirect
18. #19097796C22semidirect
22. C2×D1256C22semidirect
24. C2×#14578939C22semidirect
26. #19097778C22semidirect
27. #19097777C22semidirect
28. #19097767C22semidirect
29. #19097707C22semidirect
30. #19097706C22semidirect
31. #19097698C22semidirect
32. #19097698C22semidirect
33. #19097697C22semidirect
34. #19097688C22semidirect
35. #19097687C22semidirect
36. C8×Dic157C22semidirect
37. H(5,𝔽2)×C157C22semidirect
39. AGL(1,ℤ/8)×C157C22semidirect
43. C2×D8×C157C22semidirect
44. #131×C157C22semidirect
53. #14578944D4semidirect
55. #14578942D4semidirect
58. #14578941D4semidirect
66. #14578939D4semidirect
83. Q16×C157D4semidirect
85. SD16×C157D4semidirect
89. D8×C157D4semidirect
153. C157#3292semidirect
157. #576D157semidirect
167. H(5,𝔽2)C2×D157semidirect
175. AGL(1,ℤ/8)C2×D157semidirect
186. C2×D8C2×D157semidirect
189. #131C2×D157semidirect
216. Q16C22⋊D157semidirect
223. SD16C22⋊D157semidirect
235. D8C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237580762
#237580752
#237576482
#237575432
#237575412
#237575402
#237574412
#237574392
#237573902
#237573882
#576×C1572
#3292157