Finite Group Databasebeta

← #34110619

Subgroups — #34110619

Order 20096.

596 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#19097797C211yesmaximal, characteristic, complemented
2C2×#19097796C211yesmaximal, characteristic, complemented
3C2×#19097791C211yesmaximal, characteristic, complemented
4C22×#14578939C211yesmaximal, characteristic, complemented
5#23757842C211yesmaximal, characteristic, complemented
6#23757663C212yesmaximal, complemented
7#23757660C212yesmaximal, complemented
8#23757655C212yesmaximal, complemented
9#23757649C212yesmaximal, complemented
10#23757229C211yesmaximal, characteristic, complemented
11#535×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×D628C2211yescharacteristic
14#19097797—24no—
15#19097796—24no—
16#19097791—24no—
17C2×#14578944C2211yescharacteristic, complemented
18C2×#14578942C2212yescomplemented
19C2×#14578942C2211yescharacteristic, complemented
20C2×#14578941C2212yescomplemented
21C2×#14578941C2211yescharacteristic, complemented
22C2×#14578939—24no—
23C2×#14578939C2212yes—
24C2×#14578939C2212yes—
25C2×#14578939C2211yescharacteristic, complemented
26C2×#14578939C2212yescomplemented
27C2×#14578939C2212yescomplemented
28C2×#14578938C2211yescharacteristic
29C22×#11929434C2211yescharacteristic
30#19097768C2212yescomplemented
31#19097759C2212yescomplemented
32#19097746—22no—
33#19097741—22no—
34#19097682C2212yescomplemented
35C4×#11929434C2212yescomplemented
36C2×#41×C157C2211yescharacteristic
37C22×D4×C157C2211yescharacteristic
38#127×C157C2212yescomplemented
39#124×C157C2212yescomplemented
40#121×C157—22no—
41#120×C157—22no—
42#117×C157C2211yescharacteristic
43C23×D157—21no—
44#14578954—22no—
45#14578954—22no—
46C2×Dic314C2311yescharacteristic
47C2×D628—22no—
48C2×D628—22no—
49C2×D628—21no—
50C2×D628—24no—
51C2×D628C2312yes—
52C2×D628C2311yescharacteristic
53C2×C22⋊D157—21no—
54C2×C4×D157—21no—
55#14578944—22no—
56#14578942—22no—
57#14578942—24no—
58#14578941—22no—
59#14578941—24no—
60#14578939—24no—
61#14578939—24no—
62#14578939—22no—
63#14578939—24no—
64#14578939—24no—
65#14578938D414yescomplemented
66C2×#11929434D414yescomplemented
67C2×#11929434C2311yescharacteristic
68C2×#11929434C2312yes—
69C2×#11929434C2311yescharacteristic
70#14578931—24no—
71C24×C157—21no—
72#41×C157—22no—
73#41×C157—22no—
74C2×Q8×C157C2311yescharacteristic
75C2×D4×C157—21no—
76C2×D4×C157—22no—
77C2×D4×C157—22no—
78C2×D4×C157—21no—
79C2×D4×C157—24no—
80C2×D4×C157C2312yes—
81C2×D4×C157C2311yescharacteristic
82C22×C4×C157—21no—
83C22×C4×C157C2311yescharacteristic
84C4⋊C4×C157D412yes—
85C22⋊C4×C157—24no—
86C22⋊C4×C157D412yes—
87C42×C157C2312yes—
88C22×D157—21no—
89C22×D157—22no—
90C22×D157—42no—
91C22×D157—82no—
92Dic314—21no—
93Dic314—21no—
94D628—22no—
95D628—22no—
96D628—21no—
97D628—21no—
98D628—42no—
99D628—84no—
100C22⋊D157—42no—
101C2×Dic157—21no—
102C4×D157—42no—
103#11929434C2×D412yes—
104#11929434C2×D412yes—
105#11929434C2×D414yes—
106C23×C157—22no—
107C23×C157—21no—
108C23×C157—42no—
109C23×C157—82no—
110C23×C157C2×D411yescharacteristic
111Q8×C157—21no—
112Q8×C157—21no—
113D4×C157—22no—
114D4×C157—22no—
115D4×C157—21no—
116D4×C157—21no—
117D4×C157—42no—
118D4×C157—42no—
119D4×C157—84no—
120C2×C4×C157—21no—
121C2×C4×C157—42no—
122C2×C4×C157C2×D412yes—
123C2×C4×C157C2×D412yes—
124C2×C4×C157C2×D414yes—
125C2×C4×C157C2×D411yescharacteristic
126C2×C4×C157C2411yescharacteristic
127C22×C157—22no—
128C22×C157—41no—
129C22×C157—42no—
130C22×C157—42no—
131C22×C157—44no—
132C22×C157—84no—
133C22×C157#12712yes—
134C22×C157C22×D411yescharacteristic
135C2×D157—41no—
136C2×D157—42no—
137C2×D157—42no—
138C2×D157—44no—
139C2×D157—84no—
140C4×C157—24no—
141C4×C157—41no—
142C4×C157#12712yes—
143C4×C157C22×D411yescharacteristic
144C4×C157C22×D411yescharacteristic
145Dic157—41no—
146C2×C157—22no—
147C2×C157—81no—
148C2×C157—82no—
149C2×C157C2×#12711yescharacteristic
150C2×C157#57612yes—
151D157—81no—
152D157—82no—
153C157#315511yescharacteristic, complemented, Sylow
154#3155—1571nomaximal, Sylow
155C2×AGL(1,ℤ/8)—1571no—
156C2×AGL(1,ℤ/8)—1571no—
157C2×#135—1571no—
158C22×D8—1571no—
159#535—1571no—
160#535D15711yescharacteristic, complemented
161#496—1572no—
162#496—1572no—
163#495—1572no—
164#493—1572no—
165#405—1571no—
166C2×#41—1571no—
167C2×#41C2×D15711yescharacteristic
168C22×D4—1571no—
169C22×D4C2×D15711yescharacteristic
170AGL(1,ℤ/8)—3144no—
171AGL(1,ℤ/8)—3144no—
172#135—3144no—
173C2×Q16—1571no—
174C2×SD16—1571no—
175C2×SD16—1572no—
176C2×SD16—1571no—
177C2×SD16—1572no—
178C2×D8—3144no—
179C2×D8—1572no—
180C2×D8—1572no—
181C2×D8—1571no—
182C2×D8—1572no—
183C2×D8—1572no—
184C2×M16—1571no—
185C22×C8—1571no—
186#127—1572no—
187#127C2×D15712yescomplemented
188#124—1572no—
189#124C2×D15712yescomplemented
190#121—22no—
191#121—3142no—
192#120—22no—
193#120—3142no—
194#117C2×D15711yescharacteristic
195#97—1572no—
196C4×C8—1572no—
197C24—21no—
198C24—3141no—
199#41—22no—
200#41—22no—
201#41—3142no—
202#41—3142no—
203C2×Q8—1571no—
204C2×Q8C22×D15711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—21no—
209C2×D4—24no—
210C2×D4—3141no—
211C2×D4—3142no—
212C2×D4—3142no—
213C2×D4—3141no—
214C2×D4—3144no—
215C2×D4—1572no—
216C2×D4—1571no—
217C2×D4C22×D15712yes—
218C2×D4C22×D15711yescharacteristic
219C22×C4—21no—
220C22×C4—3141no—
221C22×C4C22×D15711yescharacteristic
222Q16—3142no—
223SD16—3142no—
224SD16—3144no—
225SD16—3142no—
226SD16—3144no—
227D8—3144no—
228D8—3144no—
229D8—3142no—
230D8—3144no—
231D8—3144no—
232M16—1574no—
233C2×C8—1574no—
234C2×C8—1571no—
235C2×C8—1572no—
236C2×C8—1571no—
237C4⋊C4C22⋊D15712yes—
238C22⋊C4—24no—
239C22⋊C4—3144no—
240C22⋊C4C22⋊D15712yes—
241C42C22×D15712yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—82no—
246C23—3142no—
247C23—3141no—
248C23—6282no—
249C23—12562no—
250C23C2×C22⋊D15711yescharacteristic
251Q8—21no—
252Q8—21no—
253Q8—3141no—
254Q8—3141no—
255D4—22no—
256D4—21no—
257D4—22no—
258D4—21no—
259D4—42no—
260D4—42no—
261D4—84no—
262D4—3142no—
263D4—3141no—
264D4—3142no—
265D4—3141no—
266D4—6282no—
267D4—6282no—
268D4—12564no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—3141no—
272C2×C4—6282no—
273C2×C4D4×D15712yes—
274C2×C4D4×D15712yes—
275C2×C4C2×C22⋊D15714yes—
276C2×C4C2×C22⋊D15711yescharacteristic
277C2×C4C23×D15711yescharacteristic
278C8—1572no—
279C8—1572no—
280C8—1574no—
281C22—22no—
282C22—41no—
283C22—42no—
284C22—42no—
285C22—44no—
286C22—84no—
287C22—6281no—
288C22—6282no—
289C22—6282no—
290C22—6284no—
291C22—12564no—
292C22#1909776712yes—
293C22C22×C22⋊D15711yescharacteristic
294C4—24no—
295C4—41no—
296C4—6281no—
297C4#1909776712yes—
298C4C2×D4×D15711yescharacteristic
299C4C2×D4×D15711yescharacteristic
300C2—22no—
301C2—82no—
302C2—81no—
303C2—12562no—
304C2—12561no—
305C2C2×#1909776711yescharacteristic
306C2#2375807612yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#535×C157C2
socleC22×C157C22×D4
hypercenter#535D157
hypocenterC157#3155
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#19097797C2semidirect
2. C2×#19097796C2semidirect
3. C2×#19097791C2semidirect
4. C22×#14578939C2semidirect
5. #23757842C2semidirect
6. #23757663C2semidirect
7. #23757660C2semidirect
8. #23757655C2semidirect
9. #23757649C2semidirect
10. #23757229C2semidirect
11. #535×C157C2semidirect
17. C2×#14578944C22semidirect
18. C2×#14578942C22semidirect
19. C2×#14578942C22semidirect
20. C2×#14578941C22semidirect
21. C2×#14578941C22semidirect
25. C2×#14578939C22semidirect
26. C2×#14578939C22semidirect
27. C2×#14578939C22semidirect
30. #19097768C22semidirect
31. #19097759C22semidirect
34. #19097682C22semidirect
35. C4×#11929434C22semidirect
38. #127×C157C22semidirect
39. #124×C157C22semidirect
65. #14578938D4semidirect
66. C2×#11929434D4semidirect
153. C157#3155semidirect
160. #535D157semidirect
187. #127C2×D157semidirect
189. #124C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977972
C2×#190977962
C2×#190977912
C22×#145789392
#237578422
#237576632
#237576602
#237576552
#237576492
#237572292
#535×C1572
#3155157