Finite Group Databasebeta

← #34110608

Subgroups — #34110608

Order 20096.

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

normal subgroups only

This group has 564 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×#19097801C211yesmaximal, characteristic, complemented
2C2×#19097797C211yesmaximal, characteristic, complemented
3C2×#19097791C211yesmaximal, characteristic, complemented
4C22×#14578942C211yesmaximal, characteristic, complemented
5#23757849C211yesmaximal, characteristic, complemented
6#23757668C212yesmaximal, complemented
7#23757660C212yesmaximal, complemented
8#23757655C212yesmaximal, complemented
9#23757645C212yesmaximal, complemented
10#23757229C211yesmaximal, characteristic, complemented
11#535×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×Dic314C2211yescharacteristic
14#19097801—24no—
15#19097797—24no—
16#19097791—24no—
17C2×#14578944C2212yescomplemented
18C2×#14578944C2211yescharacteristic, complemented
19C2×#14578942—24no—
20C2×#14578942C2212yes—
21C2×#14578942C2212yes—
22C2×#14578942C2212yes—
23C2×#14578942C2211yescharacteristic, complemented
24C2×#14578942C2212yescomplemented
25C2×#14578941C2211yescharacteristic, complemented
26C2×#14578939C2212yes—
27C2×#14578939C2211yescharacteristic, complemented
28C2×#14578938C2211yescharacteristic
29C22×#11929434C2211yescharacteristic
30#19097771C2212yescomplemented
31#19097759C2212yescomplemented
32#19097755—22no—
33#19097751—22no—
34#19097682C2212yescomplemented
35C4×#11929434C2212yescomplemented
36C2×#41×C157C2211yescharacteristic
37C22×D4×C157C2211yescharacteristic
38#127×C157C2212yes—
39#124×C157C2212yes—
40#121×C157—22no—
41#120×C157—22no—
42#117×C157C2211yescharacteristic
43#14578954—22no—
44#14578954—22no—
45C2×Dic314—21no—
46C2×Dic314—22no—
47C2×Dic314—22no—
48C2×Dic314C2312yes—
49C2×Dic314C2311yescharacteristic
50C2×D628C2311yescharacteristic
51C2×C22⋊D157—21no—
52C22×Dic157—21no—
53C2×C4×D157—21no—
54#14578944—22no—
55#14578944—24no—
56#14578942—24no—
57#14578942—24no—
58#14578942—24no—
59#14578942—22no—
60#14578942—24no—
61#14578941—22no—
62#14578939—24no—
63#14578939—22no—
64#14578938D414yescomplemented
65C2×#11929434D414yescomplemented
66C2×#11929434C2311yescharacteristic
67C2×#11929434C2312yes—
68C2×#11929434C2311yescharacteristic
69C4⋊Dic157—24no—
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—
89Dic314—22no—
90Dic314—21no—
91Dic314—22no—
92Dic314—21no—
93Dic314—42no—
94D628—21no—
95D628—21no—
96C22⋊D157—42no—
97C2×Dic157—21no—
98C2×Dic157—22no—
99C2×Dic157—42no—
100C4×D157—42no—
101#11929434C2×D412yes—
102#11929434C2×D414yes—
103#11929434C2×D412yes—
104C23×C157—22no—
105C23×C157—21no—
106C23×C157—42no—
107C23×C157—82no—
108C23×C157C2×D411yescharacteristic
109Q8×C157—21no—
110Q8×C157—21no—
111D4×C157—22no—
112D4×C157—22no—
113D4×C157—21no—
114D4×C157—21no—
115D4×C157—42no—
116D4×C157—42no—
117D4×C157—84no—
118C2×C4×C157—21no—
119C2×C4×C157—42no—
120C2×C4×C157C2×D412yes—
121C2×C4×C157C2×D412yes—
122C2×C4×C157C2×D414yes—
123C2×C4×C157C2×D411yescharacteristic
124C2×C4×C157C2411yescharacteristic
125C22×C157—22no—
126C22×C157—41no—
127C22×C157—42no—
128C22×C157—42no—
129C22×C157—44no—
130C22×C157—84no—
131C22×C157#12712yes—
132C22×C157C22×D411yescharacteristic
133C2×D157—41no—
134C2×D157—42no—
135C4×C157—24no—
136C4×C157—41no—
137C4×C157#12712yes—
138C4×C157C22×D411yescharacteristic
139C4×C157C22×D411yescharacteristic
140Dic157—41no—
141Dic157—42no—
142C2×C157—22no—
143C2×C157—81no—
144C2×C157—82no—
145C2×C157C2×#12711yescharacteristic
146C2×C157#57712yes—
147D157—81no—
148C157#315411yescharacteristic, complemented, Sylow
149#3154—1571nomaximal, Sylow
150C2×#137—1571no—
151C2×AGL(1,ℤ/8)—1571no—
152C2×#135—1571no—
153C22×SD16—1571no—
154#536—1571no—
155#535D15711yescharacteristic, complemented
156#497—1572no—
157#496—1572no—
158#495—1572no—
159#492—1572no—
160#405—1571no—
161C2×#41—1571no—
162C2×#41C2×D15711yescharacteristic
163C22×Q8—1571no—
164C22×D4C2×D15711yescharacteristic
165#137—3144no—
166AGL(1,ℤ/8)—3144no—
167#135—3144no—
168C2×Q16—1571no—
169C2×Q16—1572no—
170C2×SD16—3144no—
171C2×SD16—1571no—
172C2×SD16—1572no—
173C2×SD16—1572no—
174C2×SD16—1571no—
175C2×SD16—1572no—
176C2×SD16—1572no—
177C2×D8—1571no—
178C2×D8—1572no—
179C2×M16—1571no—
180C22×C8—1571no—
181C4⋊Q8—1572no—
182#127C2×D15712yes—
183#124—1572no—
184#124C2×D15712yes—
185#123—3142no—
186C22⋊Q8—3142no—
187#121—22no—
188#120—22no—
189#117C2×D15711yescharacteristic
190#97—1572no—
191C4×C8—1572no—
192C24—21no—
193#41—22no—
194#41—22no—
195#41—3142no—
196#41—3142no—
197C2×Q8—3141no—
198C2×Q8—3142no—
199C2×Q8—3142no—
200C2×Q8—1572no—
201C2×Q8—1571no—
202C2×Q8C22×D15711yescharacteristic
203C2×D4—21no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—24no—
208C2×D4—3141no—
209C2×D4—1571no—
210C2×D4C22×D15712yes—
211C2×D4C22×D15711yescharacteristic
212C22×C4—21no—
213C22×C4—3141no—
214C22×C4—3141no—
215C22×C4C22×D15711yescharacteristic
216Q16—3142no—
217Q16—3144no—
218SD16—3144no—
219SD16—3144no—
220SD16—3142no—
221SD16—3144no—
222SD16—3144no—
223SD16—3142no—
224D8—3142no—
225D8—3144no—
226M16—1574no—
227C2×C8—1574no—
228C2×C8—1571no—
229C2×C8—1572no—
230C2×C8—1571no—
231C4⋊C4—3144no—
232C4⋊C4C22⋊D15712yes—
233C22⋊C4—24no—
234C22⋊C4—3144no—
235C22⋊C4C22⋊D15712yes—
236C42C22×D15712yes—
237C23—22no—
238C23—21no—
239C23—42no—
240C23—82no—
241C23—3141no—
242C23C2×C22⋊D15711yescharacteristic
243Q8—21no—
244Q8—21no—
245Q8—3142no—
246Q8—3141no—
247Q8—3142no—
248Q8—3141no—
249Q8—6282no—
250D4—22no—
251D4—21no—
252D4—22no—
253D4—21no—
254D4—42no—
255D4—42no—
256D4—84no—
257D4—3141no—
258D4—3141no—
259D4—6282no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3142no—
263C2×C4—3141no—
264C2×C4—6282no—
265C2×C4—6282no—
266C2×C4D4×D15712yes—
267C2×C4D4×D15712yes—
268C2×C4C2×C22⋊D15714yes—
269C2×C4C2×C22⋊D15711yescharacteristic
270C2×C4C23×D15711yescharacteristic
271C8—1572no—
272C8—1572no—
273C8—1574no—
274C22—22no—
275C22—42no—
276C22—44no—
277C22—41no—
278C22—42no—
279C22—84no—
280C22—6282no—
281C22—6281no—
282C22#1909776712yes—
283C22C22×C22⋊D15711yescharacteristic
284C4—24no—
285C4—41no—
286C4—6281no—
287C4—6282no—
288C4#1909776712yes—
289C4C2×D4×D15711yescharacteristic
290C4C2×D4×D15711yescharacteristic
291C2—22no—
292C2—82no—
293C2—81no—
294C2—12561no—
295C2C2×#1909776711yescharacteristic
296C2#2375808412yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#535×C157C2
socleC22×C157C22×D4
hypercenter#535D157
hypocenterC157#3154
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×#19097801C2semidirect
2. C2×#19097797C2semidirect
3. C2×#19097791C2semidirect
4. C22×#14578942C2semidirect
5. #23757849C2semidirect
6. #23757668C2semidirect
7. #23757660C2semidirect
8. #23757655C2semidirect
9. #23757645C2semidirect
10. #23757229C2semidirect
11. #535×C157C2semidirect
17. C2×#14578944C22semidirect
18. C2×#14578944C22semidirect
23. C2×#14578942C22semidirect
24. C2×#14578942C22semidirect
25. C2×#14578941C22semidirect
27. C2×#14578939C22semidirect
30. #19097771C22semidirect
31. #19097759C22semidirect
34. #19097682C22semidirect
35. C4×#11929434C22semidirect
64. #14578938D4semidirect
65. C2×#11929434D4semidirect
148. C157#3154semidirect
155. #535D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978012
C2×#190977972
C2×#190977912
C22×#145789422
#237578492
#237576682
#237576602
#237576552
#237576452
#237572292
#535×C1572
#3154157