Finite Group Databasebeta

← #34109714

Subgroups — #34109714

Order 20096.

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

normal subgroups only

This group has 520 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×#19097791C211yesmaximal, characteristic, complemented
2C23×#11929434C211yesmaximal, characteristic, complemented
3#23757784C211yesmaximal, characteristic, complemented
4#23757512C212yesmaximal, complemented
5#23757507C212yesmaximal, complemented
6#23757504C212yesmaximal, complemented
7#23757500C212yesmaximal, complemented
8#23757309C211yesmaximal, characteristic, complemented
9#23757269C211yesmaximal, characteristic, complemented
10#23757267C211yesmaximal, characteristic, complemented
11#525×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13#19097791—24no—
14C2×#14578944C2211yescharacteristic, complemented
15C2×#14578942C2211yescharacteristic, complemented
16C2×#14578941C2211yescharacteristic, complemented
17C2×#14578939C2211yescharacteristic, complemented
18C22×#11929434—24no—
19C22×#11929434C2212yes—
20C22×#11929434C2212yes—
21C22×#11929434C2211yescharacteristic
22C22×#11929434C2211yescharacteristic
23#19097754—21no—
24#19097750C2212yes—
25#19097745C2212yes—
26#19097740—21no—
27C4×C22⋊D157—22no—
28#19097730C2211yescharacteristic
29#19097704C2212yescomplemented
30#19097702C2212yescomplemented
31#19097695C2212yescomplemented
32#19097694C2212yescomplemented
33#19097692C2212yescomplemented
34#19097691C2212yescomplemented
35C2×#41×C157C2211yescharacteristic
36C23×C4×C157C2211yescharacteristic
37#123×C157—21no—
38C22⋊Q8×C157C2212yes—
39#121×C157C2212yes—
40#120×C157—21no—
41C4×D4×C157—22no—
42#117×C157C2211yescharacteristic
43#14578954—22no—
44#14578954—22no—
45C2×Dic314C2311yescharacteristic
46C2×D628C2311yescharacteristic
47C2×C22⋊D157—22no—
48C2×C22⋊D157—21no—
49C2×C4×D157—21no—
50#14578944—22no—
51#14578942—22no—
52#14578941—22no—
53#14578939—22no—
54C2×#11929434—24no—
55C2×#11929434—24no—
56C2×#11929434—22no—
57C2×#11929434D414yescomplemented
58C2×#11929434C2312yes—
59C2×#11929434C2311yescharacteristic
60C2×#11929434C2311yescharacteristic
61C4⋊Dic157C2312yes—
62#14578933—22no—
63C22⋊Dic157—21no—
64C22⋊Dic157—22no—
65#14578931—22no—
66C4×Dic157—21no—
67C24×C157D411yescharacteristic
68#41×C157—22no—
69#41×C157—22no—
70C2×Q8×C157C2311yescharacteristic
71C2×D4×C157—22no—
72C2×D4×C157—21no—
73C2×D4×C157C2311yescharacteristic
74C22×C4×C157—22no—
75C22×C4×C157—22no—
76C22×C4×C157—21no—
77C22×C4×C157D412yes—
78C22×C4×C157D411yescharacteristic
79C22×C4×C157C2312yes—
80C22×C4×C157C2311yescharacteristic
81C4⋊C4×C157—22no—
82C4⋊C4×C157C2312yes—
83C22⋊C4×C157—21no—
84C22⋊C4×C157—22no—
85C22⋊C4×C157—22no—
86C42×C157—21no—
87C22×D157—21no—
88Dic314—21no—
89Dic314—21no—
90D628—21no—
91D628—21no—
92C22⋊D157—42no—
93C22⋊D157—44no—
94C2×Dic157—21no—
95C2×Dic157—22no—
96C4×D157—42no—
97#11929434—22no—
98#11929434C2×D412yes—
99#11929434C2×D412yes—
100C23×C157—22no—
101C23×C157—24no—
102C23×C157—21no—
103C23×C157C2×D411yescharacteristic
104C23×C157C2×D412yes—
105Q8×C157—21no—
106Q8×C157—21no—
107D4×C157—21no—
108D4×C157—21no—
109D4×C157—42no—
110D4×C157—44no—
111C2×C4×C157—22no—
112C2×C4×C157—24no—
113C2×C4×C157—22no—
114C2×C4×C157—22no—
115C2×C4×C157—22no—
116C2×C4×C157—21no—
117C2×C4×C157—42no—
118C2×C4×C157C2×D412yes—
119C2×C4×C157C2×D412yes—
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157#4112yes—
122C2×C4×C157C2411yescharacteristic
123C22×C157—28no—
124C22×C157—22no—
125C22×C157—22no—
126C22×C157—22no—
127C22×C157—41no—
128C22×C157—42no—
129C22×C157#12112yes—
130C22×C157#13514yescomplemented
131C22×C157C22×D411yescharacteristic
132C2×D157—41no—
133C2×D157—42no—
134C4×C157—22no—
135C4×C157—41no—
136C4×C157—42no—
137C4×C157#12112yes—
138C4×C157C22×D411yescharacteristic
139C4×C157C2×#4111yescharacteristic
140Dic157—41no—
141Dic157—42no—
142C2×C157—22no—
143C2×C157—24no—
144C2×C157—81no—
145C2×C157C2×#12111yescharacteristic
146C2×C157C2×#13512yes—
147D157—81no—
148C157#305011yescharacteristic, complemented, Sylow
149#3050—1571nomaximal, Sylow
150C2×#135—1571no—
151C23×C8—1571no—
152#525—1571no—
153#525D15711yescharacteristic, complemented
154#467—1572no—
155#466—1572no—
156#465—1572no—
157#465—1572no—
158#427—1571no—
159#416—1571no—
160#416—1571no—
161C2×#41—1571no—
162C2×#41C2×D15711yescharacteristic
163C23×C4C2×D15711yescharacteristic
164#135—3144no—
165C2×Q16—1571no—
166C2×SD16—1571no—
167C2×SD16—1571no—
168C2×D8—1571no—
169C22×C8—3144no—
170C22×C8—1571no—
171C22×C8—1572no—
172C22×C8—1572no—
173C22×C8—1571no—
174#123—21no—
175#123—3141no—
176C22⋊Q8—1572no—
177C22⋊Q8C2×D15712yes—
178#121—1572no—
179#121C2×D15712yes—
180#120—21no—
181#120—3141no—
182C4×D4—22no—
183C4×D4—3142no—
184#117—1571no—
185#117C2×D15711yescharacteristic
186#107—1572no—
187#106—1572no—
188#103—1572no—
189#103—1572no—
190#102—1572no—
191#102—1572no—
192C24C22⋊D15711yescharacteristic
193#41—22no—
194#41—22no—
195#41—3142no—
196#41—3142no—
197C2×Q8—1571no—
198C2×Q8C22×D15711yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—3142no—
202C2×D4—3141no—
203C2×D4—1571no—
204C2×D4C22×D15711yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—3141no—
209C22×C4C22⋊D15711yescharacteristic
210C22×C4C22⋊D15712yes—
211C22×C4C22×D15712yes—
212C22×C4C22×D15711yescharacteristic
213Q16—3142no—
214SD16—3142no—
215SD16—3142no—
216D8—3142no—
217C2×C8—3142no—
218C2×C8—3144no—
219C2×C8—3144no—
220C2×C8—1572no—
221C2×C8—1574no—
222C2×C8—1571no—
223C2×C8—1571no—
224C4⋊C4—22no—
225C4⋊C4—3142no—
226C4⋊C4—1572no—
227C4⋊C4C22×D15712yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—3141no—
232C22⋊C4—3142no—
233C22⋊C4—3142no—
234C42—21no—
235C42—3141no—
236C23—22no—
237C23—24no—
238C23—21no—
239C23—3141no—
240C23C2×C22⋊D15711yescharacteristic
241C23C2×C22⋊D15712yes—
242Q8—21no—
243Q8—21no—
244Q8—3141no—
245Q8—3141no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—44no—
250D4—3141no—
251D4—3141no—
252D4—6282no—
253D4—6284no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—3142no—
262C2×C4—3141no—
263C2×C4—6282no—
264C2×C4D4×D15712yes—
265C2×C4C2×C22⋊D15712yes—
266C2×C4C2×C22⋊D15711yescharacteristic
267C2×C4#1457895312yes—
268C2×C4C23×D15711yescharacteristic
269C8—3142no—
270C8—1572no—
271C8—1572no—
272C22—28no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—42no—
278C22—6281no—
279C22—6282no—
280C22#1909774312yes—
281C22#1909779114yescomplemented
282C22C22×C22⋊D15711yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—6281no—
287C4—6282no—
288C4#1909774312yes—
289C4C2×D4×D15711yescharacteristic
290C4C2×#1457895311yescharacteristic
291C2—22no—
292C2—24no—
293C2—81no—
294C2—12561no—
295C2C2×#1909774311yescharacteristic
296C2C2×#1909779112yes—
subgroupG/H
centerC2×C4C2×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#525×C157C2
socleC22×C157C22×D4
hypercenter#525D157
hypocenterC157#3050
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#19097791C2semidirect
2. C23×#11929434C2semidirect
3. #23757784C2semidirect
4. #23757512C2semidirect
5. #23757507C2semidirect
6. #23757504C2semidirect
7. #23757500C2semidirect
8. #23757309C2semidirect
9. #23757269C2semidirect
10. #23757267C2semidirect
11. #525×C157C2semidirect
14. C2×#14578944C22semidirect
15. C2×#14578942C22semidirect
16. C2×#14578941C22semidirect
17. C2×#14578939C22semidirect
29. #19097704C22semidirect
30. #19097702C22semidirect
31. #19097695C22semidirect
32. #19097694C22semidirect
33. #19097692C22semidirect
34. #19097691C22semidirect
57. C2×#11929434D4semidirect
130. C22×C157#135semidirect
148. C157#3050semidirect
153. #525D157semidirect
281. C22#19097791semidirect

Maximal subgroups

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

subgroupindex
C2×#190977912
C23×#119294342
#237577842
#237575122
#237575072
#237575042
#237575002
#237573092
#237572692
#237572672
#525×C1572
#3050157