Finite Group Databasebeta

← #34110133

Subgroups — #34110133

Order 20096.

492 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 492 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#23757818C211yesmaximal, characteristic, complemented
2C2×#19097733C211yesmaximal, characteristic, complemented
3#23757586C211yesmaximal, characteristic, complemented
4#23757552C212yesmaximal, complemented
5#23757467C212yesmaximal, complemented
6#23757409C211yesmaximal, characteristic, complemented
7#23757406C211yesmaximal, characteristic, complemented
8#23757355C212yesmaximal, complemented
9#23757353C212yesmaximal, complemented
10#23757323C211yesmaximal, characteristic, complemented
11#480×C157C211yesmaximal, characteristic, complemented
12C2×#14578953—21no—
13C22×D628C2211yescharacteristic
14C22×C4×D157—21no—
15C2×D1256C2212yes—
16C2×#14578939C2212yes—
17#19097770C2211yescharacteristic, complemented
18#19097767C2211yescharacteristic, complemented
19#19097757—21no—
20#19097745C2211yescharacteristic
21#19097743—21no—
22D4×Dic157C2211yescharacteristic, complemented
23#19097733—21no—
24#19097733—21no—
25#19097733C2212yes—
26#19097733C2212yes—
27#19097733C2211yescharacteristic, complemented
28#19097731—21no—
29C2×#14578931—21no—
30C2×C4×Dic157C2211yescharacteristic
31#19097704C2212yescomplemented
32#19097700C2212yescomplemented
33#19097693C2212yes—
34#19097691C2212yes—
35#19097691C2212yescomplemented
36C22⋊#11929434C2211yescharacteristic
37C8×Dic157C2212yescomplemented
38#121×C157C2211yescharacteristic
39C2×C4⋊C4×C157C2211yescharacteristic
40#107×C157C2212yescomplemented
41#102×C157C2212yescomplemented
42C22⋊C8×C157C2211yescharacteristic
43C23×D157—21no—
44#14578953—44no—
45C2×Dic314—21no—
46C2×D628—24no—
47C2×D628—21no—
48C2×D628C2311yescharacteristic
49C2×D628C2312yes—
50C2×C22⋊D157—21no—
51C2×C22⋊D157—21no—
52C2×C22⋊D157—21no—
53C22×Dic157—21no—
54C22×Dic157D411yescharacteristic
55C22×Dic157D411yescharacteristic
56C2×C4×D157—21no—
57C2×C4×D157—22no—
58C2×C4×D157—22no—
59C2×C4×D157—44no—
60D1256—24no—
61#14578939—24no—
62C2×#11929434C2312yes—
63C4⋊Dic157C2311yescharacteristic
64#14578933—21no—
65C22⋊Dic157—21no—
66#14578931—22no—
67#14578931—22no—
68#14578931—21no—
69#14578931—21no—
70C4×Dic157D411yescharacteristic
71C4×Dic157D411yescharacteristic
72C4×Dic157C2311yescharacteristic
73C4×Dic157C2311yescharacteristic
74C2×D4×C157—21no—
75C2×D4×C157C2311yescharacteristic
76C22×C4×C157—21no—
77C22×C4×C157C2311yescharacteristic
78C2×C8×C157C2312yes—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2312yes—
81C4⋊C4×C157C2311yescharacteristic
82C22⋊C4×C157—21no—
83C22×D157—24no—
84C22×D157—22no—
85C22×D157—21no—
86C22×D157—42no—
87C22×D157—42no—
88Dic314—42no—
89D628—24no—
90D628—22no—
91D628—22no—
92D628#4114yescomplemented
93C22⋊D157—42no—
94C22⋊D157—42no—
95C22⋊D157—42no—
96C2×Dic157—21no—
97C2×Dic157—41no—
98C2×Dic157—42no—
99C2×Dic157D814yescomplemented
100C2×Dic157C2×D411yescharacteristic
101C2×Dic157C2×D411yescharacteristic
102C2×Dic157C2×D411yescharacteristic
103C2×Dic157C2×D411yescharacteristic
104C4×D157—44no—
105C4×D157—44no—
106C4×D157—42no—
107#11929434—22no—
108C23×C157—21no—
109C23×C157C2×D411yescharacteristic
110D4×C157—22no—
111D4×C157—42no—
112C2×C4×C157—22no—
113C2×C4×C157—22no—
114C2×C4×C157—21no—
115C2×C4×C157—42no—
116C2×C4×C157C2×D411yescharacteristic
117C2×C4×C157C2411yescharacteristic
118C8×C157—22no—
119C22×C157—21no—
120C22×C157—21no—
121C22×C157—41no—
122C22×C157—41no—
123C22×C157—41no—
124C22×C157C2×D812yes—
125C22×C157C22×D411yescharacteristic
126C2×D157—24no—
127C2×D157—48no—
128C2×D157—42no—
129C2×D157—42no—
130C2×D157—41no—
131C2×D157—41no—
132C2×D157—41no—
133C4×C157—21no—
134C4×C157—42no—
135C4×C157—41no—
136C4×C157C2×#4112yes—
137Dic157—21no—
138Dic157—21no—
139Dic157—41no—
140Dic157C2×D812yes—
141Dic157C2×D812yes—
142C2×C157—22no—
143C2×C157—81no—
144C2×C157#52511yescharacteristic
145C2×C157C22×D811yescharacteristic
146C2×C157#57511yescharacteristic
147D157—44no—
148D157—81no—
149C157#310011yescharacteristic, complemented, Sylow
150#3100—1571nomaximal, Sylow
151#532—1571no—
152C2×C4×D4—1571no—
153#480—1571no—
154#480D15711yescharacteristic, complemented
155#474—1572no—
156#459—1572no—
157#447—1571no—
158#447—1571no—
159C4×D8—1572no—
160C4×D8—1572no—
161#431—1571no—
162C2×#41—3141no—
163C22×D4—1571no—
164C23×C4—3141no—
165C2×D8—1572no—
166C2×D8—1572no—
167C4⋊Q8—1571no—
168#127—1571no—
169#124—3141no—
170#121—3141no—
171#121—1571no—
172#121C2×D15711yescharacteristic
173C4×D4—3141no—
174C4×D4—3141no—
175C4×D4—3141no—
176C4×D4—1571no—
177C4×D4—1572no—
178C4×D4—1572no—
179C4×D4—1571no—
180C2×C4⋊C4C2×D15711yescharacteristic
181C2×C22⋊C4—3141no—
182C2×C42—1571no—
183#107—1572no—
184#107C2×D15712yescomplemented
185C4⋊C8—1572no—
186#102—1572no—
187#102—1572no—
188#102—1572no—
189#102C2×D15712yescomplemented
190C22⋊C8—1571no—
191C22⋊C8C2×D15711yescharacteristic
192C4×C8—1572no—
193C24—3141no—
194#41—6284no—
195C2×Q8—3141no—
196C2×D4—21no—
197C2×D4—3144no—
198C2×D4—3141no—
199C2×D4—3141no—
200C2×D4—3141no—
201C2×D4—3141no—
202C2×D4—1571no—
203C2×D4—1572no—
204C2×D4C22×D15711yescharacteristic
205C22×C4—21no—
206C22×C4—3142no—
207C22×C4—3142no—
208C22×C4—3141no—
209C22×C4—3141no—
210C22×C4—6284no—
211C22×C4—1571no—
212C22×C4—1571no—
213C22×C4C22×D15711yescharacteristic
214D8—3144no—
215D8—3144no—
216C2×C8—1572no—
217C2×C8C22×D15712yes—
218C4⋊C4—21no—
219C4⋊C4—3141no—
220C4⋊C4—1571no—
221C4⋊C4C22×D15711yescharacteristic
222C4⋊C4C22×D15712yes—
223C22⋊C4—21no—
224C22⋊C4—3142no—
225C22⋊C4—3142no—
226C22⋊C4—3141no—
227C22⋊C4—3141no—
228C22⋊C4—3141no—
229C42—1571no—
230C42—1571no—
231C42—1571no—
232C42—1571no—
233C23—21no—
234C23—3144no—
235C23—3142no—
236C23—3141no—
237C23—6282no—
238C23—6282no—
239C23D4×D15711yescharacteristic
240Q8—6282no—
241D4—22no—
242D4—42no—
243D4—3144no—
244D4—3142no—
245D4—3142no—
246D4—6282no—
247D4—6282no—
248D4—6282no—
249D4—1574no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—3141no—
255C2×C4—6284no—
256C2×C4—6284no—
257C2×C4—6282no—
258C2×C4—6281no—
259C2×C4—6282no—
260C2×C4—1571no—
261C2×C4—1571no—
262C2×C4—1574no—
263C2×C4—1571no—
264C2×C4—1571no—
265C2×C4D4×D15711yescharacteristic
266C2×C4C23×D15711yescharacteristic
267C8—22no—
268C8—3142no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—3144no—
275C22—6288no—
276C22—6282no—
277C22—6282no—
278C22—6281no—
279C22—6281no—
280C22—6281no—
281C22D8×D15712yes—
282C22C2×D4×D15711yescharacteristic
283C4—21no—
284C4—42no—
285C4—41no—
286C4—3141no—
287C4—3141no—
288C4—6281no—
289C4—1572no—
290C4—1572no—
291C4#41×D15712yes—
292C2—22no—
293C2—81no—
294C2—6284no—
295C2—12561no—
296C2#2375778111yescharacteristic
297C2C2×D8×D15711yescharacteristic
298C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#480×C157C2
socleC22×C157C22×D4
hypercenter#480D157
hypocenterC157#3100
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #480

Complements

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

NH ≅ G/Ntype
1. #23757818C2semidirect
2. C2×#19097733C2semidirect
3. #23757586C2semidirect
4. #23757552C2semidirect
5. #23757467C2semidirect
6. #23757409C2semidirect
7. #23757406C2semidirect
8. #23757355C2semidirect
9. #23757353C2semidirect
10. #23757323C2semidirect
11. #480×C157C2semidirect
17. #19097770C22semidirect
18. #19097767C22semidirect
22. D4×Dic157C22semidirect
27. #19097733C22semidirect
31. #19097704C22semidirect
32. #19097700C22semidirect
35. #19097691C22semidirect
37. C8×Dic157C22semidirect
40. #107×C157C22semidirect
41. #102×C157C22semidirect
92. D628#41semidirect
99. C2×Dic157D8semidirect
149. C157#3100semidirect
154. #480D157semidirect
184. #107C2×D157semidirect
189. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578182
C2×#190977332
#237575862
#237575522
#237574672
#237574092
#237574062
#237573552
#237573532
#237573232
#480×C1572
#3100157