Finite Group Databasebeta

← #34110232

Subgroups — #34110232

Order 20096.

444 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 119 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#23757825C211yesmaximal, characteristic, complemented
2C2×#19097733C211yesmaximal, characteristic
3#23757608C211yesmaximal, characteristic, complemented
4#23757552C212yesmaximal, complemented
5#23757477C212yesmaximal, complemented
6#23757426C211yesmaximal, characteristic
7#23757423C211yesmaximal, characteristic
8#23757376C212yesmaximal, complemented
9#23757372C212yesmaximal, complemented
10#23757327C211yesmaximal, characteristic, complemented
11#484×C157C211yesmaximal, characteristic, complemented
12C22×D628C2211yescharacteristic
13C22×C4×D157—21no—
14C2×#14578943C2212yes—
15C2×#14578941C2212yes—
16#19097770C2211yescharacteristic, complemented
17#19097769C2211yescharacteristic, complemented
18#19097761—21no—
19#19097750C2211yescharacteristic
20#19097748—21no—
21Q8×Dic157C2211yescharacteristic, complemented
22#19097736—21no—
23#19097736C2211yescharacteristic, complemented
24#19097733—21no—
25#19097733—21no—
26#19097733C2212yes—
27#19097733C2212yes—
28#19097727—21no—
29C2×#14578931—21no—
30C2×C4×Dic157C2211yescharacteristic
31#19097704C2212yescomplemented
32#19097700C2212yescomplemented
33#19097694C2212yescomplemented
34#19097693C2212yes—
35#19097691C2212yes—
36C22⋊#11929434C2211yescharacteristic
37#19097681C2212yescomplemented
38C22⋊Q8×C157C2211yescharacteristic
39C2×C4⋊C4×C157C2211yescharacteristic
40#107×C157C2212yescomplemented
41#103×C157C2212yescomplemented
42C22⋊C8×C157C2211yescharacteristic
43C23×D157—21no—
44C2×Dic314—21no—
45C2×Dic314C2311yescharacteristic
46C2×D628—24no—
47C2×D628—21no—
48C2×D628C2312yes—
49C22×Dic157D411yescharacteristic
50C22×Dic157D411yescharacteristic
51C2×C4×D157—22no—
52C2×C4×D157—22no—
53C2×C4×D157—44no—
54#14578943—24no—
55#14578941—24no—
56C2×#11929434C2312yes—
57C4⋊Dic157—21no—
58C4⋊Dic157C2311yescharacteristic
59#14578933—21no—
60#14578933—21no—
61#14578933—21no—
62#14578933—21no—
63#14578933—21no—
64C22⋊Dic157—21no—
65#14578931—22no—
66#14578931—22no—
67C4×Dic157—21no—
68C4×Dic157—21no—
69C4×Dic157D411yescharacteristic
70C4×Dic157D411yescharacteristic
71C4×Dic157C2311yescharacteristic
72C4×Dic157C2311yescharacteristic
73C2×Q8×C157C2311yescharacteristic
74C22×C4×C157—21no—
75C22×C4×C157C2311yescharacteristic
76C2×C8×C157C2312yes—
77C4⋊C4×C157—21no—
78C4⋊C4×C157—21no—
79C4⋊C4×C157C2312yes—
80C4⋊C4×C157C2311yescharacteristic
81C22⋊C4×C157—21no—
82C22×D157—24no—
83C22×D157—22no—
84C22×D157—42no—
85C22×D157—42no—
86Dic314—22no—
87Dic314—42no—
88D628—24no—
89D628—22no—
90D628#4114yes—
91C2×Dic157—22no—
92C2×Dic157—22no—
93C2×Dic157—21no—
94C2×Dic157—21no—
95C2×Dic157C2×D411yescharacteristic
96C2×Dic157C2×D411yescharacteristic
97C2×Dic157C2×D411yescharacteristic
98C2×Dic157C2×D411yescharacteristic
99C4×D157—44no—
100C4×D157—44no—
101#11929434—22no—
102C23×C157C2×D411yescharacteristic
103Q8×C157—22no—
104C2×C4×C157—22no—
105C2×C4×C157—22no—
106C2×C4×C157—21no—
107C2×C4×C157—21no—
108C2×C4×C157—42no—
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2411yescharacteristic
111C8×C157—22no—
112C22×C157—21no—
113C22×C157—21no—
114C22×C157#13712yes—
115C22×C157C22×D411yescharacteristic
116C2×D157—24no—
117C2×D157—48no—
118C2×D157—42no—
119C2×D157—42no—
120C4×C157—21no—
121C4×C157—41no—
122C4×C157—42no—
123C4×C157—41no—
124C4×C157C2×#4112yes—
125Dic157—21no—
126Dic157—21no—
127Dic157—21no—
128Dic157—41no—
129Dic157—41no—
130Dic157AGL(1,ℤ/8)12yes—
131C2×C157—22no—
132C2×C157#52511yescharacteristic
133C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
134C2×C157C2×#13711yescharacteristic
135D157—44no—
136C157#311011yescharacteristic, complemented, Sylow
137#3110—1571nomaximal, Sylow
138#533—1571no—
139C2×C4×D4—1571no—
140#484—1571no—
141#484D15711yescharacteristic, complemented
142#474—1572no—
143#461—1572no—
144#450—1571no—
145#450—1571no—
146#440—1572no—
147#440—1572no—
148#432—1571no—
149C22×D4—1571no—
150C23×C4—3141no—
151C2×SD16—1572no—
152C2×SD16—1572no—
153C4⋊Q8—1571no—
154C4⋊Q8—1571no—
155#125—3141no—
156C22⋊Q8—3141no—
157C22⋊Q8—1571no—
158C22⋊Q8C2×D15711yescharacteristic
159C4×Q8—3141no—
160C4×Q8—1571no—
161C4×Q8—1571no—
162C4×D4—3141no—
163C4×D4—3141no—
164C4×D4—1572no—
165C4×D4—1572no—
166#117—3141no—
167C2×C4⋊C4C2×D15711yescharacteristic
168C2×C22⋊C4—3141no—
169C2×C42—1571no—
170#107—1572no—
171#107C2×D15712yescomplemented
172C4⋊C8—1572no—
173#103—1572no—
174#103C2×D15712yescomplemented
175#102—1572no—
176#102—1572no—
177C22⋊C8—1571no—
178C22⋊C8C2×D15711yescharacteristic
179#97—1572no—
180C24—3141no—
181C2×Q8—3141no—
182C2×Q8—1571no—
183C2×Q8C22×D15711yescharacteristic
184C2×D4—3144no—
185C2×D4—3141no—
186C2×D4—1572no—
187C22×C4—21no—
188C22×C4—3142no—
189C22×C4—3142no—
190C22×C4—6284no—
191C22×C4—1571no—
192C22×C4—1571no—
193C22×C4C22×D15711yescharacteristic
194SD16—3144no—
195SD16—3144no—
196C2×C8—1572no—
197C2×C8C22×D15712yes—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—3141no—
201C4⋊C4—3141no—
202C4⋊C4—3141no—
203C4⋊C4—3141no—
204C4⋊C4—3141no—
205C4⋊C4—3141no—
206C4⋊C4—1571no—
207C4⋊C4C22×D15711yescharacteristic
208C4⋊C4C22×D15712yes—
209C22⋊C4—21no—
210C22⋊C4—3142no—
211C22⋊C4—3142no—
212C22⋊C4—3141no—
213C42—3141no—
214C42—3141no—
215C42—1571no—
216C42—1571no—
217C42—1571no—
218C42—1571no—
219C23—3144no—
220C23—3142no—
221C23—6282no—
222C23—6282no—
223C23D4×D15711yescharacteristic
224Q8—22no—
225Q8—3142no—
226Q8—6282no—
227D4—3144no—
228D4—3142no—
229D4—1574no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—3142no—
236C2×C4—3142no—
237C2×C4—3141no—
238C2×C4—3141no—
239C2×C4—6284no—
240C2×C4—6284no—
241C2×C4—1571no—
242C2×C4—1571no—
243C2×C4—1571no—
244C2×C4—1571no—
245C2×C4D4×D15711yescharacteristic
246C2×C4C23×D15711yescharacteristic
247C8—22no—
248C8—3142no—
249C22—21no—
250C22—21no—
251C22—3144no—
252C22—6288no—
253C22—6282no—
254C22—6282no—
255C22#1909780312yes—
256C22C2×D4×D15711yescharacteristic
257C4—21no—
258C4—41no—
259C4—42no—
260C4—41no—
261C4—3141no—
262C4—3141no—
263C4—3141no—
264C4—6281no—
265C4—6281no—
266C4—1572no—
267C4#41×D15712yes—
268C2—22no—
269C2—6284no—
270C2#2375778111yescharacteristic
271C2AGL(1,ℤ/8)×D15711yescharacteristic
272C2C2×#1909780311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#484×C157C2
socleC22×C157C22×D4
prime radicalC22×D628C22
hypercenter#484D157
hypocenterC157#3110
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
1. #23757825C2semidirect
3. #23757608C2semidirect
4. #23757552C2semidirect
5. #23757477C2semidirect
8. #23757376C2semidirect
9. #23757372C2semidirect
10. #23757327C2semidirect
11. #484×C157C2semidirect
16. #19097770C22semidirect
17. #19097769C22semidirect
21. Q8×Dic157C22semidirect
23. #19097736C22semidirect
31. #19097704C22semidirect
32. #19097700C22semidirect
33. #19097694C22semidirect
37. #19097681C22semidirect
40. #107×C157C22semidirect
41. #103×C157C22semidirect
136. C157#3110semidirect
141. #484D157semidirect
171. #107C2×D157semidirect
174. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578252
C2×#190977332
#237576082
#237575522
#237574772
#237574262
#237574232
#237573762
#237573722
#237573272
#484×C1572
#3110157