Finite Group Databasebeta

← #34110270

Subgroups — #34110270

Order 20096.

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

normal subgroups only

This group has 450 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
1C2×C4×D628C211yesmaximal, characteristic, complemented
2#23757601C211yesmaximal, characteristic, complemented
3#23757584C211yesmaximal, characteristic, complemented
4#23757570C212yesmaximal, complemented
5#23757485C212yesmaximal, complemented
6#23757423C211yesmaximal, characteristic, complemented
7#23757406C211yesmaximal, characteristic
8#23757383C212yesmaximal, complemented
9#23757371C212yesmaximal, complemented
10#23757330C211yesmaximal, characteristic, complemented
11#529×C157C211yesmaximal, characteristic, complemented
12C22×D628C2211yescharacteristic
13C22×C4×D157—21no—
14C2×#14578941C2212yes—
15C2×#14578939C2212yes—
16C4×D628—21no—
17C4×D628—21no—
18C4×D628C2212yes—
19C4×D628C2212yes—
20C2×C4⋊Dic157C2211yescharacteristic
21C2×#14578931—21no—
22#19097704C2212yescomplemented
23#19097702C2212yescomplemented
24C4⋊#11929434C2212yescomplemented
25#19097695C2212yescomplemented
26#19097692C2212yescomplemented
27#19097691C2212yes—
28#19097691C2212yes—
29C22⋊#11929434C2211yescharacteristic
30C22⋊#11929434C2211yescharacteristic
31#19097682C2212yescomplemented
32#126×C157—21no—
33#125×C157C2211yescharacteristic, complemented
34#124×C157C2211yescharacteristic, complemented
35#123×C157—21no—
36C22⋊Q8×C157C2211yescharacteristic
37#121×C157C2211yescharacteristic
38C4×Q8×C157C2211yescharacteristic, complemented
39C4×D4×C157—21no—
40C4×D4×C157C2211yescharacteristic, complemented
41#117×C157—21no—
42C2×C42×C157C2211yescharacteristic
43C23×D157—21no—
44C2×D628—24no—
45C2×D628—21no—
46C2×D628C2312yes—
47C22×Dic157—21no—
48C2×C4×D157—22no—
49C2×C4×D157—22no—
50C2×C4×D157—44no—
51#14578941—24no—
52#14578939—24no—
53C2×#11929434C2312yes—
54C2×#11929434C2312yes—
55C4⋊Dic157—21no—
56C4⋊Dic157C2312yes—
57#14578931—22no—
58#14578931—22no—
59C2×Q8×C157C2311yescharacteristic
60C2×D4×C157—21no—
61C2×D4×C157C2311yescharacteristic
62C22×C4×C157—21no—
63C22×C4×C157D411yescharacteristic
64C22×C4×C157D411yescharacteristic
65C22×C4×C157C2311yescharacteristic
66C4⋊C4×C157—21no—
67C4⋊C4×C157—21no—
68C4⋊C4×C157—21no—
69C4⋊C4×C157—21no—
70C4⋊C4×C157C2311yescharacteristic
71C4⋊C4×C157C2311yescharacteristic
72C22⋊C4×C157—21no—
73C22⋊C4×C157—21no—
74C22⋊C4×C157—21no—
75C22⋊C4×C157—21no—
76C22⋊C4×C157—21no—
77C42×C157—21no—
78C42×C157D411yescharacteristic
79C42×C157D411yescharacteristic
80C42×C157C2311yescharacteristic
81C42×C157C2311yescharacteristic
82C22×D157—24no—
83C22×D157—22no—
84C22×D157—42no—
85C22×D157—42no—
86D628—24no—
87D628—22no—
88D628#4114yes—
89C2×Dic157—22no—
90C2×Dic157—42no—
91C4×D157—44no—
92C4×D157—44no—
93#11929434—22no—
94#11929434—22no—
95C23×C157—21no—
96C23×C157C2×D411yescharacteristic
97Q8×C157—22no—
98D4×C157—22no—
99D4×C157—42no—
100C2×C4×C157—22no—
101C2×C4×C157—22no—
102C2×C4×C157—21no—
103C2×C4×C157—21no—
104C2×C4×C157—21no—
105C2×C4×C157—41no—
106C2×C4×C157—42no—
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2×D411yescharacteristic
112C2×C4×C157C2411yescharacteristic
113C22×C157—21no—
114C22×C157—21no—
115C22×C157—41no—
116C22×C157—41no—
117C22×C157—41no—
118C22×C157AGL(1,ℤ/8)12yes—
119C22×C157C22×D411yescharacteristic
120C2×D157—24no—
121C2×D157—48no—
122C2×D157—42no—
123C2×D157—42no—
124C4×C157—21no—
125C4×C157—21no—
126C4×C157—21no—
127C4×C157—21no—
128C4×C157—21no—
129C4×C157—41no—
130C4×C157—41no—
131C4×C157—41no—
132C4×C157C2×#4112yes—
133Dic157—42no—
134C2×C157—22no—
135C2×C157—81no—
136C2×C157#52511yescharacteristic
137C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C157#57511yescharacteristic
139D157—44no—
140C157#311411yescharacteristic, complemented, Sylow
141#3114—1571nomaximal, Sylow
142#529D15711yescharacteristic, complemented
143C2×C4×D4—1571no—
144#483—1571no—
145#480—1571no—
146#478—1572no—
147#463—1572no—
148#450—1571no—
149#447—1571no—
150#442—1572no—
151#440—1572no—
152#432—1571no—
153C22×D4—1571no—
154C23×C4—3141no—
155C2×SD16—1572no—
156C2×D8—1572no—
157#126—21no—
158#125C2×D15711yescharacteristic, complemented
159#124C2×D15711yescharacteristic, complemented
160#123—21no—
161C22⋊Q8C2×D15711yescharacteristic
162#121C2×D15711yescharacteristic
163C4×Q8C2×D15711yescharacteristic, complemented
164C4×D4—21no—
165C4×D4—3141no—
166C4×D4—3141no—
167C4×D4—1572no—
168C4×D4—1572no—
169C4×D4C2×D15711yescharacteristic, complemented
170#117—21no—
171C2×C4⋊C4—1571no—
172C2×C22⋊C4—3141no—
173C2×C42C2×D15711yescharacteristic
174#107—1572no—
175#106—1572no—
176C4⋊C8—1572no—
177#103—1572no—
178#102—1572no—
179#102—1572no—
180#102—1572no—
181C22⋊C8—1571no—
182C22⋊C8—1571no—
183#97—1572no—
184C24—3141no—
185C2×Q8C22×D15711yescharacteristic
186C2×D4—21no—
187C2×D4—3144no—
188C2×D4—3141no—
189C2×D4—1572no—
190C2×D4C22×D15711yescharacteristic
191C22×C4—21no—
192C22×C4—3141no—
193C22×C4—3142no—
194C22×C4—3142no—
195C22×C4—6284no—
196C22×C4C22⋊D15711yescharacteristic
197C22×C4C22⋊D15711yescharacteristic
198C22×C4C22×D15711yescharacteristic
199SD16—3144no—
200D8—3144no—
201C2×C8—1572no—
202C2×C8—1572no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—3141no—
208C4⋊C4—1572no—
209C4⋊C4C22×D15711yescharacteristic
210C4⋊C4C22×D15711yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—3142no—
217C22⋊C4—3142no—
218C42—21no—
219C42C22⋊D15711yescharacteristic
220C42C22⋊D15711yescharacteristic
221C42C22×D15711yescharacteristic
222C42C22×D15711yescharacteristic
223C23—21no—
224C23—3144no—
225C23—3142no—
226C23—6282no—
227C23—6282no—
228C23C2×C22⋊D15711yescharacteristic
229Q8—22no—
230D4—22no—
231D4—42no—
232D4—3144no—
233D4—3142no—
234D4—1574no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—3142no—
243C2×C4—6284no—
244C2×C4—6284no—
245C2×C4—6282no—
246C2×C4C2×C22⋊D15711yescharacteristic
247C2×C4C2×C22⋊D15711yescharacteristic
248C2×C4C2×C22⋊D15711yescharacteristic
249C2×C4C2×C22⋊D15711yescharacteristic
250C2×C4C2×C22⋊D15711yescharacteristic
251C2×C4C23×D15711yescharacteristic
252C8—3142no—
253C8—3142no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—3144no—
260C22—6288no—
261C22—6282no—
262C22—6282no—
263C22#1909779612yes—
264C22C22×C22⋊D15711yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—21no—
269C4—21no—
270C4—41no—
271C4—41no—
272C4—41no—
273C4—6282no—
274C4#41×D15712yes—
275C2—22no—
276C2—81no—
277C2—6284no—
278C2#2375778011yescharacteristic
279C2C2×#1909779611yescharacteristic
280C2#2375807111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#529×C157C2
socleC22×C157C22×D4
prime radical#23757406C2
hypercenter#529D157
hypocenterC157#3114
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C2×C4×D628C2semidirect
2. #23757601C2semidirect
3. #23757584C2semidirect
4. #23757570C2semidirect
5. #23757485C2semidirect
6. #23757423C2semidirect
8. #23757383C2semidirect
9. #23757371C2semidirect
10. #23757330C2semidirect
11. #529×C157C2semidirect
22. #19097704C22semidirect
23. #19097702C22semidirect
24. C4⋊#11929434C22semidirect
25. #19097695C22semidirect
26. #19097692C22semidirect
31. #19097682C22semidirect
33. #125×C157C22semidirect
34. #124×C157C22semidirect
38. C4×Q8×C157C22semidirect
40. C4×D4×C157C22semidirect
140. C157#3114semidirect
142. #529D157semidirect
158. #125C2×D157semidirect
159. #124C2×D157semidirect
163. C4×Q8C2×D157semidirect
169. C4×D4C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D6282
#237576012
#237575842
#237575702
#237574852
#237574232
#237574062
#237573832
#237573712
#237573302
#529×C1572
#3114157