Finite Group Databasebeta

← #34110275

Subgroups — #34110275

Order 20096.

450 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 121 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#23757809C211yesmaximal, characteristic, complemented
2C2×D4×Dic157C211yesmaximal, characteristic, complemented
3#23757601C211yesmaximal, characteristic, complemented
4#23757587C211yesmaximal, characteristic, complemented
5#23757571C212yesmaximal, complemented
6#23757490C212yesmaximal, complemented
7#23757407C211yesmaximal, characteristic
8#23757382C212yesmaximal, complemented
9#23757374C212yesmaximal, complemented
10#23757327C211yesmaximal, characteristic, complemented
11#450×C157C211yesmaximal, characteristic, complemented
12C23×Dic157—21no—
13C2×#14578939C2212yes—
14#19097766—21no—
15#19097762C2211yescharacteristic, complemented
16#19097760C2211yescharacteristic, complemented
17#19097755—21no—
18#19097745C2211yescharacteristic
19Q8×Dic157C2211yescharacteristic, complemented
20D4×Dic157—21no—
21D4×Dic157—21no—
22D4×Dic157C2212yes—
23D4×Dic157C2212yes—
24#19097733C2211yescharacteristic, complemented
25#19097731—21no—
26#19097729—21no—
27C2×C4⋊Dic157C2211yescharacteristic
28C2×C22⋊Dic157—21no—
29C2×C4×Dic157C2211yescharacteristic
30#19097705C2212yescomplemented
31#19097702C2212yescomplemented
32#19097700C2212yescomplemented
33#19097695C2212yescomplemented
34#19097693C2212yescomplemented
35#19097692C2212yes—
36C22⋊#11929434C2211yescharacteristic
37#19097681C2212yescomplemented
38C22×D4×C157C2211yescharacteristic
39C2×SD16×C157C2212yes—
40C22⋊Q8×C157C2211yescharacteristic
41#102×C157C2212yes—
42C22⋊C8×C157C2211yescharacteristic
43C2×D628C2311yescharacteristic
44C2×C22⋊D157—21no—
45C22×Dic157—22no—
46C22×Dic157—22no—
47C22×Dic157—21no—
48C22×Dic157—44no—
49C22×Dic157D411yescharacteristic
50C22×Dic157D411yescharacteristic
51C2×C4×D157—21no—
52#14578939—24no—
53C2×#11929434C2312yes—
54C4⋊Dic157—21no—
55C4⋊Dic157—21no—
56C4⋊Dic157—21no—
57C4⋊Dic157C2311yescharacteristic
58C4⋊Dic157C2312yes—
59#14578933—21no—
60C22⋊Dic157—22no—
61C22⋊Dic157—22no—
62#14578931—21no—
63#14578931—21no—
64#14578931—21no—
65#14578931—21no—
66C4×Dic157—21no—
67C4×Dic157D411yescharacteristic
68C4×Dic157D411yescharacteristic
69C4×Dic157C2311yescharacteristic
70C4×Dic157C2311yescharacteristic
71C24×C157—21no—
72C2×Q8×C157C2311yescharacteristic
73C2×D4×C157—24no—
74C2×D4×C157—21no—
75C2×D4×C157C2312yes—
76C22×C4×C157C2311yescharacteristic
77SD16×C157—24no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2311yescharacteristic
81C22⋊C4×C157—21no—
82C22×D157—21no—
83D628—22no—
84C22⋊D157—42no—
85C2×Dic157—22no—
86C2×Dic157—22no—
87C2×Dic157—21no—
88C2×Dic157—41no—
89C2×Dic157—44no—
90C2×Dic157—44no—
91C2×Dic157—42no—
92C2×Dic157C2×D411yescharacteristic
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96C4×D157—42no—
97#11929434—22no—
98C23×C157—24no—
99C23×C157—22no—
100C23×C157—42no—
101C23×C157—42no—
102C23×C157C2×D411yescharacteristic
103Q8×C157—22no—
104D4×C157—24no—
105D4×C157—22no—
106D4×C157#4114yes—
107C2×C4×C157—22no—
108C2×C4×C157—21no—
109C2×C4×C157—21no—
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2411yescharacteristic
112C8×C157—22no—
113C22×C157—21no—
114C22×C157—21no—
115C22×C157—24no—
116C22×C157—48no—
117C22×C157—42no—
118C22×C157—42no—
119C22×C157AGL(1,ℤ/8)12yes—
120C22×C157C22×D411yescharacteristic
121C2×D157—41no—
122C2×D157—41no—
123C2×D157—41no—
124C4×C157—21no—
125C4×C157—41no—
126C4×C157—41no—
127C4×C157C2×#4112yes—
128Dic157—21no—
129Dic157—21no—
130Dic157—21no—
131Dic157—21no—
132Dic157—42no—
133Dic157—41no—
134C2×C157—22no—
135C2×C157—44no—
136C2×C157#52511yescharacteristic
137C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C157#57511yescharacteristic
139D157—81no—
140C157#311411yescharacteristic, complemented, Sylow
141#3114—1571nomaximal, Sylow
142#529—1571no—
143C2×C4×D4—1571no—
144#483—1571no—
145#480—1571no—
146#478—1572no—
147#463—1572no—
148#450D15711yescharacteristic, complemented
149#447—1571no—
150#442—1572no—
151#440—1572no—
152#432—1571no—
153C22×D4C2×D15711yescharacteristic
154C23×C4—3141no—
155C2×SD16C2×D15712yes—
156C2×D8—1572no—
157#126—3141no—
158#125—1571no—
159#124—1571no—
160#123—3141no—
161C22⋊Q8C2×D15711yescharacteristic
162#121—1571no—
163C4×Q8—1571no—
164C4×D4—3141no—
165C4×D4—3141no—
166C4×D4—3141no—
167C4×D4—1572no—
168C4×D4—1572no—
169C4×D4—1571no—
170#117—3141no—
171C2×C4⋊C4—1571no—
172C2×C22⋊C4—3141no—
173C2×C42—1571no—
174#107—1572no—
175#106—1572no—
176C4⋊C8—1572no—
177#103—1572no—
178#102—1572no—
179#102—1572no—
180#102C2×D15712yes—
181C22⋊C8—1571no—
182C22⋊C8C2×D15711yescharacteristic
183#97—1572no—
184C24—21no—
185C2×Q8C22×D15711yescharacteristic
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—3141no—
189C2×D4—1571no—
190C2×D4C22×D15712yes—
191C22×C4—3141no—
192C22×C4—3141no—
193C22×C4—3142no—
194C22×C4—3142no—
195C22×C4—6284no—
196C22×C4—1571no—
197C22×C4—1571no—
198C22×C4C22×D15711yescharacteristic
199SD16—24no—
200D8—3144no—
201C2×C8—1572no—
202C2×C8C22×D15712yes—
203C4⋊C4—21no—
204C4⋊C4—3141no—
205C4⋊C4—3141no—
206C4⋊C4—3141no—
207C4⋊C4—3141no—
208C4⋊C4—1571no—
209C4⋊C4—1572no—
210C4⋊C4C22×D15711yescharacteristic
211C22⋊C4—21no—
212C22⋊C4—3141no—
213C22⋊C4—3141no—
214C22⋊C4—3141no—
215C22⋊C4—3142no—
216C22⋊C4—3142no—
217C22⋊C4—3141no—
218C42—3141no—
219C42—1571no—
220C42—1571no—
221C42—1571no—
222C42—1571no—
223C23—24no—
224C23—22no—
225C23—42no—
226C23—42no—
227C23—3141no—
228C23D4×D15711yescharacteristic
229Q8—22no—
230D4—24no—
231D4—22no—
232D4—3142no—
233D4—6282no—
234D4#1457895314yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—3142no—
239C2×C4—3141no—
240C2×C4—3142no—
241C2×C4—6282no—
242C2×C4—6281no—
243C2×C4—6284no—
244C2×C4—6284no—
245C2×C4—6282no—
246C2×C4—1571no—
247C2×C4—1571no—
248C2×C4—1571no—
249C2×C4—1571no—
250C2×C4D4×D15711yescharacteristic
251C2×C4C23×D15711yescharacteristic
252C8—22no—
253C8—3142no—
254C22—21no—
255C22—21no—
256C22—24no—
257C22—48no—
258C22—42no—
259C22—42no—
260C22—6281no—
261C22—6281no—
262C22—6281no—
263C22#1909779812yes—
264C22C2×D4×D15711yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—3141no—
269C4—3141no—
270C4—3141no—
271C4—3141no—
272C4—6282no—
273C4—6281no—
274C4C2×#1457895312yes—
275C2—22no—
276C2—44no—
277C2—12561no—
278C2#2375777911yescharacteristic
279C2C2×#1909779811yescharacteristic
280C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#450×C157C2
socleC22×C157C22×D4
prime radical#23757407C2
hypercenter#450D157
hypocenterC157#3114
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #23757809C2semidirect
2. C2×D4×Dic157C2semidirect
3. #23757601C2semidirect
4. #23757587C2semidirect
5. #23757571C2semidirect
6. #23757490C2semidirect
8. #23757382C2semidirect
9. #23757374C2semidirect
10. #23757327C2semidirect
11. #450×C157C2semidirect
15. #19097762C22semidirect
16. #19097760C22semidirect
19. Q8×Dic157C22semidirect
24. #19097733C22semidirect
30. #19097705C22semidirect
31. #19097702C22semidirect
32. #19097700C22semidirect
33. #19097695C22semidirect
34. #19097693C22semidirect
37. #19097681C22semidirect
140. C157#3114semidirect
148. #450D157semidirect

Maximal subgroups

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

subgroupindex
#237578092
C2×D4×Dic1572
#237576012
#237575872
#237575712
#237574902
#237574072
#237573822
#237573742
#237573272
#450×C1572
#3114157