Finite Group Databasebeta

← #34110269

Subgroups — #34110269

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#23757804C211yesmaximal, characteristic, complemented
2C2×D4×Dic157C211yesmaximal, characteristic, complemented
3#23757604C211yesmaximal, characteristic, complemented
4#23757584C211yesmaximal, characteristic, complemented
5#23757575C212yesmaximal, complemented
6#23757486C212yesmaximal, complemented
7#23757424C211yesmaximal, characteristic, complemented
8#23757385C212yesmaximal, complemented
9#23757370C212yesmaximal, complemented
10#23757327C211yesmaximal, characteristic, complemented
11#447×C157C211yesmaximal, characteristic
12C23×Dic157—21no—
13C2×#14578942C2212yes—
14#19097764—21no—
15#19097762C2211yescharacteristic, complemented
16#19097758C2211yescharacteristic, complemented
17#19097752—21no—
18#19097750C2211yescharacteristic
19#19097736C2211yescharacteristic, complemented
20D4×Dic157—21no—
21D4×Dic157—21no—
22D4×Dic157—21no—
23D4×Dic157C2212yes—
24D4×Dic157C2212yes—
25D4×Dic157C2211yescharacteristic, complemented
26#19097727—21no—
27C2×C4⋊Dic157C2211yescharacteristic
28C2×C22⋊Dic157—21no—
29C2×C4×Dic157C2211yescharacteristic
30#19097704C2212yescomplemented
31#19097703C2212yescomplemented
32#19097700C2212yescomplemented
33#19097696C2212yescomplemented
34#19097692C2212yes—
35#19097692C2212yescomplemented
36C22⋊#11929434C2211yescharacteristic
37#19097681C2212yescomplemented
38C22×D4×C157C2211yescharacteristic
39C2×D8×C157C2212yes—
40#121×C157C2211yescharacteristic
41#102×C157C2212yes—
42C22⋊C8×C157C2211yescharacteristic
43C2×Dic314C2311yescharacteristic
44C22×Dic157—21no—
45C22×Dic157—22no—
46C22×Dic157—22no—
47C22×Dic157—21no—
48C22×Dic157—44no—
49C22×Dic157D411yescharacteristic
50C22×Dic157D411yescharacteristic
51#14578942—24no—
52C2×#11929434C2312yes—
53C4⋊Dic157—21no—
54C4⋊Dic157—21no—
55C4⋊Dic157C2311yescharacteristic
56C4⋊Dic157C2312yes—
57#14578933—21no—
58#14578933—21no—
59#14578933—21no—
60C22⋊Dic157—22no—
61C22⋊Dic157—22no—
62C22⋊Dic157—21no—
63C22⋊Dic157—21no—
64C22⋊Dic157—21no—
65C22⋊Dic157—21no—
66C4×Dic157—21no—
67C4×Dic157D411yescharacteristic
68C4×Dic157D411yescharacteristic
69C4×Dic157C2311yescharacteristic
70C4×Dic157C2311yescharacteristic
71C24×C157—21no—
72C2×D4×C157—24no—
73C2×D4×C157—21no—
74C2×D4×C157—21no—
75C2×D4×C157C2311yescharacteristic
76C2×D4×C157C2312yes—
77C22×C4×C157C2311yescharacteristic
78D8×C157—24no—
79C2×C8×C157C2312yes—
80C4⋊C4×C157C2311yescharacteristic
81C22⋊C4×C157—21no—
82Dic314—22no—
83C2×Dic157—22no—
84C2×Dic157—22no—
85C2×Dic157—21no—
86C2×Dic157—21no—
87C2×Dic157—41no—
88C2×Dic157—42no—
89C2×Dic157—44no—
90C2×Dic157—44no—
91C2×Dic157—42no—
92C2×Dic157C2×D411yescharacteristic
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96#11929434—22no—
97C23×C157—24no—
98C23×C157—22no—
99C23×C157—21no—
100C23×C157—42no—
101C23×C157—42no—
102C23×C157C2×D411yescharacteristic
103D4×C157—24no—
104D4×C157—22no—
105D4×C157—22no—
106D4×C157—42no—
107D4×C157#4114yes—
108C2×C4×C157—22no—
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×C157—41no—
120C22×C157—41no—
121C22×C157—41no—
122C22×C157AGL(1,ℤ/8)12yes—
123C22×C157C22×D411yescharacteristic
124C4×C157—21no—
125C4×C157—41no—
126C4×C157C2×#4112yes—
127Dic157—21no—
128Dic157—21no—
129Dic157—21no—
130Dic157—21no—
131Dic157—41no—
132Dic157—42no—
133Dic157—41no—
134C2×C157—22no—
135C2×C157—44no—
136C2×C157—81no—
137C2×C157#52511yescharacteristic
138C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
139C2×C157#57511yescharacteristic
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#450—1571no—
149#447D15711yescharacteristic
150#442—1572no—
151#440—1572no—
152#432—1571no—
153C22×D4C2×D15711yescharacteristic
154C23×C4—3141no—
155C2×SD16—1572no—
156C2×D8C2×D15712yes—
157#126—3141no—
158#125—1571no—
159#124—1571no—
160#123—3141no—
161C22⋊Q8—1571no—
162#121C2×D15711yescharacteristic
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×Q8—1571no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4C22×D15711yescharacteristic
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—3144no—
200D8—24no—
201C2×C8—1572no—
202C2×C8C22×D15712yes—
203C4⋊C4—3141no—
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—3142no—
214C22⋊C4—3142no—
215C22⋊C4—3141no—
216C22⋊C4—3141no—
217C22⋊C4—3141no—
218C42—3141no—
219C42—1571no—
220C42—1571no—
221C42—1571no—
222C42—1571no—
223C23—24no—
224C23—22no—
225C23—21no—
226C23—42no—
227C23—42no—
228C23D4×D15711yescharacteristic
229Q8—3142no—
230D4—24no—
231D4—22no—
232D4—22no—
233D4—42no—
234D4#1457895314yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—3142no—
238C2×C4—3141no—
239C2×C4—3141no—
240C2×C4—3142no—
241C2×C4—6281no—
242C2×C4—6284no—
243C2×C4—6284no—
244C2×C4—6282no—
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—41no—
261C22—41no—
262C22—41no—
263C22#1909779512yes—
264C22C2×D4×D15711yescharacteristic
265C4—21no—
266C4—41no—
267C4—3141no—
268C4—3141no—
269C4—3141no—
270C4—3141no—
271C4—6281no—
272C4—6282no—
273C4—6281no—
274C4C2×#1457895312yes—
275C2—22no—
276C2—44no—
277C2—81no—
278C2#2375777911yescharacteristic
279C2C2×#1909779511yescharacteristic
280C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting, prime radical#447×C157C2
socleC22×C157C22×D4
hypercenter#447D157
hypocenterC157#3114
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #23757804C2semidirect
2. C2×D4×Dic157C2semidirect
3. #23757604C2semidirect
4. #23757584C2semidirect
5. #23757575C2semidirect
6. #23757486C2semidirect
7. #23757424C2semidirect
8. #23757385C2semidirect
9. #23757370C2semidirect
10. #23757327C2semidirect
15. #19097762C22semidirect
16. #19097758C22semidirect
19. #19097736C22semidirect
25. D4×Dic157C22semidirect
30. #19097704C22semidirect
31. #19097703C22semidirect
32. #19097700C22semidirect
33. #19097696C22semidirect
35. #19097692C22semidirect
37. #19097681C22semidirect
140. C157#3114semidirect

Maximal subgroups

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

subgroupindex
#237578042
C2×D4×Dic1572
#237576042
#237575842
#237575752
#237574862
#237574242
#237573852
#237573702
#237573272
#447×C1572
#3114157