Finite Group Databasebeta

← #34110049

Subgroups — #34110049

Order 20096.

420 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 137 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#23757784C211yesmaximal, characteristic
2#23757614C212yesmaximal, complemented
3#23757608C211yesmaximal, characteristic, complemented
4#23757603C211yesmaximal, characteristic, complemented
5#23757597C212yesmaximal, complemented
6#23757591C211yesmaximal, characteristic, complemented
7#23757586C211yesmaximal, characteristic, complemented
8#23757309C212yesmaximal, complemented
9#23757267C212yesmaximal, complemented
10C2×C22⋊#11929434C211yesmaximal, characteristic, complemented
11C2×#117×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×#11929434C2212yes—
14#19097754—21no—
15#19097750C2211yescharacteristic
16#19097750C2211yescharacteristic
17#19097745C2211yescharacteristic
18#19097745C2211yescharacteristic
19#19097740—21no—
20C4×C22⋊D157—21no—
21C4×C22⋊D157—21no—
22#19097730C2211yescharacteristic
23#19097704C2212yescomplemented
24#19097704C2212yescomplemented
25#19097702C2212yescomplemented
26#19097702C2212yescomplemented
27#19097694C2212yescomplemented
28#19097694C2212yescomplemented
29#19097691C2212yescomplemented
30#19097691C2212yescomplemented
31C22⋊#11929434C2211yescharacteristic
32C22⋊#11929434C2211yescharacteristic
33C22⋊#11929434C2212yes—
34C23×C4×C157C2211yescharacteristic
35#117×C157—21no—
36#117×C157—21no—
37#117×C157C2212yes—
38#117×C157C2212yes—
39C2×C4⋊C4×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41C2×C22⋊C4×C157—21no—
42C2×C42×C157—21no—
43#14578954—24no—
44C2×Dic314C2311yescharacteristic
45C2×D628C2311yescharacteristic
46C2×C22⋊D157—21no—
47C2×C22⋊D157—21no—
48C2×C22⋊D157—21no—
49C2×C4×D157—21no—
50C2×#11929434—24no—
51C2×#11929434C2312yes—
52C2×#11929434C2312yes—
53C4⋊Dic157C2311yescharacteristic
54C4⋊Dic157C2311yescharacteristic
55#14578933—21no—
56#14578933—21no—
57C22⋊Dic157—21no—
58C22⋊Dic157—21no—
59C22⋊Dic157—21no—
60#14578931—21no—
61#14578931—21no—
62C4×Dic157—21no—
63C24×C157D411yescharacteristic
64C22×C4×C157—24no—
65C22×C4×C157—21no—
66C22×C4×C157—21no—
67C22×C4×C157D411yescharacteristic
68C22×C4×C157D411yescharacteristic
69C22×C4×C157D411yescharacteristic
70C22×C4×C157C2311yescharacteristic
71C22×C4×C157C2311yescharacteristic
72C22×C4×C157C2311yescharacteristic
73C4⋊C4×C157—21no—
74C4⋊C4×C157—21no—
75C4⋊C4×C157C2312yes—
76C4⋊C4×C157C2312yes—
77C22⋊C4×C157—22no—
78C22⋊C4×C157—22no—
79C42×C157—22no—
80C42×C157—22no—
81C22×D157—21no—
82Dic314—22no—
83D628—22no—
84C22⋊D157—42no—
85C22⋊D157—42no—
86C22⋊D157—42no—
87C2×Dic157—21no—
88C2×Dic157—21no—
89C2×Dic157—21no—
90C4×D157—42no—
91#11929434—22no—
92#11929434—22no—
93C23×C157—22no—
94C23×C157—22no—
95C23×C157—41no—
96C23×C157C2×D411yescharacteristic
97C23×C157C2×D411yescharacteristic
98C23×C157C2×D411yescharacteristic
99C2×C4×C157—22no—
100C2×C4×C157—22no—
101C2×C4×C157—22no—
102C2×C4×C157—22no—
103C2×C4×C157—22no—
104C2×C4×C157—22no—
105C2×C4×C157—22no—
106C2×C4×C157—42no—
107C2×C4×C157—42no—
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157#4114yes—
112C2×C4×C157C2411yescharacteristic
113C22×C157—21no—
114C22×C157—21no—
115C22×C157—21no—
116C22×C157—21no—
117C22×C157—21no—
118C22×C157—21no—
119C22×C157—21no—
120C22×C157—44no—
121C22×C157#12312yes—
122C22×C157#13512yes—
123C22×C157C22×D411yescharacteristic
124C2×D157—41no—
125C2×D157—41no—
126C2×D157—41no—
127C4×C157—21no—
128C4×C157—21no—
129C4×C157—42no—
130C4×C157—42no—
131C4×C157#12312yes—
132C4×C157C2×#4112yes—
133Dic157—41no—
134Dic157—41no—
135Dic157—41no—
136C2×C157—22no—
137C2×C157—22no—
138C2×C157—41no—
139C2×C157C2×#12311yescharacteristic
140C2×C157C2×#13511yescharacteristic
141C2×C157#57511yescharacteristic
142D157—81no—
143C157#309111yescharacteristic, complemented, Sylow
144#3091—1571nomaximal, Sylow
145#525—1571no—
146C2×#117D15711yescharacteristic, complemented
147#485—1572no—
148#484—1571no—
149#483—1571no—
150#482—1572no—
151#481—1571no—
152#480—1571no—
153#427—1572no—
154#416—1572no—
155C2×C22⋊C8—1571no—
156C2×#41—1571no—
157C23×C4C2×D15711yescharacteristic
158C22×C8—1572no—
159#123—3141no—
160C22⋊Q8—1571no—
161C22⋊Q8—1571no—
162#121—1571no—
163#121—1571no—
164#120—3141no—
165C4×D4—3141no—
166C4×D4—3141no—
167#117—21no—
168#117—21no—
169#117—1571no—
170#117C2×D15712yes—
171#117C2×D15712yes—
172C2×C4⋊C4C2×D15711yescharacteristic
173C2×C4⋊C4C2×D15711yescharacteristic
174C2×C22⋊C4—21no—
175C2×C42—21no—
176#107—1572no—
177#107—1572no—
178#106—1572no—
179#106—1572no—
180#103—1572no—
181#103—1572no—
182#102—1572no—
183#102—1572no—
184C22⋊C8—1571no—
185C22⋊C8—1571no—
186C22⋊C8—1572no—
187C24C22⋊D15711yescharacteristic
188#41—3144no—
189C2×Q8—1571no—
190C2×D4—3141no—
191C2×D4—3141no—
192C2×D4—3141no—
193C2×D4—1571no—
194C22×C4—24no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—3141no—
198C22×C4C22⋊D15711yescharacteristic
199C22×C4C22⋊D15711yescharacteristic
200C22×C4C22⋊D15711yescharacteristic
201C22×C4C22×D15711yescharacteristic
202C22×C4C22×D15711yescharacteristic
203C22×C4C22×D15711yescharacteristic
204C2×C8—3144no—
205C2×C8—1572no—
206C2×C8—1572no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—3141no—
210C4⋊C4—3141no—
211C4⋊C4—1571no—
212C4⋊C4—1571no—
213C4⋊C4C22×D15712yes—
214C4⋊C4C22×D15712yes—
215C22⋊C4—22no—
216C22⋊C4—22no—
217C22⋊C4—3141no—
218C22⋊C4—3141no—
219C22⋊C4—3141no—
220C22⋊C4—3141no—
221C22⋊C4—3141no—
222C42—22no—
223C42—22no—
224C42—3141no—
225C23—22no—
226C23—22no—
227C23—41no—
228C23—3141no—
229C23C2×C22⋊D15711yescharacteristic
230C23C2×C22⋊D15711yescharacteristic
231C23C2×C22⋊D15711yescharacteristic
232Q8—3142no—
233D4—3142no—
234D4—6282no—
235D4—6282no—
236D4—6282no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—22no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—3141no—
247C2×C4—3141no—
248C2×C4—3141no—
249C2×C4—6282no—
250C2×C4C2×C22⋊D15711yescharacteristic
251C2×C4C2×C22⋊D15711yescharacteristic
252C2×C4C2×C22⋊D15711yescharacteristic
253C2×C4#1457895414yes—
254C2×C4C23×D15711yescharacteristic
255C8—3142no—
256C8—3142no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—21no—
263C22—21no—
264C22—44no—
265C22—6281no—
266C22—6281no—
267C22—6281no—
268C22#1909775412yes—
269C22#1909779112yes—
270C22C22×C22⋊D15711yescharacteristic
271C4—21no—
272C4—21no—
273C4—42no—
274C4—42no—
275C4—6281no—
276C4—6281no—
277C4—6281no—
278C4#1909775412yes—
279C4C2×#1457895412yes—
280C2—22no—
281C2—22no—
282C2—41no—
283C2—12561no—
284C2C2×#1909775411yescharacteristic
285C2C2×#1909779111yescharacteristic
286C2#2375807111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×#117×C157C2
socleC22×C157C22×D4
prime radical#23757784C2
hypercenterC2×#117D157
hypocenterC157#3091
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #23757614C2semidirect
3. #23757608C2semidirect
4. #23757603C2semidirect
5. #23757597C2semidirect
6. #23757591C2semidirect
7. #23757586C2semidirect
8. #23757309C2semidirect
9. #23757267C2semidirect
10. C2×C22⋊#11929434C2semidirect
11. C2×#117×C157C2semidirect
23. #19097704C22semidirect
24. #19097704C22semidirect
25. #19097702C22semidirect
26. #19097702C22semidirect
27. #19097694C22semidirect
28. #19097694C22semidirect
29. #19097691C22semidirect
30. #19097691C22semidirect
143. C157#3091semidirect
146. C2×#117D157semidirect

Maximal subgroups

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

subgroupindex
#237577842
#237576142
#237576082
#237576032
#237575972
#237575912
#237575862
#237573092
#237572672
C2×C22⋊#119294342
C2×#117×C1572
#3091157