Finite Group Databasebeta

← #34110082

Subgroups — #34110082

Order 20096.

410 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 127 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×#19097747C211yesmaximal, characteristic
2#117×D157C211yesmaximal, characteristic
3#23757613C212yesmaximal, complemented
4#23757612C212yesmaximal, complemented
5#23757607C212yesmaximal, complemented
6#23757602C212yesmaximal, complemented
7#23757316C211yesmaximal, characteristic, complemented
8#23757315C211yesmaximal, characteristic, complemented
9#23757282C211yesmaximal, characteristic, complemented
10#23757236C211yesmaximal, characteristic
11#419×C157C211yesmaximal, characteristic, complemented
12C22×C4×D157C2211yescharacteristic
13C2×#14578938C2211yescharacteristic
14#19097747—21no—
15#19097747—21no—
16#19097747C2212yes—
17#19097747C2212yes—
18#19097730C2211yescharacteristic
19#19097729C2211yescharacteristic
20#19097729C2211yescharacteristic
21#19097728—21no—
22#19097727—21no—
23C2×C4⋊Dic157C2211yescharacteristic
24C2×#14578933—21no—
25C4⋊C4×D157C2211yescharacteristic
26C4⋊C4×D157C2211yescharacteristic
27C2×#14578931—21no—
28C22⋊C4×D157—21no—
29C42×D157—21no—
30#19097705C2212yescomplemented
31#19097704C2212yescomplemented
32#19097703C2212yescomplemented
33#19097702C2212yescomplemented
34#19097695C2212yescomplemented
35#19097695C2212yescomplemented
36#19097683C2212yes—
37#19097683C2212yes—
38C22×Q8×C157C2211yescharacteristic
39C2×M16×C157C2211yescharacteristic
40#117×C157C2211yescharacteristic
41#103×C157C2212yescomplemented
42#103×C157C2212yescomplemented
43C23×D157D411yescharacteristic
44C22×Dic157—21no—
45C22×Dic157D411yescharacteristic
46C2×C4×D157—21no—
47C2×C4×D157—21no—
48C2×C4×D157—22no—
49C2×C4×D157—22no—
50C2×C4×D157D411yescharacteristic
51C2×C4×D157D411yescharacteristic
52C2×C4×D157C2311yescharacteristic
53C2×C4×D157C2311yescharacteristic
54#14578938—22no—
55C2×#11929434C2312yes—
56C4⋊Dic157—21no—
57C4⋊Dic157C2311yescharacteristic
58C4⋊Dic157C2311yescharacteristic
59C4⋊Dic157C2312yes—
60#14578933—21no—
61#14578933—21no—
62#14578933—22no—
63#14578933—22no—
64C22⋊Dic157—21no—
65#14578931—21no—
66#14578931—21no—
67#14578931—22no—
68#14578931—22no—
69C4×Dic157—21no—
70C4×Dic157—21no—
71C4×Dic157—21no—
72C2×Q8×C157—21no—
73C2×Q8×C157—22no—
74C2×Q8×C157—22no—
75C2×Q8×C157C2312yes—
76C22×C4×C157—21no—
77C22×C4×C157C2311yescharacteristic
78M16×C157—22no—
79C2×C8×C157C2312yes—
80C4⋊C4×C157C2311yescharacteristic
81C4⋊C4×C157C2311yescharacteristic
82C22⋊C4×C157—21no—
83C42×C157—21no—
84C22×D157—22no—
85C22×D157—22no—
86C22×D157—41no—
87C22×D157C2×D411yescharacteristic
88C22×D157C2×D411yescharacteristic
89C2×Dic157—22no—
90C2×Dic157—21no—
91C2×Dic157—21no—
92C2×Dic157—22no—
93C2×Dic157—42no—
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96C4×D157—22no—
97C4×D157—22no—
98C4×D157—21no—
99C4×D157—21no—
100C4×D157—21no—
101C4×D157—21no—
102C4×D157—42no—
103C4×D157—42no—
104#11929434—22no—
105C23×C157C2×D411yescharacteristic
106Q8×C157—22no—
107Q8×C157—22no—
108Q8×C157—42no—
109C2×C4×C157—21no—
110C2×C4×C157—21no—
111C2×C4×C157—22no—
112C2×C4×C157—42no—
113C2×C4×C157C2×D411yescharacteristic
114C2×C4×C157#4112yes—
115C2×C4×C157#4112yes—
116C2×C4×C157C2411yescharacteristic
117C8×C157—22no—
118C22×C157—21no—
119C22×C157—21no—
120C22×C157#12312yes—
121C22×C157C22×D411yescharacteristic
122C2×D157—21no—
123C2×D157—21no—
124C2×D157—21no—
125C2×D157—21no—
126C2×D157—21no—
127C2×D157—44no—
128C2×D157#13712yes—
129C4×C157—42no—
130C4×C157—41no—
131C4×C157—41no—
132C4×C157#12312yes—
133C4×C157C2×#4111yescharacteristic
134C4×C157C2×#4111yescharacteristic
135Dic157—21no—
136Dic157—21no—
137Dic157—41no—
138Dic157—41no—
139Dic157—42no—
140C2×C157—22no—
141C2×C157C2×#12311yescharacteristic
142C2×C157C2×#13711yescharacteristic
143C2×C157#57511yescharacteristic
144D157—22no—
145D157—41no—
146C157#309511yescharacteristic, complemented, Sylow
147#3095—1571nomaximal, Sylow
148C2×C22⋊Q8—1571no—
149C2×#117—1571no—
150#485—1572no—
151#485—1572no—
152#484—1572no—
153#483—1572no—
154#428—1571no—
155#428—1571no—
156#419—1571no—
157#419D15711yescharacteristic, complemented
158#407—1571no—
159C22×Q8C2×D15711yescharacteristic
160C23×C4—1571no—
161C2×M16—1571no—
162C2×M16C2×D15711yescharacteristic
163C22⋊Q8—3141no—
164C22⋊Q8—3141no—
165C22⋊Q8—1572no—
166C22⋊Q8—1572no—
167#117—3141no—
168#117—3141no—
169#117—1571no—
170#117—1571no—
171#117—1571no—
172#117C2×D15711yescharacteristic
173C2×C4⋊C4—3141no—
174C2×C4⋊C4—1571no—
175C2×C4⋊C4—1571no—
176C2×C4⋊C4—1571no—
177C2×C22⋊C4—3141no—
178C2×C22⋊C4—3141no—
179C2×C42—3141no—
180#107—1572no—
181#107—1572no—
182#106—1572no—
183#106—1572no—
184#103—1572no—
185#103—1572no—
186#103C2×D15712yescomplemented
187#103C2×D15712yescomplemented
188C22⋊C8—1572no—
189C22⋊C8—1572no—
190C24—1571no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8C22×D15712yes—
195C22×C4—21no—
196C22×C4—3141no—
197C22×C4—3141no—
198C22×C4—3142no—
199C22×C4—3142no—
200C22×C4—3141no—
201C22×C4—1571no—
202C22×C4—1571no—
203C22×C4—1571no—
204C22×C4—1571no—
205C22×C4—1571no—
206C22×C4C22×D15711yescharacteristic
207M16—22no—
208M16—3142no—
209C2×C8—1572no—
210C2×C8C22×D15712yes—
211C4⋊C4—3142no—
212C4⋊C4—3142no—
213C4⋊C4—3141no—
214C4⋊C4—3141no—
215C4⋊C4—3141no—
216C4⋊C4—1571no—
217C4⋊C4—1571no—
218C4⋊C4—1572no—
219C4⋊C4C22×D15711yescharacteristic
220C4⋊C4C22×D15711yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—3141no—
223C22⋊C4—3141no—
224C22⋊C4—3142no—
225C22⋊C4—3142no—
226C22⋊C4—3141no—
227C42—21no—
228C42—3141no—
229C42—3141no—
230C42—3141no—
231C23—3142no—
232C23—3142no—
233C23—6281no—
234C23—1571no—
235C23—1571no—
236C23D4×D15711yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—3142no—
245C2×C4—3142no—
246C2×C4—3142no—
247C2×C4—3141no—
248C2×C4—3141no—
249C2×C4—3141no—
250C2×C4—3141no—
251C2×C4—3141no—
252C2×C4—3141no—
253C2×C4—3142no—
254C2×C4—6282no—
255C2×C4—6282no—
256C2×C4—6282no—
257C2×C4—1571no—
258C2×C4—1571no—
259C2×C4D4×D15711yescharacteristic
260C2×C4#1457895312yes—
261C2×C4#1457895412yes—
262C2×C4C23×D15711yescharacteristic
263C8—22no—
264C8—3142no—
265C22—21no—
266C22—21no—
267C22—3141no—
268C22—3141no—
269C22—3141no—
270C22—3141no—
271C22—3141no—
272C22—6284no—
273C22—1572no—
274C22#1909775312yes—
275C22C2×D4×D15711yescharacteristic
276C4—42no—
277C4—41no—
278C4—41no—
279C4—3141no—
280C4—3141no—
281C4—6281no—
282C4—6281no—
283C4—6282no—
284C4#1909775312yes—
285C4C2×#1457895311yescharacteristic
286C4C2×#1457895411yescharacteristic
287C2—22no—
288C2—3142no—
289C2—6281no—
290C2C2×#1909775311yescharacteristic
291C2#137×D15711yescharacteristic
292C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#419×C157C2
socleC22×C157C22×D4
prime radicalC23×D157D4
hypercenter#419D157
hypocenterC157#3095
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #23757613C2semidirect
4. #23757612C2semidirect
5. #23757607C2semidirect
6. #23757602C2semidirect
7. #23757316C2semidirect
8. #23757315C2semidirect
9. #23757282C2semidirect
11. #419×C157C2semidirect
30. #19097705C22semidirect
31. #19097704C22semidirect
32. #19097703C22semidirect
33. #19097702C22semidirect
34. #19097695C22semidirect
35. #19097695C22semidirect
41. #103×C157C22semidirect
42. #103×C157C22semidirect
146. C157#3095semidirect
157. #419D157semidirect
186. #103C2×D157semidirect
187. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977472
#117×D1572
#237576132
#237576122
#237576072
#237576022
#237573162
#237573152
#237572822
#237572362
#419×C1572
#3095157