Finite Group Databasebeta

← #34110074

Subgroups — #34110074

Order 20096.

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

normal subgroups only

This group has 462 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×#19097742C211yesmaximal, characteristic
2#117×D157C211yesmaximal, characteristic, complemented
3#23757596C212yesmaximal, complemented
4#23757595C212yesmaximal, complemented
5#23757590C212yesmaximal, complemented
6#23757585C212yesmaximal, complemented
7#23757316C211yesmaximal, characteristic, complemented
8#23757315C211yesmaximal, characteristic, complemented
9#23757278C211yesmaximal, characteristic, complemented
10#23757236C211yesmaximal, characteristic, complemented
11#418×C157C211yesmaximal, characteristic, complemented
12C22×C22⋊D157—21no—
13C22×C4×D157C2211yescharacteristic
14C2×#14578938C2211yescharacteristic
15#19097742—21no—
16#19097742—21no—
17#19097742C2212yes—
18#19097742C2212yes—
19#19097730C2211yescharacteristic
20#19097729C2211yescharacteristic, complemented
21#19097729C2211yescharacteristic, complemented
22#19097728—21no—
23#19097727—21no—
24C2×C4⋊Dic157C2211yescharacteristic
25C4⋊C4×D157C2211yescharacteristic, complemented
26C4⋊C4×D157C2211yescharacteristic, complemented
27C2×C22⋊Dic157—21no—
28C22⋊C4×D157—21no—
29C42×D157—21no—
30#19097705C2212yescomplemented
31#19097704C2212yescomplemented
32#19097703C2212yescomplemented
33#19097702C2212yescomplemented
34#19097692C2212yescomplemented
35#19097692C2212yescomplemented
36#19097683C2212yescomplemented
37#19097683C2212yescomplemented
38C22×D4×C157C2211yescharacteristic
39C2×M16×C157C2211yescharacteristic
40#117×C157C2211yescharacteristic
41#102×C157C2212yescomplemented
42#102×C157C2212yescomplemented
43C23×D157D411yescharacteristic
44C2×C22⋊D157—22no—
45C2×C22⋊D157—22no—
46C2×C22⋊D157—44no—
47C22×Dic157—21no—
48C22×Dic157D411yescharacteristic
49C2×C4×D157—21no—
50C2×C4×D157—21no—
51C2×C4×D157—22no—
52C2×C4×D157—22no—
53C2×C4×D157D411yescharacteristic
54C2×C4×D157D411yescharacteristic
55C2×C4×D157C2311yescharacteristic
56C2×C4×D157C2311yescharacteristic
57#14578938—22no—
58C2×#11929434C2312yes—
59C4⋊Dic157—21no—
60C4⋊Dic157C2311yescharacteristic
61C4⋊Dic157C2311yescharacteristic
62C4⋊Dic157C2312yes—
63#14578933—21no—
64#14578933—21no—
65C22⋊Dic157—22no—
66C22⋊Dic157—22no—
67C22⋊Dic157—21no—
68#14578931—21no—
69#14578931—21no—
70C4×Dic157—21no—
71C4×Dic157—21no—
72C4×Dic157—21no—
73C24×C157—21no—
74C2×D4×C157—21no—
75C2×D4×C157—22no—
76C2×D4×C157—22no—
77C2×D4×C157C2312yes—
78C22×C4×C157C2311yescharacteristic
79M16×C157—22no—
80C2×C8×C157C2312yes—
81C4⋊C4×C157C2311yescharacteristic
82C4⋊C4×C157C2311yescharacteristic
83C22⋊C4×C157—21no—
84C42×C157—21no—
85C22×D157—22no—
86C22×D157—22no—
87C22×D157—41no—
88C22×D157C2×D411yescharacteristic
89C22×D157C2×D411yescharacteristic
90C22⋊D157—44no—
91C22⋊D157—44no—
92C2×Dic157—22no—
93C2×Dic157—21no—
94C2×Dic157—21no—
95C2×Dic157—22no—
96C2×Dic157—42no—
97C2×Dic157C2×D411yescharacteristic
98C2×Dic157C2×D411yescharacteristic
99C4×D157—22no—
100C4×D157—22no—
101C4×D157—21no—
102C4×D157—21no—
103C4×D157—21no—
104C4×D157—21no—
105C4×D157—42no—
106C4×D157—42no—
107#11929434—22no—
108C23×C157—22no—
109C23×C157—42no—
110C23×C157—42no—
111C23×C157—42no—
112C23×C157C2×D411yescharacteristic
113D4×C157—22no—
114D4×C157—22no—
115D4×C157—42no—
116C2×C4×C157—21no—
117C2×C4×C157—21no—
118C2×C4×C157C2×D411yescharacteristic
119C2×C4×C157#4112yes—
120C2×C4×C157#4112yes—
121C2×C4×C157C2411yescharacteristic
122C8×C157—22no—
123C22×C157—21no—
124C22×C157—21no—
125C22×C157—42no—
126C22×C157—42no—
127C22×C157—42no—
128C22×C157—84no—
129C22×C157#12312yes—
130C22×C157C22×D411yescharacteristic
131C2×D157—21no—
132C2×D157—21no—
133C2×D157—21no—
134C2×D157—21no—
135C2×D157—21no—
136C2×D157—44no—
137C2×D157AGL(1,ℤ/8)12yes—
138C4×C157—41no—
139C4×C157—41no—
140C4×C157#12312yes—
141C4×C157C2×#4111yescharacteristic
142C4×C157C2×#4111yescharacteristic
143Dic157—21no—
144Dic157—21no—
145Dic157—41no—
146Dic157—42no—
147Dic157—41no—
148C2×C157—22no—
149C2×C157—82no—
150C2×C157C2×#12311yescharacteristic
151C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
152C2×C157#57511yescharacteristic
153D157—22no—
154D157—41no—
155C157#309411yescharacteristic, complemented, Sylow
156#3094—1571nomaximal, Sylow
157C2×#121—1571no—
158C2×#117—1571no—
159#482—1572no—
160#482—1572no—
161#481—1572no—
162#480—1572no—
163#428—1571no—
164#428—1571no—
165#418—1571no—
166#418D15711yescharacteristic, complemented
167#407—1571no—
168C22×D4—3141no—
169C22×D4C2×D15711yescharacteristic
170C23×C4—1571no—
171C2×M16—1571no—
172C2×M16C2×D15711yescharacteristic
173#121—3141no—
174#121—3141no—
175#121—1572no—
176#121—1572no—
177#117—3141no—
178#117—3141no—
179#117—1571no—
180#117—1571no—
181#117—1571no—
182#117C2×D15711yescharacteristic
183C2×C4⋊C4—1571no—
184C2×C4⋊C4—1571no—
185C2×C4⋊C4—1571no—
186C2×C22⋊C4—3141no—
187C2×C22⋊C4—3141no—
188C2×C42—3141no—
189#107—1572no—
190#107—1572no—
191#106—1572no—
192#106—1572no—
193#102—1572no—
194#102—1572no—
195#102C2×D15712yescomplemented
196#102C2×D15712yescomplemented
197C22⋊C8—1572no—
198C22⋊C8—1572no—
199C24—21no—
200C24—1571no—
201C2×D4—21no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—3142no—
205C2×D4—3142no—
206C2×D4—6284no—
207C2×D4C22×D15712yes—
208C22×C4—3141no—
209C22×C4—3141no—
210C22×C4—3142no—
211C22×C4—3142no—
212C22×C4—3141no—
213C22×C4—1571no—
214C22×C4—1571no—
215C22×C4—1571no—
216C22×C4—1571no—
217C22×C4—1571no—
218C22×C4C22×D15711yescharacteristic
219M16—22no—
220M16—3142no—
221C2×C8—1572no—
222C2×C8C22×D15712yes—
223C4⋊C4—3141no—
224C4⋊C4—3141no—
225C4⋊C4—3141no—
226C4⋊C4—1571no—
227C4⋊C4—1571no—
228C4⋊C4—1572no—
229C4⋊C4C22×D15711yescharacteristic
230C4⋊C4C22×D15711yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—3141no—
233C22⋊C4—3141no—
234C22⋊C4—3141no—
235C22⋊C4—3142no—
236C22⋊C4—3142no—
237C42—21no—
238C42—3141no—
239C42—3141no—
240C42—3141no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—42no—
245C23—3142no—
246C23—3142no—
247C23—6281no—
248C23—1571no—
249C23—1571no—
250C23D4×D15711yescharacteristic
251D4—22no—
252D4—22no—
253D4—42no—
254D4—6284no—
255D4—6284no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—3142no—
259C2×C4—3142no—
260C2×C4—3142no—
261C2×C4—3141no—
262C2×C4—3141no—
263C2×C4—3141no—
264C2×C4—3141no—
265C2×C4—3141no—
266C2×C4—3141no—
267C2×C4—3142no—
268C2×C4—6282no—
269C2×C4—6282no—
270C2×C4—6282no—
271C2×C4—1571no—
272C2×C4—1571no—
273C2×C4D4×D15711yescharacteristic
274C2×C4#1457895312yes—
275C2×C4#1457895412yes—
276C2×C4C23×D15711yescharacteristic
277C8—22no—
278C8—3142no—
279C22—21no—
280C22—21no—
281C22—42no—
282C22—42no—
283C22—42no—
284C22—84no—
285C22—3141no—
286C22—3141no—
287C22—3141no—
288C22—3141no—
289C22—3141no—
290C22—6284no—
291C22—1572no—
292C22#1909775312yes—
293C22C2×D4×D15711yescharacteristic
294C4—41no—
295C4—41no—
296C4—3141no—
297C4—3141no—
298C4—6282no—
299C4—6281no—
300C4—6281no—
301C4#1909775312yes—
302C4C2×#1457895311yescharacteristic
303C4C2×#1457895411yescharacteristic
304C2—22no—
305C2—82no—
306C2—3142no—
307C2—6281no—
308C2C2×#1909775311yescharacteristic
309C2AGL(1,ℤ/8)×D15711yescharacteristic
310C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#418×C157C2
socleC22×C157C22×D4
prime radicalC2×#19097742C2
hypercenter#418D157
hypocenterC157#3094
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #117×D157C2semidirect
3. #23757596C2semidirect
4. #23757595C2semidirect
5. #23757590C2semidirect
6. #23757585C2semidirect
7. #23757316C2semidirect
8. #23757315C2semidirect
9. #23757278C2semidirect
10. #23757236C2semidirect
11. #418×C157C2semidirect
20. #19097729C22semidirect
21. #19097729C22semidirect
25. C4⋊C4×D157C22semidirect
26. C4⋊C4×D157C22semidirect
30. #19097705C22semidirect
31. #19097704C22semidirect
32. #19097703C22semidirect
33. #19097702C22semidirect
34. #19097692C22semidirect
35. #19097692C22semidirect
36. #19097683C22semidirect
37. #19097683C22semidirect
41. #102×C157C22semidirect
42. #102×C157C22semidirect
155. C157#3094semidirect
166. #418D157semidirect
195. #102C2×D157semidirect
196. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977422
#117×D1572
#237575962
#237575952
#237575902
#237575852
#237573162
#237573152
#237572782
#237572362
#418×C1572
#3094157