Finite Group Databasebeta

← #34110078

Subgroups — #34110078

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×#19097746C211yesmaximal, characteristic
2#117×D157C211yesmaximal, characteristic, complemented
3#23757600C212yesmaximal, complemented
4#23757599C212yesmaximal, complemented
5#23757593C212yesmaximal, complemented
6#23757588C212yesmaximal, complemented
7#23757313C211yesmaximal, characteristic, complemented
8#23757279C211yesmaximal, characteristic, complemented
9#23757277C211yesmaximal, characteristic, complemented
10#23757236C211yesmaximal, characteristic, complemented
11#428×C157C211yesmaximal, characteristic, complemented
12C22×D628—21no—
13C22×D628C2211yescharacteristic
14C22×C4×D157C2211yescharacteristic
15C2×#14578938C2211yescharacteristic
16#19097746—21no—
17#19097746—21no—
18#19097746C2212yes—
19#19097746C2212yes—
20#19097730C2211yescharacteristic
21#19097729C2211yescharacteristic, complemented
22#19097729C2211yescharacteristic, complemented
23#19097728—21no—
24#19097727—21no—
25C4⋊C4×D157C2211yescharacteristic, complemented
26C4⋊C4×D157C2211yescharacteristic, complemented
27C2×#14578931—21no—
28C22⋊C4×D157—21no—
29C42×D157—21no—
30#19097704C2212yescomplemented
31#19097702C2212yescomplemented
32#19097693C2212yescomplemented
33#19097693C2212yescomplemented
34#19097691C2212yescomplemented
35#19097691C2212yescomplemented
36#19097683C2212yescomplemented
37#19097683C2212yescomplemented
38C2×M16×C157C2211yescharacteristic
39#117×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#107×C157C2212yescomplemented
42#106×C157C2212yescomplemented
43C23×D157—21no—
44C23×D157D411yescharacteristic
45C2×D628—22no—
46C2×D628—22no—
47C2×D628—21no—
48C2×D628—22no—
49C2×D628—22no—
50C2×D628—44no—
51C2×D628C2312yes—
52C22×Dic157D411yescharacteristic
53C2×C4×D157—21no—
54C2×C4×D157—21no—
55C2×C4×D157—22no—
56C2×C4×D157—22no—
57C2×C4×D157D411yescharacteristic
58C2×C4×D157D411yescharacteristic
59C2×C4×D157C2311yescharacteristic
60C2×C4×D157C2311yescharacteristic
61#14578938—22no—
62C2×#11929434C2312yes—
63C4⋊Dic157C2311yescharacteristic
64C4⋊Dic157C2311yescharacteristic
65#14578933—21no—
66#14578933—21no—
67C22⋊Dic157—21no—
68#14578931—21no—
69#14578931—21no—
70#14578931—22no—
71#14578931—22no—
72C4×Dic157—21no—
73C4×Dic157—21no—
74C4×Dic157—21no—
75C22×C4×C157—21no—
76C22×C4×C157C2311yescharacteristic
77M16×C157—22no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157—21no—
80C4⋊C4×C157C2311yescharacteristic
81C4⋊C4×C157C2312yes—
82C4⋊C4×C157C2311yescharacteristic
83C22⋊C4×C157—21no—
84C42×C157—21no—
85C22×D157—22no—
86C22×D157—22no—
87C22×D157—22no—
88C22×D157—41no—
89C22×D157—42no—
90C22×D157—42no—
91C22×D157—42no—
92C22×D157C2×D411yescharacteristic
93C22×D157C2×D411yescharacteristic
94D628—22no—
95D628—22no—
96D628—44no—
97D628—44no—
98D628—42no—
99C2×Dic157—22no—
100C2×Dic157—21no—
101C2×Dic157—21no—
102C2×Dic157C2×D411yescharacteristic
103C2×Dic157C2×D411yescharacteristic
104C4×D157—22no—
105C4×D157—22no—
106C4×D157—21no—
107C4×D157—21no—
108C4×D157—21no—
109C4×D157—21no—
110C4×D157—42no—
111C4×D157—42no—
112#11929434—22no—
113C23×C157C2×D411yescharacteristic
114C2×C4×C157—21no—
115C2×C4×C157—21no—
116C2×C4×C157—22no—
117C2×C4×C157—42no—
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#12312yes—
126C22×C157C22×D411yescharacteristic
127C2×D157—21no—
128C2×D157—21no—
129C2×D157—21no—
130C2×D157—21no—
131C2×D157—21no—
132C2×D157—44no—
133C2×D157—42no—
134C2×D157—42no—
135C2×D157—42no—
136C2×D157—84no—
137C2×D157AGL(1,ℤ/8)12yes—
138C4×C157—42no—
139C4×C157—41no—
140C4×C157—41no—
141C4×C157#12312yes—
142C4×C157C2×#4111yescharacteristic
143C4×C157C2×#4111yescharacteristic
144Dic157—21no—
145Dic157—21no—
146Dic157—41no—
147Dic157—41no—
148C2×C157—22no—
149C2×C157C2×#12311yescharacteristic
150C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
151C2×C157#57511yescharacteristic
152D157—22no—
153D157—41no—
154D157—82no—
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#428D15711yescharacteristic, complemented
165#418—1571no—
166#418—1571no—
167#407—1571no—
168C22×D4—3141no—
169C22×D4—1571no—
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⋊C4C2×D15711yescharacteristic
186C2×C22⋊C4—3141no—
187C2×C22⋊C4—3141no—
188C2×C42—3141no—
189#107—1572no—
190#107C2×D15712yescomplemented
191#106—1572no—
192#106C2×D15712yescomplemented
193#102—1572no—
194#102—1572no—
195#102—1572no—
196#102—1572no—
197C22⋊C8—1572no—
198C22⋊C8—1572no—
199C24—3141no—
200C24—1571no—
201C2×D4—3142no—
202C2×D4—3142no—
203C2×D4—3141no—
204C2×D4—3142no—
205C2×D4—3142no—
206C2×D4—6284no—
207C2×D4—1572no—
208C22×C4—21no—
209C22×C4—3141no—
210C22×C4—3141no—
211C22×C4—3142no—
212C22×C4—3142no—
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—21no—
224C4⋊C4—3141no—
225C4⋊C4—3141no—
226C4⋊C4—1571no—
227C4⋊C4—1571no—
228C4⋊C4C22×D15711yescharacteristic
229C4⋊C4C22×D15711yescharacteristic
230C4⋊C4C22×D15712yes—
231C22⋊C4—21no—
232C22⋊C4—3141no—
233C22⋊C4—3141no—
234C22⋊C4—3142no—
235C22⋊C4—3142no—
236C22⋊C4—3141no—
237C42—21no—
238C42—3141no—
239C42—3141no—
240C42—3141no—
241C23—3142no—
242C23—3142no—
243C23—3142no—
244C23—6281no—
245C23—6282no—
246C23—6282no—
247C23—6282no—
248C23—1571no—
249C23—1571no—
250C23D4×D15711yescharacteristic
251D4—3142no—
252D4—3142no—
253D4—6284no—
254D4—6284no—
255D4—6282no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—22no—
259C2×C4—42no—
260C2×C4—3142no—
261C2×C4—3142no—
262C2×C4—3142no—
263C2×C4—3141no—
264C2×C4—3141no—
265C2×C4—3141no—
266C2×C4—3141no—
267C2×C4—3141no—
268C2×C4—3141no—
269C2×C4—6282no—
270C2×C4—6282no—
271C2×C4—1571no—
272C2×C4—1571no—
273C2×C4D4×D15711yescharacteristic
274C2×C4#1457895412yes—
275C2×C4#1457895512yes—
276C2×C4C23×D15711yescharacteristic
277C8—22no—
278C8—3142no—
279C22—21no—
280C22—21no—
281C22—3141no—
282C22—3141no—
283C22—3141no—
284C22—3141no—
285C22—3141no—
286C22—6284no—
287C22—6282no—
288C22—6282no—
289C22—6282no—
290C22—1572no—
291C22—12564no—
292C22#1909775612yes—
293C22C2×D4×D15711yescharacteristic
294C4—42no—
295C4—41no—
296C4—41no—
297C4—3141no—
298C4—3141no—
299C4—6281no—
300C4—6281no—
301C4#1909775612yes—
302C4C2×#1457895411yescharacteristic
303C4C2×#1457895511yescharacteristic
304C2—22no—
305C2—3142no—
306C2—6281no—
307C2—12562no—
308C2C2×#1909775611yescharacteristic
309C2AGL(1,ℤ/8)×D15711yescharacteristic
310C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
prime radicalC2×#19097746C2
hypercenter#428D157
hypocenterC157#3094
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #117×D157C2semidirect
3. #23757600C2semidirect
4. #23757599C2semidirect
5. #23757593C2semidirect
6. #23757588C2semidirect
7. #23757313C2semidirect
8. #23757279C2semidirect
9. #23757277C2semidirect
10. #23757236C2semidirect
11. #428×C157C2semidirect
21. #19097729C22semidirect
22. #19097729C22semidirect
25. C4⋊C4×D157C22semidirect
26. C4⋊C4×D157C22semidirect
30. #19097704C22semidirect
31. #19097702C22semidirect
32. #19097693C22semidirect
33. #19097693C22semidirect
34. #19097691C22semidirect
35. #19097691C22semidirect
36. #19097683C22semidirect
37. #19097683C22semidirect
41. #107×C157C22semidirect
42. #106×C157C22semidirect
155. C157#3094semidirect
164. #428D157semidirect
190. #107C2×D157semidirect
192. #106C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977462
#117×D1572
#237576002
#237575992
#237575932
#237575882
#237573132
#237572792
#237572772
#237572362
#428×C1572
#3094157