Finite Group Databasebeta

← #34110086

Subgroups — #34110086

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×#19097751C211yesmaximal, characteristic
2#117×D157C211yesmaximal, characteristic
3#23757617C212yesmaximal, complemented
4#23757616C212yesmaximal, complemented
5#23757610C212yesmaximal, complemented
6#23757605C212yesmaximal, complemented
7#23757313C211yesmaximal, characteristic, complemented
8#23757283C211yesmaximal, characteristic, complemented
9#23757281C211yesmaximal, characteristic, complemented
10#23757236C211yesmaximal, characteristic
11#428×C157C211yesmaximal, characteristic, complemented
12C22×Dic314C2211yescharacteristic
13C22×C4×D157C2211yescharacteristic
14C2×#14578938C2211yescharacteristic
15#19097751—21no—
16#19097751—21no—
17#19097751C2212yes—
18#19097751C2212yes—
19#19097730C2211yescharacteristic
20#19097729C2211yescharacteristic
21#19097729C2211yescharacteristic
22#19097728—21no—
23#19097727—21no—
24C2×C4⋊Dic157—21no—
25C4⋊C4×D157C2211yescharacteristic
26C4⋊C4×D157C2211yescharacteristic
27C2×#14578931—21no—
28C22⋊C4×D157—21no—
29C42×D157—21no—
30#19097704C2212yescomplemented
31#19097702C2212yescomplemented
32#19097696C2212yescomplemented
33#19097696C2212yescomplemented
34#19097694C2212yescomplemented
35#19097694C2212yescomplemented
36#19097683C2212yes—
37#19097683C2212yes—
38C2×M16×C157C2211yescharacteristic
39#117×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2211yescharacteristic
41#107×C157C2212yescomplemented
42#106×C157C2212yescomplemented
43C23×D157D411yescharacteristic
44C2×Dic314—21no—
45C2×Dic314—22no—
46C2×Dic314—22no—
47C2×Dic314C2312yes—
48C22×Dic157—21no—
49C22×Dic157D411yescharacteristic
50C2×C4×D157—21no—
51C2×C4×D157—21no—
52C2×C4×D157—22no—
53C2×C4×D157—22no—
54C2×C4×D157D411yescharacteristic
55C2×C4×D157D411yescharacteristic
56C2×C4×D157C2311yescharacteristic
57C2×C4×D157C2311yescharacteristic
58#14578938—22no—
59C2×#11929434C2312yes—
60C4⋊Dic157—22no—
61C4⋊Dic157—22no—
62C4⋊Dic157C2311yescharacteristic
63C4⋊Dic157C2311yescharacteristic
64#14578933—21no—
65#14578933—21no—
66C22⋊Dic157—21no—
67#14578931—21no—
68#14578931—21no—
69#14578931—22no—
70#14578931—22no—
71C4×Dic157—21no—
72C4×Dic157—21no—
73C4×Dic157—21no—
74C22×C4×C157—21no—
75C22×C4×C157C2311yescharacteristic
76M16×C157—22no—
77C2×C8×C157C2312yes—
78C4⋊C4×C157—21no—
79C4⋊C4×C157C2311yescharacteristic
80C4⋊C4×C157C2312yes—
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
89Dic314—22no—
90Dic314—22no—
91Dic314—42no—
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×C157C2×D411yescharacteristic
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—41no—
130C4×C157—42no—
131C4×C157—41no—
132C4×C157#12312yes—
133C4×C157C2×#4111yescharacteristic
134C4×C157C2×#4111yescharacteristic
135Dic157—21no—
136Dic157—21no—
137Dic157—42no—
138Dic157—41no—
139Dic157—41no—
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#428D15711yescharacteristic, complemented
156#419—1571no—
157#419—1571no—
158#407—1571no—
159C22×Q8—1571no—
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⋊C4C2×D15711yescharacteristic
177C2×C22⋊C4—3141no—
178C2×C22⋊C4—3141no—
179C2×C42—3141no—
180#107—1572no—
181#107C2×D15712yescomplemented
182#106—1572no—
183#106C2×D15712yescomplemented
184#103—1572no—
185#103—1572no—
186#103—1572no—
187#103—1572no—
188C22⋊C8—1572no—
189C22⋊C8—1572no—
190C24—1571no—
191C2×Q8—3141no—
192C2×Q8—3142no—
193C2×Q8—3142no—
194C2×Q8—1572no—
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—21no—
212C4⋊C4—3142no—
213C4⋊C4—3142no—
214C4⋊C4—3141no—
215C4⋊C4—3141no—
216C4⋊C4—1571no—
217C4⋊C4—1571no—
218C4⋊C4C22×D15711yescharacteristic
219C4⋊C4C22×D15711yescharacteristic
220C4⋊C4C22×D15712yes—
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—3142no—
238Q8—3142no—
239Q8—6282no—
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#1457895412yes—
261C2×C4#1457895512yes—
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#1909775612yes—
275C22C2×D4×D15711yescharacteristic
276C4—41no—
277C4—42no—
278C4—41no—
279C4—3141no—
280C4—3141no—
281C4—6282no—
282C4—6281no—
283C4—6281no—
284C4#1909775612yes—
285C4C2×#1457895411yescharacteristic
286C4C2×#1457895511yescharacteristic
287C2—22no—
288C2—3142no—
289C2—6281no—
290C2C2×#1909775611yescharacteristic
291C2#137×D15711yescharacteristic
292C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
prime radicalC23×D157D4
hypercenter#428D157
hypocenterC157#3095
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
3. #23757617C2semidirect
4. #23757616C2semidirect
5. #23757610C2semidirect
6. #23757605C2semidirect
7. #23757313C2semidirect
8. #23757283C2semidirect
9. #23757281C2semidirect
11. #428×C157C2semidirect
30. #19097704C22semidirect
31. #19097702C22semidirect
32. #19097696C22semidirect
33. #19097696C22semidirect
34. #19097694C22semidirect
35. #19097694C22semidirect
41. #107×C157C22semidirect
42. #106×C157C22semidirect
146. C157#3095semidirect
155. #428D157semidirect
181. #107C2×D157semidirect
183. #106C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977512
#117×D1572
#237576172
#237576162
#237576102
#237576052
#237573132
#237572832
#237572812
#237572362
#428×C1572
#3095157