Finite Group Databasebeta

← #34109420

Subgroups — #34109420

Order 20096.

732 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 732 conjugacy classes of subgroups — too many to draw. Only its 149 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×#19097818C211yesmaximal, characteristic, complemented
2C2×#19097797C211yesmaximal, characteristic, complemented
3C22×#14578942C211yesmaximal, characteristic, complemented
4#23757416C214yesmaximal, complemented
5#23757411C214yesmaximal, complemented
6#23757281C211yesmaximal, characteristic, complemented
7#23757267C211yesmaximal, characteristic, complemented
8#23757240C211yesmaximal, characteristic, complemented
9#535×C157C211yesmaximal, characteristic, complemented
10#19097818—24no—
11#19097818—24no—
12C2×#14578954C2211yescharacteristic
13C2×#14578954C2212yescomplemented
14C2×#14578953—21no—
15C2×#14578953—22no—
16C22×Dic314C2211yescharacteristic
17C22×Dic314C2212yescomplemented
18C2×Q8×D157—21no—
19#19097797—24no—
20C2×#14578942—24no—
21C2×#14578942C2212yes—
22C2×#14578942C2214yes—
23C2×#14578939C2212yes—
24C2×#14578938C2211yescharacteristic
25C22×#11929434C2211yescharacteristic
26#19097694C2212yescomplemented
27#19097694C2214yescomplemented
28#19097691C2212yescomplemented
29C22⋊#11929434C2214yescomplemented
30C2×#41×C157C2211yescharacteristic
31C22×D4×C157C2211yescharacteristic
32#127×C157—21no—
33#124×C157—21no—
34#121×C157C2214yes—
35#120×C157—22no—
36#117×C157C2211yescharacteristic
37#14578954—24no—
38#14578954—24no—
39#14578954—22no—
40#14578954D414yescomplemented
41#14578953—216no—
42#14578953—28no—
43C2×Dic314—24no—
44C2×Dic314—24no—
45C2×Dic314—21no—
46C2×Dic314—22no—
47C2×Dic314—22no—
48C2×Dic314D414yescomplemented
49C2×Dic314C2312yes—
50C2×Dic314C2311yescharacteristic
51Q8×D157—28no—
52C2×D628C2311yescharacteristic
53C2×C22⋊D157—21no—
54C2×C22⋊D157—22no—
55C22×Dic157—21no—
56C22×Dic157—22no—
57C2×C4×D157—21no—
58C2×C4×D157—22no—
59#14578942—28no—
60#14578942—24no—
61#14578939—24no—
62#14578938—22no—
63C2×#11929434—22no—
64C2×#11929434C2312yes—
65C2×#11929434C2312yes—
66C24×C157—21no—
67#41×C157—22no—
68#41×C157—22no—
69C2×Q8×C157D411yescharacteristic
70C2×D4×C157—24no—
71C2×D4×C157—22no—
72C2×D4×C157—22no—
73C2×D4×C157—21no—
74C2×D4×C157D412yes—
75C2×D4×C157D411yescharacteristic
76C2×D4×C157C2312yes—
77C22×C4×C157C2312yes—
78C22×C4×C157C2311yescharacteristic
79C4⋊C4×C157C2312yes—
80C22⋊C4×C157—21no—
81C22⋊C4×C157—24no—
82C42×C157—21no—
83C22×D157—21no—
84Dic314—28no—
85Dic314—24no—
86Dic314—21no—
87Dic314—22no—
88Dic314C2×D412yes—
89Dic314C2×D414yes—
90D628—21no—
91D628C2×D412yes—
92C22⋊D157—28no—
93C22⋊D157—24no—
94C2×Dic157—28no—
95C2×Dic157—24no—
96C2×Dic157—21no—
97C2×Dic157—22no—
98C4×D157—28no—
99C4×D157—24no—
100#11929434—22no—
101#11929434—22no—
102C23×C157—22no—
103C23×C157—42no—
104C23×C157—82no—
105C23×C157C2×D412yes—
106C23×C157C2×D411yescharacteristic
107Q8×C157—22no—
108D4×C157—24no—
109D4×C157—22no—
110D4×C157—22no—
111D4×C157—22no—
112D4×C157—42no—
113D4×C157—84no—
114C2×C4×C157—22no—
115C2×C4×C157—22no—
116C2×C4×C157—22no—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2×D412yes—
119C2×C4×C157C2×D412yes—
120C2×C4×C157C2×D411yescharacteristic
121C2×C4×C157C2411yescharacteristic
122C22×C157—22no—
123C22×C157—22no—
124C22×C157—24no—
125C22×C157—42no—
126C22×C157—44no—
127C22×C157—84no—
128C22×C157#12012yes—
129C22×C157C22×D411yescharacteristic
130C2×D157—22no—
131C2×D157—42no—
132C4×C157—22no—
133C4×C157—42no—
134C4×C157#12012yes—
135C4×C157C22×D411yescharacteristic
136C4×C157C22×D411yescharacteristic
137Dic157—24no—
138Dic157—22no—
139C2×C157—22no—
140C2×C157—42no—
141C2×C157—82no—
142C2×C157C2×#12011yescharacteristic
143C2×C157#57712yes—
144D157—42no—
145C157#301011yescharacteristic, complemented, Sylow
146#3010—1571nomaximal, Sylow
147C2×2−1+4—1571no—
148C2×AGL(1,ℤ/8)—1571no—
149C22×SD16—1571no—
150#535D15711yescharacteristic, complemented
151#449—1574no—
152#448—1574no—
153#419—1571no—
154#416—1571no—
155#408—1571no—
1562−1+4—3144no—
1572−1+4—3144no—
158C2×#41—3142no—
159C2×#41—3141no—
160C2×#41—1572no—
161C2×#41—1571no—
162C2×#41C2×D15711yescharacteristic
163C22×Q8—3141no—
164C22×Q8—1572no—
165C22×Q8—1571no—
166C22×D4C2×D15711yescharacteristic
167AGL(1,ℤ/8)—3144no—
168C2×SD16—3144no—
169C2×SD16—1574no—
170C2×SD16—1572no—
171C2×D8—1572no—
172C2×M16—1571no—
173C22×C8—1571no—
174#127—21no—
175#124—21no—
176#121C2×D15714yes—
177#120—22no—
178#117C2×D15711yescharacteristic
179#103—1572no—
180#103—1574no—
181#102—1572no—
182C22⋊C8—1574no—
183C24—21no—
184#41—22no—
185#41—22no—
186#41—3144no—
187#41—3144no—
188#41—31416no—
189#41—3148no—
190#41—3142no—
191#41—1574no—
192C2×Q8—3144no—
193C2×Q8—3144no—
194C2×Q8—3148no—
195C2×Q8—3142no—
196C2×Q8—3142no—
197C2×Q8—3141no—
198C2×Q8—1574no—
199C2×Q8—1571no—
200C2×Q8—1572no—
201C2×Q8C22⋊D15711yescharacteristic
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—3142no—
207C2×D4—3141no—
208C2×D4—1571no—
209C2×D4C22⋊D15712yes—
210C2×D4C22⋊D15711yescharacteristic
211C2×D4C22×D15712yes—
212C22×C4—3142no—
213C22×C4—3141no—
214C22×C4—3142no—
215C22×C4—3141no—
216C22×C4C22×D15712yes—
217C22×C4C22×D15711yescharacteristic
218SD16—3148no—
219SD16—3144no—
220D8—3144no—
221M16—3142no—
222C2×C8—3142no—
223C2×C8—1572no—
224C2×C8—1572no—
225C4⋊C4C22×D15712yes—
226C22⋊C4—21no—
227C22⋊C4—24no—
228C42—21no—
229C23—22no—
230C23—42no—
231C23—82no—
232C23—3141no—
233C23C2×C22⋊D15712yes—
234C23C2×C22⋊D15711yescharacteristic
235Q8—22no—
236Q8—3148no—
237Q8—3144no—
238Q8—3141no—
239Q8—3142no—
240Q8—1572no—
241Q8—1574no—
242D4—24no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—42no—
247D4—84no—
248D4—3148no—
249D4—3144no—
250D4—3141no—
251D4—1572no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—3148no—
256C2×C4—3148no—
257C2×C4—3144no—
258C2×C4—3142no—
259C2×C4—3141no—
260C2×C4—3144no—
261C2×C4D4×D15712yes—
262C2×C4D4×D15712yes—
263C2×C4C2×C22⋊D15712yes—
264C2×C4C2×C22⋊D15711yescharacteristic
265C2×C4C23×D15711yescharacteristic
266C8—3142no—
267C8—3142no—
268C22—22no—
269C22—24no—
270C22—22no—
271C22—44no—
272C22—42no—
273C22—84no—
274C22—3142no—
275C22—6282no—
276C22#1909773912yes—
277C22C22×C22⋊D15711yescharacteristic
278C4—22no—
279C4—42no—
280C4—3144no—
281C4—3142no—
282C4#1909773912yes—
283C4C2×D4×D15711yescharacteristic
284C4C2×D4×D15711yescharacteristic
285C2—22no—
286C2—42no—
287C2—82no—
288C2—6282no—
289C2C2×#1909773911yescharacteristic
290C2#2375808412yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#535×C157C2
socleC22×C157C22×D4
hypercenter#535D157
hypocenterC157#3010
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#19097818C2semidirect
2. C2×#19097797C2semidirect
3. C22×#14578942C2semidirect
4. #23757416C2semidirect
5. #23757411C2semidirect
6. #23757281C2semidirect
7. #23757267C2semidirect
8. #23757240C2semidirect
9. #535×C157C2semidirect
13. C2×#14578954C22semidirect
17. C22×Dic314C22semidirect
26. #19097694C22semidirect
27. #19097694C22semidirect
28. #19097691C22semidirect
29. C22⋊#11929434C22semidirect
40. #14578954D4semidirect
48. C2×Dic314D4semidirect
145. C157#3010semidirect
150. #535D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978182
C2×#190977972
C22×#145789422
#237574162
#237574112
#237572812
#237572672
#237572402
#535×C1572
#3010157