Finite Group Databasebeta

← #34109378

Subgroups — #34109378

Order 20096.

758 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19097802C211yesmaximal, characteristic, complemented
2C2×#19097797C211yesmaximal, characteristic, complemented
3#23757784C211yesmaximal, characteristic, complemented
4#23757435C214yesmaximal, complemented
5#23757418C214yesmaximal, complemented
6#23757282C211yesmaximal, characteristic, complemented
7#23757278C211yesmaximal, characteristic, complemented
8#23757237C211yesmaximal, characteristic, complemented
9C22×#41×C157C211yesmaximal, characteristic, complemented
10C2×#14578954C2211yescharacteristic
11#19097802—24no—
12#19097797—24no—
13C2×#14578944C2212yescomplemented
14C2×#14578942C2212yescomplemented
15C2×#14578941C2212yescomplemented
16C2×#14578939C2212yescomplemented
17C2×#14578938C2211yescharacteristic
18C2×#14578938C2211yescharacteristic
19#19097754—21no—
20#19097750C2212yescomplemented
21#19097745C2212yescomplemented
22#19097740—21no—
23C4×C22⋊D157—22no—
24#19097730C2211yescharacteristic
25#19097695C2214yescomplemented
26#19097692C2214yescomplemented
27C22⋊#11929434C2214yescomplemented
28C2×#41×C157—24no—
29C2×#41×C157—24no—
30C2×#41×C157—22no—
31C2×#41×C157C2214yes—
32C22×Q8×C157C2211yescharacteristic
33C22×D4×C157—21no—
34C22×D4×C157C2211yescharacteristic
35C23×C4×C157—21no—
36C23×C4×C157C2211yescharacteristic
37#14578954—22no—
38#14578954—22no—
39C2×Dic314C2311yescharacteristic
40C2×D628C2311yescharacteristic
41C2×C22⋊D157—22no—
42C2×C22⋊D157—21no—
43C2×C4×D157—21no—
44#14578944—24no—
45#14578942—24no—
46#14578941—24no—
47#14578939—24no—
48#14578938—22no—
49#14578938—22no—
50C2×#11929434C2312yes—
51C2×#11929434C2312yes—
52C4⋊Dic157C2312yes—
53#14578933—22no—
54C22⋊Dic157—21no—
55C22⋊Dic157—22no—
56#14578931—22no—
57C4×Dic157—21no—
58C24×C157—21no—
59C24×C157D411yescharacteristic
60#41×C157—216no—
61#41×C157—28no—
62#41×C157—28no—
63C2×Q8×C157—22no—
64C2×Q8×C157—21no—
65C2×Q8×C157D414yescomplemented
66C2×Q8×C157C2312yes—
67C2×D4×C157—28no—
68C2×D4×C157—24no—
69C2×D4×C157—21no—
70C2×D4×C157—22no—
71C2×D4×C157D414yescomplemented
72C2×D4×C157C2312yes—
73C22×C4×C157—28no—
74C22×C4×C157—21no—
75C22×C4×C157—24no—
76C22×C4×C157—22no—
77C22×C4×C157—22no—
78C22×C4×C157D411yescharacteristic
79C22×C4×C157D412yes—
80C22×C4×C157C2312yes—
81C22×C4×C157C2311yescharacteristic
82C22×D157—21no—
83Dic314—21no—
84Dic314—21no—
85D628—21no—
86D628—21no—
87C22⋊D157—44no—
88C22⋊D157—42no—
89C2×Dic157—21no—
90C2×Dic157—22no—
91C4×D157—42no—
92#11929434—22no—
93#11929434—22no—
94C23×C157—22no—
95C23×C157—24no—
96C23×C157—22no—
97C23×C157—42no—
98C23×C157—44no—
99C23×C157C2×D412yes—
100C23×C157C2×D411yescharacteristic
101Q8×C157—24no—
102Q8×C157—22no—
103Q8×C157C2×D414yes—
104D4×C157—216no—
105D4×C157—24no—
106D4×C157—22no—
107D4×C157C2×D414yes—
108C2×C4×C157—216no—
109C2×C4×C157—22no—
110C2×C4×C157—24no—
111C2×C4×C157—22no—
112C2×C4×C157—24no—
113C2×C4×C157—22no—
114C2×C4×C157—22no—
115C2×C4×C157C2×D412yes—
116C2×C4×C157C2×D411yescharacteristic
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2×D412yes—
119C2×C4×C157C2411yescharacteristic
120C22×C157—22no—
121C22×C157—22no—
122C22×C157—24no—
123C22×C157—22no—
124C22×C157—22no—
125C22×C157—44no—
126C22×C157—48no—
127C22×C157—44no—
128C22×C157#12012yes—
129C22×C157C22×D411yescharacteristic
130C2×D157—41no—
131C2×D157—42no—
132C4×C157—22no—
133C4×C157—24no—
134C4×C157#12012yes—
135C4×C157C22×D411yescharacteristic
136C4×C157C22×D411yescharacteristic
137Dic157—41no—
138Dic157—42no—
139C2×C157—22no—
140C2×C157—42no—
141C2×C157—44no—
142C2×C157C2×#12011yescharacteristic
143C2×C157#57512yes—
144D157—81no—
145C157#300511yescharacteristic, complemented, Sylow
146#3005—1571nomaximal, Sylow
147C22×#41D15711yescharacteristic, complemented
148C2×#137—1571no—
149C2×AGL(1,ℤ/8)—1571no—
150#525—1571no—
151#452—1574no—
152#449—1574no—
153#419—1571no—
154#418—1571no—
155#407—1571no—
156C2×#41—24no—
157C2×#41—24no—
158C2×#41—22no—
159C2×#41—1571no—
160C2×#41C2×D15714yes—
161C22×Q8C2×D15711yescharacteristic
162C22×D4—21no—
163C22×D4C2×D15711yescharacteristic
164C23×C4—21no—
165C23×C4C2×D15711yescharacteristic
166#137—3144no—
167AGL(1,ℤ/8)—3144no—
168C2×Q16—1572no—
169C2×SD16—1572no—
170C2×SD16—1572no—
171C2×D8—1572no—
172C2×M16—1571no—
173C2×M16—1571no—
174#123—3141no—
175C22⋊Q8—1572no—
176#121—1572no—
177#120—3141no—
178C4×D4—3142no—
179#117—1571no—
180#103—1574no—
181#102—1574no—
182C22⋊C8—1574no—
183C24—21no—
184C24C22⋊D15711yescharacteristic
185#41—216no—
186#41—28no—
187#41—28no—
188#41—3142no—
189#41—3142no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—1571no—
193C2×Q8C22⋊D15714yescomplemented
194C2×Q8C22×D15712yes—
195C2×D4—28no—
196C2×D4—24no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—3142no—
200C2×D4—3141no—
201C2×D4—1571no—
202C2×D4C22⋊D15714yescomplemented
203C2×D4C22×D15712yes—
204C22×C4—28no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—3141no—
210C22×C4C22⋊D15712yes—
211C22×C4C22⋊D15711yescharacteristic
212C22×C4C22×D15712yes—
213C22×C4C22×D15711yescharacteristic
214Q16—3144no—
215SD16—3144no—
216SD16—3144no—
217D8—3144no—
218M16—3142no—
219M16—3142no—
220C2×C8—1572no—
221C2×C8—1572no—
222C4⋊C4—3142no—
223C4⋊C4—1572no—
224C22⋊C4—3141no—
225C22⋊C4—3142no—
226C22⋊C4—3142no—
227C42—3141no—
228C23—22no—
229C23—24no—
230C23—22no—
231C23—42no—
232C23—44no—
233C23—3141no—
234C23C2×C22⋊D15712yes—
235C23C2×C22⋊D15711yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—3141no—
239Q8—3141no—
240Q8C2×C22⋊D15714yes—
241D4—216no—
242D4—24no—
243D4—22no—
244D4—3141no—
245D4—3141no—
246D4—6284no—
247D4—6282no—
248D4C2×C22⋊D15714yes—
249C2×C4—216no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—24no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—3142no—
257C2×C4—3141no—
258C2×C4—6282no—
259C2×C4C2×C22⋊D15712yes—
260C2×C4C2×C22⋊D15711yescharacteristic
261C2×C4C2×C22⋊D15712yes—
262C2×C4C2×C22⋊D15712yes—
263C2×C4C23×D15711yescharacteristic
264C8—3142no—
265C8—3142no—
266C22—22no—
267C22—22no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—44no—
272C22—48no—
273C22—44no—
274C22—6281no—
275C22—6282no—
276C22#1909774012yes—
277C22C22×C22⋊D15711yescharacteristic
278C4—22no—
279C4—24no—
280C4—6281no—
281C4—6282no—
282C4#1909774012yes—
283C4C22×C22⋊D15711yescharacteristic
284C4C22×C22⋊D15711yescharacteristic
285C2—22no—
286C2—42no—
287C2—44no—
288C2—12561no—
289C2C2×#1909774011yescharacteristic
290C2#2375807112yes—
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC22×#41×C157C2
socleC22×C157C22×D4
hypercenterC22×#41D157
hypocenterC157#3005
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#19097802C2semidirect
2. C2×#19097797C2semidirect
3. #23757784C2semidirect
4. #23757435C2semidirect
5. #23757418C2semidirect
6. #23757282C2semidirect
7. #23757278C2semidirect
8. #23757237C2semidirect
9. C22×#41×C157C2semidirect
13. C2×#14578944C22semidirect
14. C2×#14578942C22semidirect
15. C2×#14578941C22semidirect
16. C2×#14578939C22semidirect
20. #19097750C22semidirect
21. #19097745C22semidirect
25. #19097695C22semidirect
26. #19097692C22semidirect
27. C22⋊#11929434C22semidirect
65. C2×Q8×C157D4semidirect
71. C2×D4×C157D4semidirect
145. C157#3005semidirect
147. C22×#41D157semidirect
193. C2×Q8C22⋊D157semidirect
202. C2×D4C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978022
C2×#190977972
#237577842
#237574352
#237574182
#237572822
#237572782
#237572372
C22×#41×C1572
#3005157