Finite Group Databasebeta

← #34109527

Subgroups — #34109527

Order 20096.

766 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#23758109C211yesmaximal, characteristic, complemented
2#23757784C212yesmaximal, complemented
3#23757462C214yesmaximal, complemented
4#23757459C214yesmaximal, complemented
5#23757247C211yesmaximal, characteristic, complemented
6#23757246C212yesmaximal, complemented
7#525×C157C211yesmaximal, characteristic, complemented
8#19097819—21no—
9#19097818—22no—
10#19097817—22no—
11#19097816—21no—
12#19097816C2214yes—
13C2×#14578954—22no—
14C2×#14578954C2212yes—
15#41×D157—24no—
16#19097754C2214yescomplemented
17#19097750—22no—
18#19097745—22no—
19#19097740C2214yescomplemented
20C4×C22⋊D157—24no—
21#19097730C2212yescomplemented
22#19097686C2214yescomplemented
23#19097685C2218yescomplemented
24C2×#41×C157C2211yescharacteristic
25C23×C4×C157C2211yescharacteristic, complemented
26#123×C157C2212yescomplemented
27C22⋊Q8×C157—21no—
28#121×C157—21no—
29#120×C157C2212yescomplemented
30C4×D4×C157—22no—
31#117×C157C2211yescharacteristic, complemented
32#14578955—24no—
33#14578954—24no—
34#14578954—24no—
35#14578954—22no—
36#14578954—22no—
37#14578954—24no—
38#14578954—22no—
39#14578954—22no—
40#14578953—24no—
41#14578953—28no—
42C2×Dic314—22no—
43C2×Dic314D412yescomplemented
44Q8×D157—24no—
45C2×D628—22no—
46C2×D628D412yescomplemented
47C2×C22⋊D157—24no—
48C2×C22⋊D157—24no—
49C2×C22⋊D157C2314yes—
50D4×D157—24no—
51D4×D157—28no—
52C2×C4×D157—24no—
53C2×C4×D157D414yescomplemented
54C4⋊Dic157—22no—
55#14578933—24no—
56C22⋊Dic157—24no—
57C22⋊Dic157C2314yes—
58#14578931—24no—
59C4×Dic157—22no—
60C24×C157C2311yescharacteristic
61#41×C157—22no—
62#41×C157—22no—
63C2×Q8×C157D411yescharacteristic, complemented
64C2×D4×C157—22no—
65C2×D4×C157D411yescharacteristic, complemented
66C2×D4×C157C2312yes—
67C22×C4×C157—21no—
68C22×C4×C157—21no—
69C22×C4×C157—22no—
70C22×C4×C157—22no—
71C22×C4×C157D412yescomplemented
72C22×C4×C157C2311yescharacteristic
73C22×C4×C157C2311yescharacteristic
74C4⋊C4×C157—21no—
75C4⋊C4×C157—22no—
76C22⋊C4×C157—22no—
77C22⋊C4×C157—22no—
78C22⋊C4×C157C2312yes—
79C42×C157—21no—
80C22×D157—24no—
81C22×D157C2×D414yes—
82Dic314—24no—
83Dic314—22no—
84Dic314—22no—
85D628—24no—
86D628—22no—
87D628—22no—
88C22⋊D157—28no—
89C22⋊D157—24no—
90C22⋊D157—24no—
91C22⋊D157—48no—
92C2×Dic157—24no—
93C2×Dic157—24no—
94C2×Dic157C2×D414yes—
95C4×D157—28no—
96C4×D157—24no—
97C4×D157—24no—
98C23×C157—22no—
99C23×C157—21no—
100C23×C157—42no—
101C23×C157C2×D412yes—
102C23×C157C2411yescharacteristic
103Q8×C157—22no—
104D4×C157—24no—
105D4×C157—22no—
106D4×C157—44no—
107C2×C4×C157—22no—
108C2×C4×C157—22no—
109C2×C4×C157—22no—
110C2×C4×C157—21no—
111C2×C4×C157—22no—
112C2×C4×C157—41no—
113C2×C4×C157—42no—
114C2×C4×C157—41no—
115C2×C4×C157C2×D412yes—
116C2×C4×C157C2×D412yes—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2×D411yescharacteristic
119C2×C4×C157C2×D411yescharacteristic
120C22×C157—22no—
121C22×C157—24no—
122C22×C157—22no—
123C22×C157—41no—
124C22×C157—42no—
125C22×C157—42no—
126C22×C157—41no—
127C22×C157—41no—
128C22×C157C22×D412yes—
129C22×C157C22×D411yescharacteristic
130C2×D157—24no—
131C2×D157—44no—
132C2×D157—84no—
133C4×C157—22no—
134C4×C157—42no—
135C4×C157—41no—
136C4×C157#12011yescharacteristic
137C4×C157#12012yes—
138C4×C157#12011yescharacteristic
139Dic157—24no—
140Dic157—44no—
141C2×C157—22no—
142C2×C157—21no—
143C2×C157—42no—
144C2×C157—42no—
145C2×C157C2×#12011yescharacteristic
146D157—44no—
147C157#302511yescharacteristic, complemented, Sylow
148#3025—1571nomaximal, Sylow
149#585—1571no—
150#525—1572no—
151#525D15711yescharacteristic, complemented
152#458—1574no—
153B(4,𝔽2)—1574no—
154#410—1571no—
155#410—1572no—
1562−1+4—3142no—
1572−1+4—3141no—
158H(5,𝔽2)—3142no—
159H(5,𝔽2)—3141no—
160H(5,𝔽2)—1574no—
161C2×#41—3144no—
162C2×#41—3142no—
163C2×#41—1572no—
164C2×#41C2×D15711yescharacteristic
165C23×C4C2×D15711yescharacteristic, complemented
166#123—1574no—
167#123C2×D15712yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—3142no—
170#121—21no—
171#121—3142no—
172#120—1574no—
173#120C2×D15712yescomplemented
174C4×D4—22no—
175C4×D4—3144no—
176#117—1572no—
177#117C2×D15711yescharacteristic, complemented
178#99—1574no—
179#99—1578no—
180C24C22×D15711yescharacteristic
181#41—22no—
182#41—22no—
183#41—3144no—
184#41—3144no—
185#41—3144no—
186#41—3144no—
187#41—3148no—
188#41—3142no—
189#41—3142no—
190#41—3142no—
191#41—3144no—
192#41—3142no—
193C2×Q8—3142no—
194C2×Q8—3144no—
195C2×Q8—1572no—
196C2×Q8C22⋊D15711yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—3142no—
199C2×D4—3144no—
200C2×D4—3148no—
201C2×D4—3144no—
202C2×D4—3144no—
203C2×D4—1574no—
204C2×D4—1572no—
205C2×D4C22⋊D15711yescharacteristic, complemented
206C2×D4C22×D15712yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—3144no—
212C22×C4—1574no—
213C22×C4C22⋊D15712yescomplemented
214C22×C4C22×D15711yescharacteristic
215C22×C4C22×D15711yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—3144no—
219C4⋊C4—3142no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—3144no—
223C22⋊C4—3144no—
224C22⋊C4—1574no—
225C22⋊C4C22×D15712yes—
226C42—21no—
227C42—3142no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—3144no—
232C23—1574no—
233C23C2×C22⋊D15712yes—
234C23C23×D15711yescharacteristic
235Q8—22no—
236Q8—3144no—
237Q8—3142no—
238Q8—3142no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—3144no—
243D4—3148no—
244D4—3144no—
245D4—3144no—
246D4—3142no—
247D4—3142no—
248D4—6288no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—3148no—
258C2×C4—3144no—
259C2×C4—3144no—
260C2×C4—3144no—
261C2×C4—3144no—
262C2×C4—1574no—
263C2×C4D4×D15712yes—
264C2×C4D4×D15712yes—
265C2×C4C2×C22⋊D15712yes—
266C2×C4C2×C22⋊D15711yescharacteristic
267C2×C4C2×C22⋊D15711yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—3144no—
277C22—6284no—
278C22—12564no—
279C22C2×D4×D15712yes—
280C22C22×C22⋊D15711yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—3144no—
285C4—6284no—
286C4#1909773911yescharacteristic
287C4#1909773912yes—
288C4#1909773911yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—6284no—
294C2C2×#1909773911yescharacteristic
subgroupG/H
centerC4#19097739
commutator, cosocleC23×C157C24
FrattiniC23C23×D157
Fitting#525×C157C2
socleC2×C157C2×#120
hypercenter#525D157
hypocenterC157#3025
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #23758109C2semidirect
2. #23757784C2semidirect
3. #23757462C2semidirect
4. #23757459C2semidirect
5. #23757247C2semidirect
6. #23757246C2semidirect
7. #525×C157C2semidirect
16. #19097754C22semidirect
19. #19097740C22semidirect
21. #19097730C22semidirect
22. #19097686C22semidirect
23. #19097685C22semidirect
25. C23×C4×C157C22semidirect
26. #123×C157C22semidirect
29. #120×C157C22semidirect
31. #117×C157C22semidirect
43. C2×Dic314D4semidirect
46. C2×D628D4semidirect
53. C2×C4×D157D4semidirect
63. C2×Q8×C157D4semidirect
65. C2×D4×C157D4semidirect
71. C22×C4×C157D4semidirect
147. C157#3025semidirect
151. #525D157semidirect
165. C23×C4C2×D157semidirect
167. #123C2×D157semidirect
173. #120C2×D157semidirect
177. #117C2×D157semidirect
196. C2×Q8C22⋊D157semidirect
205. C2×D4C22⋊D157semidirect
213. C22×C4C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
#237577842
#237574622
#237574592
#237572472
#237572462
#525×C1572
#3025157