Finite Group Databasebeta

← #34109530

Subgroups — #34109530

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#23757785C212yesmaximal, complemented
3#23757780C211yesmaximal, characteristic, complemented
4#23757464C214yesmaximal, complemented
5#23757461C214yesmaximal, complemented
6#23757246C212yesmaximal, complemented
7#410×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—
16C22×C4×D157C2211yescharacteristic, complemented
17#19097755C2214yescomplemented
18#19097752C2212yescomplemented
19#19097751—22no—
20#19097747—21no—
21#19097746—22no—
22#19097742—21no—
23#19097741C2214yescomplemented
24#19097739C2212yescomplemented
25C4×D628—24no—
26C4×C22⋊D157—22no—
27#19097730C2211yescharacteristic, complemented
28#19097685C2218yescomplemented
29C2×#41×C157C2211yescharacteristic
30#117×C157C2212yescomplemented
31#99×C157C2214yescomplemented
32C23×D157C2311yescharacteristic
33#14578955—24no—
34#14578954—24no—
35#14578954—24no—
36#14578954—22no—
37#14578954—22no—
38#14578954—24no—
39#14578954—22no—
40#14578954—22no—
41#14578953—24no—
42#14578953—28no—
43C2×Dic314—22no—
44C2×Dic314D412yescomplemented
45Q8×D157—24no—
46C2×D628—22no—
47C2×D628—24no—
48C2×D628D412yescomplemented
49C2×C22⋊D157—24no—
50C2×C22⋊D157—22no—
51C2×C22⋊D157C2314yes—
52D4×D157—24no—
53D4×D157—28no—
54C22×Dic157C2311yescharacteristic
55C2×C4×D157—24no—
56C2×C4×D157—21no—
57C2×C4×D157—22no—
58C2×C4×D157—21no—
59C2×C4×D157—22no—
60C2×C4×D157D414yescomplemented
61C4⋊Dic157—24no—
62C4⋊Dic157—21no—
63#14578933—22no—
64C22⋊Dic157—22no—
65C22⋊Dic157C2312yes—
66#14578931—24no—
67#14578931—24no—
68#14578931—22no—
69C4×Dic157—21no—
70#41×C157—22no—
71#41×C157—22no—
72C2×Q8×C157D411yescharacteristic, complemented
73C2×D4×C157D411yescharacteristic, complemented
74C2×D4×C157C2312yes—
75C22×C4×C157D412yescomplemented
76C22×C4×C157C2311yescharacteristic
77C4⋊C4×C157—22no—
78C22⋊C4×C157C2314yes—
79C42×C157—22no—
80C22×D157—21no—
81C22×D157—22no—
82C22×D157—24no—
83C22×D157—42no—
84C22×D157C2×D414yes—
85Dic314—24no—
86Dic314—22no—
87Dic314—22no—
88D628—24no—
89D628—22no—
90D628—22no—
91D628—48no—
92C22⋊D157—28no—
93C22⋊D157—24no—
94C22⋊D157—24no—
95C22⋊D157—44no—
96C2×Dic157—24no—
97C2×Dic157—22no—
98C2×Dic157—21no—
99C2×Dic157—22no—
100C2×Dic157C2×D414yes—
101C4×D157—28no—
102C4×D157—24no—
103C4×D157—24no—
104C4×D157—41no—
105C4×D157—42no—
106C4×D157—41no—
107C23×C157C2×D412yes—
108C23×C157C2411yescharacteristic
109Q8×C157—22no—
110D4×C157—24no—
111D4×C157—22no—
112C2×C4×C157—22no—
113C2×C4×C157—22no—
114C2×C4×C157—24no—
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—41no—
122C22×C157—42no—
123C22×C157C22×D412yes—
124C22×C157C22×D411yescharacteristic
125C2×D157—22no—
126C2×D157—24no—
127C2×D157—24no—
128C2×D157—41no—
129C2×D157—42no—
130C2×D157—41no—
131C2×D157—44no—
132C2×D157—84no—
133C4×C157—22no—
134C4×C157—44no—
135C4×C157#12011yescharacteristic
136C4×C157#12012yes—
137C4×C157#12011yescharacteristic
138Dic157—24no—
139Dic157—42no—
140Dic157—41no—
141C2×C157—21no—
142C2×C157—22no—
143C2×C157—42no—
144C2×C157C2×#12011yescharacteristic
145D157—42no—
146D157—44no—
147C157#302511yescharacteristic, complemented, Sylow
148#3025—1571nomaximal, Sylow
149#585—1571no—
150#525—1571no—
151#525—1572no—
152#458—1574no—
153B(4,𝔽2)—1574no—
154#410—1572no—
155#410D15711yescharacteristic, complemented
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×C4—1571no—
166#123—1572no—
167#123—1574no—
168C22⋊Q8—3142no—
169C22⋊Q8—3141no—
170#121—3142no—
171#121—3141no—
172#120—1572no—
173#120—1574no—
174C4×D4—3144no—
175C4×D4—3142no—
176#117—1571no—
177#117C2×D15712yescomplemented
178#99—1578no—
179#99C2×D15714yescomplemented
180C24—1571no—
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×Q8D62811yescharacteristic, complemented
197C2×D4—3142no—
198C2×D4—3144no—
199C2×D4—3144no—
200C2×D4—3148no—
201C2×D4—3144no—
202C2×D4—3142no—
203C2×D4—1574no—
204C2×D4—1572no—
205C2×D4D62811yescharacteristic, complemented
206C2×D4C22×D15712yes—
207C22×C4—3144no—
208C22×C4—3142no—
209C22×C4—3142no—
210C22×C4—3141no—
211C22×C4—3141no—
212C22×C4—1571no—
213C22×C4—1574no—
214C22×C4D62812yescomplemented
215C22×C4C22×D15711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—3144no—
218C4⋊C4—3141no—
219C4⋊C4—3142no—
220C22⋊C4—3144no—
221C22⋊C4—3144no—
222C22⋊C4—3142no—
223C22⋊C4—3142no—
224C22⋊C4—1572no—
225C22⋊C4C22×D15714yes—
226C42—22no—
227C42—3141no—
228C23—3144no—
229C23—3142no—
230C23—3141no—
231C23—6282no—
232C23—1574no—
233C23C2×D62812yes—
234C23C23×D15711yescharacteristic
235Q8—22no—
236Q8—3144no—
237Q8—3142no—
238Q8—3142no—
239D4—24no—
240D4—22no—
241D4—3144no—
242D4—3148no—
243D4—3144no—
244D4—3144no—
245D4—3142no—
246D4—3142no—
247D4—6288no—
248D4—6284no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—3148no—
253C2×C4—3144no—
254C2×C4—3144no—
255C2×C4—3144no—
256C2×C4—3142no—
257C2×C4—3142no—
258C2×C4—3141no—
259C2×C4—6281no—
260C2×C4—6282no—
261C2×C4—6281no—
262C2×C4—1574no—
263C2×C4D4×D15712yes—
264C2×C4D4×D15712yes—
265C2×C4C2×D62812yes—
266C2×C4C2×D62811yescharacteristic
267C2×C4C2×D62811yescharacteristic
268C22—22no—
269C22—41no—
270C22—42no—
271C22—3144no—
272C22—3142no—
273C22—3144no—
274C22—6284no—
275C22—6282no—
276C22—6281no—
277C22—6281no—
278C22—12564no—
279C22C2×D4×D15712yes—
280C22C22×D62811yescharacteristic
281C4—22no—
282C4—44no—
283C4—3144no—
284C4—6282no—
285C4—6281no—
286C4#1909774111yescharacteristic
287C4#1909774112yes—
288C4#1909774111yescharacteristic
289C2—22no—
290C2—21no—
291C2—42no—
292C2—6282no—
293C2—6284no—
294C2C2×#1909774111yescharacteristic
subgroupG/H
centerC4#19097741
commutator, cosocleC23×C157C24
FrattiniC23C23×D157
Fitting#410×C157C2
socleC2×C157C2×#120
hypercenter#410D157
hypocenterC157#3025
LCSG ▷ C23×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #23758109C2semidirect
2. #23757785C2semidirect
3. #23757780C2semidirect
4. #23757464C2semidirect
5. #23757461C2semidirect
6. #23757246C2semidirect
7. #410×C157C2semidirect
16. C22×C4×D157C22semidirect
17. #19097755C22semidirect
18. #19097752C22semidirect
23. #19097741C22semidirect
24. #19097739C22semidirect
27. #19097730C22semidirect
28. #19097685C22semidirect
30. #117×C157C22semidirect
31. #99×C157C22semidirect
44. C2×Dic314D4semidirect
48. C2×D628D4semidirect
60. C2×C4×D157D4semidirect
72. C2×Q8×C157D4semidirect
73. C2×D4×C157D4semidirect
75. C22×C4×C157D4semidirect
147. C157#3025semidirect
155. #410D157semidirect
177. #117C2×D157semidirect
179. #99C2×D157semidirect
196. C2×Q8D628semidirect
205. C2×D4D628semidirect
214. C22×C4D628semidirect

Maximal subgroups

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

subgroupindex
#237581092
#237577852
#237577802
#237574642
#237574612
#237572462
#410×C1572
#3025157