Finite Group Databasebeta

← #34103493

Subgroups — #34103493

Order 20096.

1232 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 1232 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C23×D628C211yesmaximal, characteristic, complemented
2C2×#19097746C212yesmaximal, complemented
3C2×#19097741C212yesmaximal, complemented
4C2×C4×C22⋊D157C211yesmaximal, characteristic, complemented
5#23757191C212yesmaximal, complemented
6#23757183C212yesmaximal, complemented
7#23757174C212yesmaximal, complemented
8#23757128C212yesmaximal, complemented
9#380×C157C211yesmaximal, characteristic, complemented
10C24×D157C2212yes—
11C22×D628—24no—
12C22×D628—28no—
13C22×D628C2212yes—
14C22×D628C2212yes—
15C22×D628C2214yescomplemented
16C22×C22⋊D157C2211yescharacteristic
17C22×C4×D157C2211yescharacteristic, complemented
18#19097746—28no—
19#19097741—28no—
20C4×C22⋊D157—24no—
21C2×#14578933C2211yescharacteristic, complemented
22C2×C22⋊Dic157C2211yescharacteristic
23C2×#14578931C2214yescomplemented
24C2×#14578931C2214yescomplemented
25C2×#14578931C2211yescharacteristic, complemented
26C2×#14578931C2214yescomplemented
27C2×C4×Dic157C2211yescharacteristic, complemented
28C23×C4×C157C2211yescharacteristic
29C2×C4⋊C4×C157C2212yescomplemented
30C2×C22⋊C4×C157C2212yes—
31#95×C157C2212yescomplemented
32C23×D157—28no—
33C23×D157—24no—
34C23×D157—24no—
35C23×D157—24no—
36C23×D157—44no—
37C23×D157C2314yes—
38C23×D157C2311yescharacteristic
39C2×D628—232no—
40C2×D628—28no—
41C2×D628—28no—
42C2×D628—28no—
43C2×D628—24no—
44C2×D628—24no—
45C2×D628—24no—
46C2×D628—44no—
47C2×D628D418yescomplemented
48C2×C22⋊D157—22no—
49C2×C22⋊D157—22no—
50C2×C22⋊D157D414yescomplemented
51C22×Dic157C2311yescharacteristic
52C22×Dic157C2311yescharacteristic
53C22×Dic157C2311yescharacteristic
54C2×C4×D157—22no—
55C2×C4×D157—24no—
56#14578933—22no—
57C22⋊Dic157—22no—
58#14578931—28no—
59#14578931—28no—
60#14578931—22no—
61#14578931—44no—
62C4×Dic157—22no—
63C24×C157C2311yescharacteristic
64C22×C4×C157—24no—
65C22×C4×C157D414yescomplemented
66C22×C4×C157C2314yes—
67C22×C4×C157C2311yescharacteristic
68C22×C4×C157C2311yescharacteristic
69C4⋊C4×C157—24no—
70C22⋊C4×C157—24no—
71C22×D157—216no—
72C22×D157—21no—
73C22×D157—22no—
74C22×D157—24no—
75C22×D157—24no—
76C22×D157—24no—
77C22×D157—21no—
78C22×D157—24no—
79C22×D157—22no—
80C22×D157—24no—
81C22×D157—24no—
82C22×D157—416no—
83C22×D157—416no—
84C22×D157—48no—
85C22×D157—48no—
86C22×D157C2×D418yes—
87C22×D157C2×D412yescomplemented
88D628—216no—
89D628—216no—
90D628—416no—
91C22⋊D157—28no—
92C2×Dic157—21no—
93C2×Dic157—21no—
94C2×Dic157—22no—
95C2×Dic157—21no—
96C2×Dic157—21no—
97C2×Dic157—22no—
98C2×Dic157C2×D412yescomplemented
99C4×D157—44no—
100C23×C157—22no—
101C23×C157—22no—
102C23×C157C2×D412yes—
103C23×C157C2×D412yes—
104C23×C157#4112yes—
105C23×C157C2411yescharacteristic
106C2×C4×C157—22no—
107C2×C4×C157—28no—
108C2×C4×C157—24no—
109C2×C4×C157—24no—
110C2×C4×C157—24no—
111C2×C4×C157C2×D414yes—
112C2×C4×C157C2×D412yes—
113C2×C4×C157C2×D412yes—
114C22×C157—22no—
115C22×C157—22no—
116C22×C157—24no—
117C22×C157—24no—
118C22×C157#12114yescomplemented
119C22×C157C22×D412yes—
120C22×C157C22×D411yescharacteristic
121C22×C157C22×D411yescharacteristic
122C22×C157C2×#4111yescharacteristic
123C22×C157H(5,𝔽2)12yes—
124C2×D157—28no—
125C2×D157—22no—
126C2×D157—24no—
127C2×D157—28no—
128C2×D157—28no—
129C2×D157—432no—
130C2×D157—41no—
131C2×D157—44no—
132C2×D157—41no—
133C2×D157—44no—
134C2×D157—44no—
135C2×D157—42no—
136C2×D157—44no—
137C4×C157—22no—
138C4×C157—44no—
139C4×C157#12014yescomplemented
140Dic157—22no—
141Dic157—41no—
142Dic157—41no—
143C2×C157—24no—
144C2×C157C2×#12011yescharacteristic
145C2×C157C2×#12111yescharacteristic
146C2×C157#53511yescharacteristic
147C2×C157D4212yes—
148C2×C157#54612yes—
149D157—42no—
150D157—48no—
151C157#240811yescharacteristic, complemented, Sylow
152#2408—1571nomaximal, Sylow
153C23×D4—1571no—
154C2×#121—1572no—
155C2×#120—1572no—
156C2×C4×D4—1571no—
157#394—1572no—
158#392—1572no—
159#390—1572no—
160#380D15711yescharacteristic, complemented
161#379—1572no—
162C25—1572no—
163C22×D4—3148no—
164C22×D4—3144no—
165C22×D4—1571no—
166C22×D4—1572no—
167C22×D4—1574no—
168C22×D4—1572no—
169C23×C4—1571no—
170C23×C4C2×D15711yescharacteristic
171#121—3148no—
172#120—3148no—
173C4×D4—3144no—
174C2×C4⋊C4—1571no—
175C2×C4⋊C4C2×D15712yescomplemented
176C2×C22⋊C4—1571no—
177C2×C22⋊C4—1571no—
178C2×C22⋊C4—1574no—
179C2×C22⋊C4—1574no—
180C2×C22⋊C4—1574no—
181C2×C22⋊C4C2×D15712yes—
182C2×C42—1571no—
183#95C2×D15712yescomplemented
184C24—3148no—
185C24—3144no—
186C24—3144no—
187C24—3144no—
188C24—6284no—
189C24—1574no—
190C24—1571no—
191C24C22×D15711yescharacteristic
192C2×D4—31432no—
193C2×D4—3148no—
194C2×D4—3148no—
195C2×D4—3148no—
196C2×D4—3142no—
197C2×D4—3142no—
198C2×D4—3144no—
199C2×D4—3144no—
200C2×D4—3144no—
201C2×D4—6284no—
202C2×D4—1574no—
203C2×D4—1578no—
204C22×C4—24no—
205C22×C4—3144no—
206C22×C4—3142no—
207C22×C4—1571no—
208C22×C4—1571no—
209C22×C4—1571no—
210C22×C4D62814yescomplemented
211C22×C4C22×D15714yes—
212C22×C4C22×D15711yescharacteristic
213C22×C4C22×D15711yescharacteristic
214C4⋊C4—24no—
215C4⋊C4—3142no—
216C22⋊C4—24no—
217C22⋊C4—3148no—
218C22⋊C4—3148no—
219C22⋊C4—3142no—
220C22⋊C4—3142no—
221C22⋊C4—6284no—
222C42—3142no—
223C23—22no—
224C23—22no—
225C23—31416no—
226C23—3144no—
227C23—3142no—
228C23—3144no—
229C23—3141no—
230C23—3144no—
231C23—3141no—
232C23—3144no—
233C23—3144no—
234C23—3142no—
235C23—3144no—
236C23—62816no—
237C23—62816no—
238C23—6288no—
239C23—6288no—
240C23—1578no—
241C23—1572no—
242C23D4×D15712yes—
243C23C2×D62812yes—
244C23#1457895512yes—
245C23C23×D15711yescharacteristic
246D4—31416no—
247D4—3148no—
248D4—31416no—
249D4—62816no—
250C2×C4—22no—
251C2×C4—28no—
252C2×C4—24no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—3141no—
256C2×C4—3141no—
257C2×C4—3142no—
258C2×C4—3141no—
259C2×C4—3141no—
260C2×C4—3142no—
261C2×C4—6284no—
262C2×C4—1572no—
263C2×C4D4×D15714yes—
264C2×C4C2×D62812yes—
265C2×C4C2×D62812yes—
266C22—22no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—3144no—
271C22—3148no—
272C22—3148no—
273C22—3148no—
274C22—3142no—
275C22—62832no—
276C22—6284no—
277C22—6281no—
278C22—6284no—
279C22—6284no—
280C22—6282no—
281C22—6281no—
282C22—6284no—
283C22#1909774614yescomplemented
284C22C2×D4×D15711yescharacteristic
285C22C2×D4×D15712yes—
286C22C22×D62811yescharacteristic
287C22C2×#1457895511yescharacteristic
288C22#1909781712yes—
289C4—22no—
290C4—44no—
291C4—3142no—
292C4—6281no—
293C4—6281no—
294C4#1909774114yescomplemented
295C2—24no—
296C2—6288no—
297C2—6282no—
298C2C2×#1909774111yescharacteristic
299C2C2×#1909774611yescharacteristic
300C2#2375784211yescharacteristic
301C2#2375790712yes—
302C2#2375791412yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#380×C157C2
hypercenter#380D157
hypocenterC157#2408
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
1. C23×D628C2semidirect
2. C2×#19097746C2semidirect
3. C2×#19097741C2semidirect
4. C2×C4×C22⋊D157C2semidirect
5. #23757191C2semidirect
6. #23757183C2semidirect
7. #23757174C2semidirect
8. #23757128C2semidirect
9. #380×C157C2semidirect
15. C22×D628C22semidirect
17. C22×C4×D157C22semidirect
21. C2×#14578933C22semidirect
23. C2×#14578931C22semidirect
24. C2×#14578931C22semidirect
25. C2×#14578931C22semidirect
26. C2×#14578931C22semidirect
27. C2×C4×Dic157C22semidirect
29. C2×C4⋊C4×C157C22semidirect
31. #95×C157C22semidirect
47. C2×D628D4semidirect
50. C2×C22⋊D157D4semidirect
65. C22×C4×C157D4semidirect
87. C22×D157C2×D4semidirect
98. C2×Dic157C2×D4semidirect
118. C22×C157#121semidirect
139. C4×C157#120semidirect
151. C157#2408semidirect
160. #380D157semidirect
175. C2×C4⋊C4C2×D157semidirect
183. #95C2×D157semidirect
210. C22×C4D628semidirect
283. C22#19097746semidirect
294. C4#19097741semidirect

Maximal subgroups

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

subgroupindex
C23×D6282
C2×#190977462
C2×#190977412
C2×C4×C22⋊D1572
#237571912
#237571832
#237571742
#237571282
#380×C1572
#2408157