Finite Group Databasebeta

← #34103377

Subgroups — #34103377

Order 20096.

696 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 155 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×#19097733C212yesmaximal, complemented
2C2×#19097733C211yesmaximal, characteristic, complemented
3C2×#19097729C211yesmaximal, characteristic, complemented
4#23757199C211yesmaximal, characteristic, complemented
5#23757195C212yesmaximal, complemented
6#23757191C212yesmaximal, complemented
7#23757190C211yesmaximal, characteristic, complemented
8#23757183C211yesmaximal, characteristic
9#23757124C211yesmaximal, characteristic, complemented
10#23757120C212yesmaximal, complemented
11#398×C157C211yesmaximal, characteristic, complemented
12C22×D628C2211yescharacteristic
13C22×D628C2212yes—
14C22×C4×D157C2212yes—
15C22×C4×D157C2211yescharacteristic
16#19097733—24no—
17#19097733—28no—
18#19097729—24no—
19C2×C4⋊Dic157C2211yescharacteristic
20C2×#14578933C2212yes—
21C2×#14578931C2212yes—
22C2×#14578931C2212yes—
23C2×#14578931C2211yescharacteristic
24C2×#14578931C2212yes—
25C2×#14578931C2211yescharacteristic
26C2×#14578931C2212yes—
27C2×#14578931C2212yes—
28C2×C4×Dic157C2211yescharacteristic
29C2×C4×Dic157C2212yes—
30C2×C4×Dic157C2211yescharacteristic
31#19097678C2212yes—
32#19097678C2211yescharacteristic
33C2×C4⋊C4×C157C2211yescharacteristic
34C2×C4⋊C4×C157C2212yes—
35C2×C4⋊C4×C157C2211yescharacteristic
36#95×C157C2212yes—
37#95×C157C2211yescharacteristic
38C23×D157C2311yescharacteristic
39C23×D157C2312yes—
40C2×D628—24no—
41C2×D628—24no—
42C2×D628—24no—
43C2×D628—22no—
44C2×D628—24no—
45C22×Dic157C2312yes—
46C22×Dic157C2311yescharacteristic
47C22×Dic157C2311yescharacteristic
48C22×Dic157C2311yescharacteristic
49C2×C4×D157—22no—
50C2×C4×D157—22no—
51C2×C4×D157—24no—
52C2×C4×D157—24no—
53C2×C4×D157—24no—
54C2×C4×D157D414yescomplemented
55C4⋊Dic157—22no—
56#14578933—24no—
57#14578931—24no—
58#14578931—24no—
59#14578931—22no—
60#14578931—24no—
61#14578931—22no—
62#14578931—24no—
63#14578931—24no—
64C4×Dic157—22no—
65C4×Dic157—24no—
66C4×Dic157D414yescomplemented
67C22×C4×C157C2311yescharacteristic
68C22×C4×C157C2312yes—
69C22×C4×C157C2311yescharacteristic
70C22×C4×C157C2311yescharacteristic
71C22×C4×C157C2312yes—
72C4⋊C4×C157—22no—
73C4⋊C4×C157—24no—
74C4⋊C4×C157—22no—
75C22×D157—22no—
76C22×D157—22no—
77C22×D157—21no—
78C22×D157—21no—
79C22×D157—22no—
80C22×D157—22no—
81C22×D157—22no—
82C22×D157—22no—
83C22×D157—22no—
84C22×D157—22no—
85C22×D157C2×D412yes—
86C22×D157#4114yes—
87D628—28no—
88D628—216no—
89C2×Dic157—22no—
90C2×Dic157—22no—
91C2×Dic157—21no—
92C2×Dic157—22no—
93C2×Dic157—22no—
94C2×Dic157—22no—
95C2×Dic157C2×D412yes—
96C2×Dic157C2×D412yes—
97C2×Dic157C2×D412yes—
98C2×Dic157#4112yes—
99C4×D157—28no—
100C4×D157—48no—
101C23×C157C2411yescharacteristic
102C2×C4×C157—21no—
103C2×C4×C157—22no—
104C2×C4×C157—21no—
105C2×C4×C157—22no—
106C2×C4×C157—22no—
107C2×C4×C157—22no—
108C2×C4×C157—22no—
109C2×C4×C157—21no—
110C2×C4×C157—22no—
111C2×C4×C157—22no—
112C2×C4×C157C2×D412yes—
113C2×C4×C157C2×D412yes—
114C2×C4×C157#4114yes—
115C22×C157C22×D411yescharacteristic
116C22×C157C22×D411yescharacteristic
117C22×C157C2×#4111yescharacteristic
118C22×C157C2×#4112yes—
119C22×C157C2×#4112yes—
120C2×D157—22no—
121C2×D157—24no—
122C2×D157—24no—
123C2×D157—24no—
124C2×D157—24no—
125C2×D157—42no—
126C2×D157—41no—
127C2×D157—42no—
128C2×D157—42no—
129C2×D157—42no—
130C2×D157—42no—
131C2×D157—41no—
132C4×C157—22no—
133C4×C157—22no—
134C4×C157—24no—
135C4×C157—42no—
136C4×C157—41no—
137Dic157—22no—
138Dic157—42no—
139Dic157—41no—
140Dic157#12114yes—
141C2×C157C2×#12111yescharacteristic
142C2×C157#52511yescharacteristic
143C2×C157#52512yes—
144C2×C157#52912yes—
145C2×C157#53211yescharacteristic
146D157—42no—
147D157—44no—
148C157#239211yescharacteristic, complemented, Sylow
149#2392—1571nomaximal, Sylow
150C2×C4×D4—1572no—
151C2×C4×D4—1571no—
152C2×#117—1571no—
153#398D15711yescharacteristic, complemented
154#397—1571no—
155#396—1572no—
156#394—1571no—
157#394—1572no—
158#392—1571no—
159C4×C4⋊C4—1571no—
160C4×C22⋊C4—1572no—
161C22×D4—1571no—
162C22×D4—1572no—
163C23×C4—1572no—
164C23×C4—1571no—
165C4×D4—3148no—
166C4×D4—3144no—
167#117—3144no—
168C2×C4⋊C4—1571no—
169C2×C4⋊C4—1572no—
170C2×C4⋊C4C2×D15711yescharacteristic
171C2×C4⋊C4C2×D15712yes—
172C2×C4⋊C4C2×D15711yescharacteristic
173C2×C22⋊C4—1571no—
174C2×C22⋊C4—1571no—
175C2×C22⋊C4—1572no—
176C2×C22⋊C4—1572no—
177C2×C22⋊C4—1572no—
178C2×C22⋊C4—1572no—
179C2×C22⋊C4—1572no—
180C2×C42—1571no—
181C2×C42—1572no—
182C2×C42—1571no—
183#95—1572no—
184#95—1571no—
185#95C2×D15712yes—
186#95C2×D15711yescharacteristic
187C24—1571no—
188C24—1572no—
189C2×D4—3144no—
190C2×D4—3144no—
191C2×D4—3144no—
192C2×D4—3142no—
193C2×D4—3144no—
194C22×C4—3142no—
195C22×C4—3142no—
196C22×C4—3144no—
197C22×C4—3144no—
198C22×C4—3144no—
199C22×C4—1572no—
200C22×C4—1571no—
201C22×C4—1571no—
202C22×C4—1574no—
203C22×C4—1571no—
204C22×C4C22×D15712yes—
205C22×C4C22×D15711yescharacteristic
206C22×C4C22×D15711yescharacteristic
207C22×C4C22×D15711yescharacteristic
208C22×C4C22×D15712yes—
209C4⋊C4—22no—
210C4⋊C4—24no—
211C4⋊C4—22no—
212C4⋊C4—3142no—
213C4⋊C4—3144no—
214C22⋊C4—3144no—
215C22⋊C4—3144no—
216C22⋊C4—3142no—
217C22⋊C4—3144no—
218C22⋊C4—3142no—
219C22⋊C4—3144no—
220C22⋊C4—3144no—
221C42—3142no—
222C42—3144no—
223C42—1574no—
224C23—3141no—
225C23—3142no—
226C23—3142no—
227C23—3141no—
228C23—3142no—
229C23—3142no—
230C23—3142no—
231C23—3142no—
232C23—3142no—
233C23—3142no—
234C23—1572no—
235C23—1574no—
236C23C23×D15711yescharacteristic
237D4—3148no—
238D4—31416no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—3148no—
250C2×C4—3142no—
251C2×C4—3142no—
252C2×C4—3141no—
253C2×C4—3142no—
254C2×C4—3142no—
255C2×C4—3142no—
256C2×C4—6288no—
257C2×C4—1572no—
258C2×C4—1572no—
259C2×C4—1572no—
260C2×C4—1572no—
261C2×C4D4×D15712yes—
262C2×C4D4×D15712yes—
263C2×C4#1457895514yes—
264C22—3142no—
265C22—3144no—
266C22—3144no—
267C22—3144no—
268C22—3144no—
269C22—6281no—
270C22—6282no—
271C22—6282no—
272C22—6281no—
273C22—6282no—
274C22—6282no—
275C22—6282no—
276C22C2×D4×D15711yescharacteristic
277C22C2×D4×D15711yescharacteristic
278C22#41×D15711yescharacteristic
279C22#41×D15712yes—
280C22C2×#1457895512yes—
281C4—22no—
282C4—22no—
283C4—24no—
284C4—42no—
285C4—41no—
286C4—3142no—
287C4—6281no—
288C4—6282no—
289C4—1574no—
290C2—6282no—
291C2—6284no—
292C2#121×D15711yescharacteristic
293C2#2375778111yescharacteristic
294C2#2375778612yes—
295C2#2375780812yes—
296C2#2375782411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#398×C157C2
prime radical#23757183C2
hypercenter#398D157
hypocenterC157#2392
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
1. C2×#19097733C2semidirect
2. C2×#19097733C2semidirect
3. C2×#19097729C2semidirect
4. #23757199C2semidirect
5. #23757195C2semidirect
6. #23757191C2semidirect
7. #23757190C2semidirect
9. #23757124C2semidirect
10. #23757120C2semidirect
11. #398×C157C2semidirect
54. C2×C4×D157D4semidirect
66. C4×Dic157D4semidirect
148. C157#2392semidirect
153. #398D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977332
C2×#190977292
#237571992
#237571952
#237571912
#237571902
#237571832
#237571242
#237571202
#398×C1572
#2392157