Finite Group Databasebeta

← #34103494

Subgroups — #34103494

Order 20096.

840 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 171 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
1C22×Q8×D157C211yesmaximal, characteristic, complemented
2C2×#19097748C212yesmaximal, complemented
3C2×#19097731C211yesmaximal, characteristic
4#23757194C211yesmaximal, characteristic, complemented
5#23757192C211yesmaximal, characteristic, complemented
6#23757185C211yesmaximal, characteristic, complemented
7#23757177C212yesmaximal, complemented
8#23757149C212yesmaximal
9#23757147C212yesmaximal, complemented
10#23757132C211yesmaximal, characteristic, complemented
11#393×C157C211yesmaximal, characteristic, complemented
12C22×Dic314C2211yescharacteristic
13C22×Dic314C2212yes—
14C2×Q8×D157—28no—
15C2×Q8×D157—24no—
16C22×C22⋊D157C2211yescharacteristic
17C23×Dic157C2211yescharacteristic
18C22×C4×D157C2212yes—
19C22×C4×D157C2211yescharacteristic
20#19097748—28no—
21#19097731—24no—
22C2×C4⋊Dic157C2212yes—
23C2×#14578933C2212yes—
24C2×#14578933C2212yes—
25C2×#14578933C2211yescharacteristic
26C2×C22⋊Dic157C2212yes—
27C2×#14578931C2212yes—
28C2×#14578931C2211yescharacteristic
29C2×C4×Dic157C2211yescharacteristic
30#19097678C2211yescharacteristic
31#19097678C2212yes—
32#19097678C2212yes—
33#19097678C2212yes—
34C22×Q8×C157C2211yescharacteristic
35C2×C22⋊C4×C157C2212yes—
36C2×C22⋊C4×C157C2211yescharacteristic
37#95×C157C2211yescharacteristic
38#95×C157C2212yes—
39C23×D157C2311yescharacteristic
40C2×Dic314—24no—
41C2×Dic314—24no—
42C2×Dic314—24no—
43C2×Dic314—22no—
44C2×Dic314D418yescomplemented
45Q8×D157—232no—
46C2×C22⋊D157—22no—
47C2×C22⋊D157—22no—
48C2×C22⋊D157Q814yescomplemented
49C22×Dic157—22no—
50C22×Dic157—24no—
51C22×Dic157C2312yes—
52C22×Dic157C2311yescharacteristic
53C22×Dic157C2312yes—
54C22×Dic157C2311yescharacteristic
55C22×Dic157C2311yescharacteristic
56C2×C4×D157—24no—
57C2×C4×D157—24no—
58C2×C4×D157—24no—
59C2×C4×D157—24no—
60C2×C4×D157D414yes—
61C4⋊Dic157—24no—
62#14578933—24no—
63#14578933—22no—
64#14578933—24no—
65C22⋊Dic157—24no—
66#14578931—22no—
67#14578931—24no—
68C4×Dic157—22no—
69C24×C157C2311yescharacteristic
70C2×Q8×C157—24no—
71C2×Q8×C157—22no—
72C22×C4×C157C2311yescharacteristic
73C22×C4×C157C2312yes—
74C22×C4×C157C2312yes—
75C22×C4×C157C2311yescharacteristic
76C22⋊C4×C157—22no—
77C22⋊C4×C157—24no—
78C22×D157—22no—
79C22×D157—22no—
80C22×D157C2×D412yes—
81C22×D157C2×Q812yes—
82C22×D157#4112yes—
83Dic314—28no—
84Dic314—216no—
85C22⋊D157—28no—
86C2×Dic157—22no—
87C2×Dic157—22no—
88C2×Dic157—21no—
89C2×Dic157—21no—
90C2×Dic157—21no—
91C2×Dic157—22no—
92C2×Dic157—22no—
93C2×Dic157—22no—
94C2×Dic157—22no—
95C2×Dic157—44no—
96C2×Dic157C2×D414yes—
97C2×Dic157C2×D414yes—
98C2×Dic157C2×D412yes—
99C2×Dic157C2×Q812yescomplemented
100C4×D157—216no—
101C4×D157—28no—
102C23×C157—21no—
103C23×C157—22no—
104C23×C157—21no—
105C23×C157—22no—
106C23×C157C2×Q812yescomplemented
107C23×C157C2411yescharacteristic
108Q8×C157—28no—
109C2×C4×C157—22no—
110C2×C4×C157—22no—
111C2×C4×C157—22no—
112C2×C4×C157—21no—
113C2×C4×C157—22no—
114C2×C4×C157—22no—
115C2×C4×C157—22no—
116C2×C4×C157—22no—
117C2×C4×C157C2×D414yes—
118C2×C4×C157C2×D412yes—
119C22×C157—22no—
120C22×C157—24no—
121C22×C157—41no—
122C22×C157—41no—
123C22×C157—42no—
124C22×C157C22×D411yescharacteristic
125C22×C157C22×D412yes—
126C22×C157C22×Q811yescharacteristic
127C22×C157C2×#4111yescharacteristic
128C22×C1572−1+412yes—
129C2×D157—22no—
130C2×D157—22no—
131C2×D157—24no—
132C2×D157—24no—
133C2×D157C22⋊Q814yescomplemented
134C4×C157—24no—
135C4×C157—22no—
136C4×C157—41no—
137C4×C157—42no—
138Dic157—24no—
139Dic157—22no—
140Dic157—41no—
141Dic157—42no—
142Dic157#12014yes—
143C2×C157—42no—
144C2×C157C2×#12011yescharacteristic
145C2×C157C2×C22⋊Q811yescharacteristic
146C2×C157#53611yescharacteristic
147C2×C157#54812yes—
148C2×C157D4×Q812yes—
149D157—24no—
150C157#240911yescharacteristic, complemented, Sylow
151#2409—1571nomaximal, Sylow
152C23×Q8—1571no—
153C2×C22⋊Q8—1572no—
154C2×C4×D4—1571no—
155#395—1571no—
156#395—1571no—
157#393—1571no—
158#393D15711yescharacteristic, complemented
159#391—1572no—
160#386—1572no—
161#385—1572no—
162#380—1571no—
163C22×Q8—3148no—
164C22×Q8—3144no—
165C22×Q8—1571no—
166C22×Q8—1572no—
167C22×Q8C2×D15711yescharacteristic
168C22×D4—1571no—
169C23×C4—1571no—
170C23×C4—1572no—
171C23×C4—1571no—
172C22⋊Q8—3148no—
173C4×D4—3144no—
174C2×C4⋊C4—1571no—
175C2×C4⋊C4—1572no—
176C2×C4⋊C4—1572no—
177C2×C4⋊C4—1572no—
178C2×C22⋊C4—1572no—
179C2×C22⋊C4—1571no—
180C2×C22⋊C4—1572no—
181C2×C22⋊C4C2×D15711yescharacteristic
182C2×C22⋊C4C2×D15712yes—
183C2×C42—1571no—
184#95—1571no—
185#95—1572no—
186#95—1572no—
187#95—1572no—
188#95C2×D15711yescharacteristic
189#95C2×D15712yes—
190C24—1571no—
191C24C22×D15711yescharacteristic
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—3144no—
195C2×Q8—3144no—
196C2×Q8—3144no—
197C2×Q8—31432no—
198C2×Q8—3142no—
199C2×Q8—1578no—
200C2×D4—3142no—
201C2×D4—3142no—
202C2×D4—1574no—
203C22×C4—3144no—
204C22×C4—3144no—
205C22×C4—3144no—
206C22×C4—3144no—
207C22×C4—3142no—
208C22×C4—3144no—
209C22×C4—1572no—
210C22×C4—1571no—
211C22×C4—1571no—
212C22×C4—1572no—
213C22×C4—1574no—
214C22×C4—1571no—
215C22×C4C22×D15711yescharacteristic
216C22×C4C22×D15712yes—
217C22×C4C22×D15712yes—
218C22×C4C22×D15711yescharacteristic
219C4⋊C4—3144no—
220C4⋊C4—3144no—
221C4⋊C4—3142no—
222C4⋊C4—3144no—
223C22⋊C4—22no—
224C22⋊C4—24no—
225C22⋊C4—3142no—
226C22⋊C4—3144no—
227C22⋊C4—3144no—
228C42—3142no—
229C23—22no—
230C23—21no—
231C23—21no—
232C23—22no—
233C23—3142no—
234C23—3142no—
235C23—1572no—
236C23—1572no—
237C23—1572no—
238C23Q8×D15712yescomplemented
239C23C23×D15711yescharacteristic
240Q8—28no—
241Q8—3148no—
242Q8—31416no—
243D4—3148no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—31416no—
253C2×C4—3148no—
254C2×C4—3141no—
255C2×C4—3142no—
256C2×C4—3142no—
257C2×C4—3141no—
258C2×C4—3142no—
259C2×C4—3141no—
260C2×C4—3142no—
261C2×C4—3142no—
262C2×C4—3142no—
263C2×C4—6284no—
264C2×C4—1574no—
265C2×C4—1572no—
266C2×C4—1572no—
267C2×C4—1574no—
268C2×C4D4×D15714yes—
269C2×C4D4×D15712yes—
270C22—24no—
271C22—22no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—3142no—
276C22—3144no—
277C22—3144no—
278C22—3142no—
279C22—1574no—
280C22C2×D4×D15711yescharacteristic
281C22C2×D4×D15712yes—
282C22C2×Q8×D15711yescharacteristic
283C22#41×D15711yescharacteristic
284C22#1909781812yes—
285C4—24no—
286C4—22no—
287C4—41no—
288C4—42no—
289C4—3144no—
290C4—3142no—
291C4—6282no—
292C4—6281no—
293C4—1574no—
294C2—42no—
295C2—3144no—
296C2#120×D15711yescharacteristic
297C2C22⋊Q8×D15711yescharacteristic
298C2#2375784311yescharacteristic
299C2#2375793112yes—
300C2#2375793612yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#393×C157C2
prime radicalC22×C22⋊D157C22
hypercenter#393D157
hypocenterC157#2409
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C22×Q8×D157C2semidirect
2. C2×#19097748C2semidirect
4. #23757194C2semidirect
5. #23757192C2semidirect
6. #23757185C2semidirect
7. #23757177C2semidirect
9. #23757147C2semidirect
10. #23757132C2semidirect
11. #393×C157C2semidirect
44. C2×Dic314D4semidirect
48. C2×C22⋊D157Q8semidirect
99. C2×Dic157C2×Q8semidirect
106. C23×C157C2×Q8semidirect
133. C2×D157C22⋊Q8semidirect
150. C157#2409semidirect
158. #393D157semidirect
238. C23Q8×D157semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×D1572
C2×#190977482
C2×#190977312
#237571942
#237571922
#237571852
#237571772
#237571492
#237571472
#237571322
#393×C1572
#2409157