Finite Group Databasebeta

← #34110612

Subgroups — #34110612

Order 20096.

564 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19097804C211yesmaximal, characteristic, complemented
2C2×#19097799C211yesmaximal, characteristic, complemented
3C2×#19097793C211yesmaximal, characteristic, complemented
4C22×#14578943C211yesmaximal, characteristic, complemented
5#23757849C211yesmaximal, characteristic, complemented
6#23757842C211yesmaximal, characteristic, complemented
7#23757667C212yesmaximal, complemented
8#23757662C212yesmaximal, complemented
9#23757657C212yesmaximal, complemented
10#23757646C212yesmaximal, complemented
11#405×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C2×#14578954C2211yescharacteristic
14C22×Dic314C2211yescharacteristic
15C22×D628C2211yescharacteristic
16#19097804—24no—
17#19097799—24no—
18#19097793—24no—
19C2×Dic628C2212yescomplemented
20C2×Dic628C2211yescharacteristic, complemented
21C2×#14578943—24no—
22C2×#14578943C2212yes—
23C2×#14578943C2212yes—
24C2×#14578943C2212yes—
25C2×#14578943C2211yescharacteristic, complemented
26C2×#14578943C2212yescomplemented
27C2×#14578943C2211yescharacteristic, complemented
28C2×D1256C2212yes—
29C2×D1256C2211yescharacteristic, complemented
30#19097771C2212yescomplemented
31#19097768C2212yes—
32#19097759C2212yes—
33#19097759C2212yescomplemented
34#19097755—22no—
35#19097751—22no—
36#19097746—22no—
37#19097741—22no—
38C2×M16×C157C2211yescharacteristic
39C22×C8×C157C2211yescharacteristic
40#117×C157C2211yescharacteristic
41#97×C157C2212yescomplemented
42C4×C8×C157C2212yescomplemented
43C23×D157—21no—
44#14578954—22no—
45#14578954—22no—
46#14578954—22no—
47#14578954—22no—
48C2×Dic314—21no—
49C2×Dic314—22no—
50C2×Dic314—22no—
51C2×Dic314C2311yescharacteristic
52C2×Dic314C2312yes—
53C2×Dic314C2311yescharacteristic
54C2×D628—22no—
55C2×D628—22no—
56C2×D628—21no—
57C2×D628—24no—
58C2×D628C2312yes—
59C2×D628C2311yescharacteristic
60C2×D628C2311yescharacteristic
61C2×C22⋊D157—21no—
62C2×C22⋊D157—21no—
63C22×Dic157—21no—
64C2×C4×D157—21no—
65C2×C4×D157—21no—
66Dic628—22no—
67Dic628—24no—
68#14578943—24no—
69#14578943—24no—
70#14578943—24no—
71#14578943—22no—
72#14578943—22no—
73#14578943—24no—
74D1256—24no—
75D1256—22no—
76C4⋊Dic157—24no—
77#14578931—24no—
78#14578931—24no—
79C22×C4×C157C2311yescharacteristic
80M16×C157D414yescomplemented
81C2×C8×C157D414yescomplemented
82C2×C8×C157C2311yescharacteristic
83C2×C8×C157C2312yes—
84C2×C8×C157C2311yescharacteristic
85C4⋊C4×C157D412yes—
86C22⋊C4×C157D412yes—
87C42×C157C2312yes—
88C22×D157—21no—
89C22×D157—21no—
90C22×D157—22no—
91C22×D157—42no—
92C22×D157—82no—
93Dic314—21no—
94Dic314—22no—
95Dic314—21no—
96Dic314—21no—
97Dic314—22no—
98Dic314—21no—
99Dic314—42no—
100D628—22no—
101D628—22no—
102D628—21no—
103D628—21no—
104D628—21no—
105D628—21no—
106D628—42no—
107D628—84no—
108C22⋊D157—42no—
109C22⋊D157—42no—
110C2×Dic157—21no—
111C2×Dic157—21no—
112C2×Dic157—22no—
113C2×Dic157—42no—
114C4×D157—42no—
115C4×D157—42no—
116C23×C157C2×D411yescharacteristic
117C2×C4×C157C2×D414yes—
118C2×C4×C157C2×D411yescharacteristic
119C2×C4×C157C2×D412yes—
120C2×C4×C157C2×D412yes—
121C2×C4×C157C2411yescharacteristic
122C8×C157C2×D412yes—
123C8×C157C2×D414yes—
124C8×C157C2×D412yes—
125C22×C157—22no—
126C22×C157#12712yes—
127C22×C157C22×D411yescharacteristic
128C2×D157—41no—
129C2×D157—41no—
130C2×D157—42no—
131C2×D157—42no—
132C2×D157—44no—
133C2×D157—42no—
134C2×D157—84no—
135C4×C157—24no—
136C4×C157#12712yes—
137C4×C157C22×D411yescharacteristic
138C4×C157C22×D411yescharacteristic
139Dic157—41no—
140Dic157—41no—
141Dic157—42no—
142C2×C157—22no—
143C2×C157C2×#12711yescharacteristic
144C2×C157#57712yes—
145D157—81no—
146D157—81no—
147D157—82no—
148C157#315411yescharacteristic, complemented, Sylow
149#3154—1571nomaximal, Sylow
150C2×#137—1571no—
151C2×AGL(1,ℤ/8)—1571no—
152C2×#135—1571no—
153C22×SD16—1571no—
154#536—1571no—
155#535—1571no—
156#497—1572no—
157#496—1572no—
158#495—1572no—
159#492—1572no—
160#405D15711yescharacteristic, complemented
161C2×#41—1571no—
162C2×#41—1571no—
163C22×Q8—1571no—
164C22×D4—1571no—
165#137—3144no—
166AGL(1,ℤ/8)—3144no—
167#135—3144no—
168C2×Q16—1571no—
169C2×Q16—1572no—
170C2×SD16—3144no—
171C2×SD16—1572no—
172C2×SD16—1572no—
173C2×SD16—1571no—
174C2×SD16—1571no—
175C2×SD16—1572no—
176C2×SD16—1572no—
177C2×D8—1571no—
178C2×D8—1572no—
179C2×M16C2×D15711yescharacteristic
180C22×C8C2×D15711yescharacteristic
181C4⋊Q8—1572no—
182#127—1572no—
183#124—1572no—
184#124—1572no—
185#123—3142no—
186C22⋊Q8—3142no—
187#121—3142no—
188#120—3142no—
189#117C2×D15711yescharacteristic
190#97C2×D15712yescomplemented
191C4×C8C2×D15712yescomplemented
192C24—3141no—
193#41—3142no—
194#41—3142no—
195#41—3142no—
196#41—3142no—
197C2×Q8—3141no—
198C2×Q8—3142no—
199C2×Q8—3142no—
200C2×Q8—1571no—
201C2×Q8—1572no—
202C2×Q8—1571no—
203C2×D4—3141no—
204C2×D4—3141no—
205C2×D4—3142no—
206C2×D4—3142no—
207C2×D4—3141no—
208C2×D4—3144no—
209C2×D4—1571no—
210C2×D4—1572no—
211C2×D4—1571no—
212C22×C4—3141no—
213C22×C4—3141no—
214C22×C4—3141no—
215C22×C4C22×D15711yescharacteristic
216Q16—3142no—
217Q16—3144no—
218SD16—3144no—
219SD16—3144no—
220SD16—3142no—
221SD16—3142no—
222SD16—3144no—
223SD16—3144no—
224D8—3142no—
225D8—3144no—
226M16D62814yescomplemented
227C2×C8D62814yescomplemented
228C2×C8C22×D15711yescharacteristic
229C2×C8C22×D15712yes—
230C2×C8C22×D15711yescharacteristic
231C4⋊C4—3144no—
232C4⋊C4D62812yes—
233C22⋊C4—3144no—
234C22⋊C4—3144no—
235C22⋊C4D62812yes—
236C42C22×D15712yes—
237C23—3141no—
238C23—3142no—
239C23—3141no—
240C23—6282no—
241C23—12562no—
242C23C2×D62811yescharacteristic
243Q8—3142no—
244Q8—3141no—
245Q8—3142no—
246Q8—3141no—
247Q8—3141no—
248Q8—3141no—
249Q8—6282no—
250D4—3141no—
251D4—3142no—
252D4—3141no—
253D4—3141no—
254D4—3142no—
255D4—3141no—
256D4—6282no—
257D4—6282no—
258D4—6282no—
259D4—12564no—
260C2×C4—3141no—
261C2×C4—3142no—
262C2×C4—3141no—
263C2×C4—6282no—
264C2×C4—6282no—
265C2×C4—6282no—
266C2×C4C2×D62814yes—
267C2×C4C2×D62811yescharacteristic
268C2×C4C2×D62812yes—
269C2×C4C2×D62812yes—
270C2×C4C23×D15711yescharacteristic
271C8C2×D62812yes—
272C8C2×D62812yes—
273C8C2×D62814yes—
274C22—22no—
275C22—6282no—
276C22—6284no—
277C22—6282no—
278C22—6281no—
279C22—6281no—
280C22—6282no—
281C22—12564no—
282C22#1909776812yes—
283C22C22×D62811yescharacteristic
284C4—24no—
285C4—6281no—
286C4—6281no—
287C4—6282no—
288C4#1909776812yes—
289C4C22×D62811yescharacteristic
290C4C22×D62811yescharacteristic
291C2—22no—
292C2—12562no—
293C2—12561no—
294C2—12561no—
295C2C2×#1909776811yescharacteristic
296C2#2375808512yes—
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#405×C157C2
socleC22×C157C22×D4
hypercenter#405D157
hypocenterC157#3154
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ #117 ◁ #405

Complements

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

NH ≅ G/Ntype
1. C2×#19097804C2semidirect
2. C2×#19097799C2semidirect
3. C2×#19097793C2semidirect
4. C22×#14578943C2semidirect
5. #23757849C2semidirect
6. #23757842C2semidirect
7. #23757667C2semidirect
8. #23757662C2semidirect
9. #23757657C2semidirect
10. #23757646C2semidirect
11. #405×C157C2semidirect
19. C2×Dic628C22semidirect
20. C2×Dic628C22semidirect
25. C2×#14578943C22semidirect
26. C2×#14578943C22semidirect
27. C2×#14578943C22semidirect
29. C2×D1256C22semidirect
30. #19097771C22semidirect
33. #19097759C22semidirect
41. #97×C157C22semidirect
42. C4×C8×C157C22semidirect
80. M16×C157D4semidirect
81. C2×C8×C157D4semidirect
148. C157#3154semidirect
160. #405D157semidirect
190. #97C2×D157semidirect
191. C4×C8C2×D157semidirect
226. M16D628semidirect
227. C2×C8D628semidirect

Maximal subgroups

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

subgroupindex
C2×#190978042
C2×#190977992
C2×#190977932
C22×#145789432
#237578492
#237578422
#237576672
#237576622
#237576572
#237576462
#405×C1572
#3154157