Finite Group Databasebeta

← #34110584

Subgroups — #34110584

Order 20096.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#19097799C211yesmaximal, characteristic, complemented
2C2×#19097797C211yesmaximal, characteristic, complemented
3C2×#19097796C211yesmaximal, characteristic, complemented
4#23757819C211yesmaximal, characteristic, complemented
5C2×#19097767C211yesmaximal, characteristic, complemented
6#23757661C212yesmaximal, complemented
7#23757659C212yesmaximal, complemented
8#23757648C212yesmaximal, complemented
9#23757644C212yesmaximal, complemented
10M16×Dic157C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×D628C2211yescharacteristic
14C22×C22⋊D157—22no—
15#19097799—24no—
16#19097797—24no—
17#19097796—24no—
18C2×#14578943C2212yescomplemented
19C2×#14578942C2212yescomplemented
20C2×#14578941C2212yescomplemented
21C2×D1256C2212yescomplemented
22C2×#14578939C2212yescomplemented
23C2×#14578939C2212yescomplemented
24C2×#14578938C2211yescharacteristic
25#19097769C2211yescharacteristic, complemented
26#19097767—22no—
27#19097767C2212yes—
28#19097767C2212yes—
29#19097767C2211yescharacteristic, complemented
30#19097760C2211yescharacteristic, complemented
31#19097758C2211yescharacteristic, complemented
32#19097743—22no—
33C4×C22⋊D157—22no—
34C2×C4×Dic157C2211yescharacteristic
35#19097681C2212yescomplemented
36C8×Dic157C2212yescomplemented
37C2×#41×C157C2211yescharacteristic
38C22×D4×C157C2211yescharacteristic
39AGL(1,ℤ/8)×C157—24no—
40C2×SD16×C157C2212yescomplemented
41C2×D8×C157C2212yescomplemented
42C2×M16×C157C2211yescharacteristic
43C23×D157—21no—
44#14578954—22no—
45#14578954—22no—
46C2×Dic314C2311yescharacteristic
47C2×D628—22no—
48C2×D628—22no—
49C2×D628—21no—
50C2×D628C2311yescharacteristic
51C2×D628C2312yes—
52C2×C22⋊D157—22no—
53C2×C22⋊D157—24no—
54C2×C22⋊D157—24no—
55C2×C22⋊D157—21no—
56C2×C22⋊D157—48no—
57C22×Dic157D412yes—
58C2×C4×D157—21no—
59#14578943—24no—
60#14578942—24no—
61#14578941—24no—
62D1256—24no—
63#14578939—24no—
64#14578939—24no—
65#14578938D414yescomplemented
66C2×#11929434C2312yes—
67#14578933—22no—
68C22⋊Dic157—22no—
69#14578931—22no—
70C4×Dic157D412yes—
71C4×Dic157C2311yescharacteristic
72C4×Dic157C2311yescharacteristic
73C24×C157—21no—
74#41×C157—22no—
75#41×C157—22no—
76C2×Q8×C157C2311yescharacteristic
77C2×D4×C157—21no—
78C2×D4×C157—22no—
79C2×D4×C157—22no—
80C2×D4×C157—21no—
81C2×D4×C157C2311yescharacteristic
82C2×D4×C157C2312yes—
83C22×C4×C157—21no—
84C22×C4×C157C2311yescharacteristic
85SD16×C157—24no—
86D8×C157—24no—
87M16×C157D414yescomplemented
88C2×C8×C157C2312yes—
89C22×D157—21no—
90C22×D157—22no—
91C22×D157—44no—
92C22×D157—42no—
93Dic314—21no—
94Dic314—21no—
95D628—22no—
96D628—22no—
97D628—21no—
98D628—21no—
99D628—42no—
100C22⋊D157—48no—
101C22⋊D157—42no—
102C22⋊D157—44no—
103C22⋊D157—48no—
104C2×Dic157—24no—
105C2×Dic157—21no—
106C2×Dic157C2×D412yes—
107C2×Dic157C2×D412yes—
108C4×D157—42no—
109#11929434C2×D414yescomplemented
110C23×C157—22no—
111C23×C157—21no—
112C23×C157—44no—
113C23×C157—42no—
114C23×C157C2×D411yescharacteristic
115Q8×C157—21no—
116Q8×C157—21no—
117D4×C157—21no—
118D4×C157—22no—
119D4×C157—22no—
120D4×C157—21no—
121D4×C157—42no—
122D4×C157—42no—
123C2×C4×C157—21no—
124C2×C4×C157—42no—
125C2×C4×C157C2×D411yescharacteristic
126C2×C4×C157C2×D412yes—
127C2×C4×C157C2×D412yes—
128C2×C4×C157C2411yescharacteristic
129C8×C157C2×D414yescomplemented
130C22×C157—22no—
131C22×C157—41no—
132C22×C157—42no—
133C22×C157—44no—
134C22×C157—42no—
135C22×C157—84no—
136C22×C157#12712yes—
137C22×C157C22×D411yescharacteristic
138C2×D157—41no—
139C2×D157—42no—
140C2×D157—44no—
141C2×D157—42no—
142C2×D157—84no—
143C4×C157—41no—
144C4×C157#12712yes—
145C4×C157C22×D411yescharacteristic
146C4×C157C22×D411yescharacteristic
147Dic157—22no—
148Dic157—41no—
149Dic157AGL(1,ℤ/8)14yescomplemented
150C2×C157—22no—
151C2×C157—81no—
152C2×C157—82no—
153C2×C157C2×#12711yescharacteristic
154C2×C157C2×AGL(1,ℤ/8)12yes—
155D157—81no—
156D157—82no—
157C157#315111yescharacteristic, complemented, Sylow
158#3151—1571nomaximal, Sylow
159C2×AGL(1,ℤ/8)—1571no—
160C2×AGL(1,ℤ/8)—1571no—
161C2×AGL(1,ℤ/8)—1571no—
162C2×AGL(1,ℤ/8)D15711yescharacteristic, complemented
163#532—1571no—
164C2×#127—1571no—
165#496—1572no—
166#496—1572no—
167#493—1572no—
168#492—1572no—
169C4×M16—1571no—
170C2×#41—1571no—
171C2×#41C2×D15711yescharacteristic
172C22×D4—3142no—
173C22×D4—1571no—
174C22×D4C2×D15711yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—3144no—
177AGL(1,ℤ/8)—3144no—
178AGL(1,ℤ/8)—3144no—
179C2×SD16—1572no—
180C2×SD16—1572no—
181C2×SD16—1572no—
182C2×SD16C2×D15712yescomplemented
183C2×D8—1572no—
184C2×D8—1572no—
185C2×D8—1572no—
186C2×D8C2×D15712yescomplemented
187C2×M16—1571no—
188C2×M16C2×D15711yescharacteristic
189C4⋊Q8—1571no—
190#127—3142no—
191#127—1572no—
192#127—1572no—
193#127—1571no—
194#124—1571no—
195#124—1571no—
196#121—3142no—
197C4×D4—3142no—
198C2×C42—1571no—
199#97—1572no—
200C4×C8—1572no—
201C24—21no—
202C24—3141no—
203#41—22no—
204#41—22no—
205#41—3142no—
206#41—3142no—
207C2×Q8—1571no—
208C2×Q8C22×D15711yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—3141no—
214C2×D4—3142no—
215C2×D4—3144no—
216C2×D4—3144no—
217C2×D4—3142no—
218C2×D4—3141no—
219C2×D4—3142no—
220C2×D4—6288no—
221C2×D4—1571no—
222C2×D4—1572no—
223C2×D4C22×D15711yescharacteristic
224C2×D4C22×D15712yes—
225C22×C4—21no—
226C22×C4—3141no—
227C22×C4—1572no—
228C22×C4C22×D15711yescharacteristic
229SD16—24no—
230SD16—3144no—
231SD16—3144no—
232SD16—3144no—
233D8—24no—
234D8—3144no—
235D8—3144no—
236D8—3144no—
237M16—1574no—
238M16C22⋊D15714yescomplemented
239C2×C8—1572no—
240C2×C8C22×D15712yes—
241C4⋊C4—3142no—
242C22⋊C4—3142no—
243C22⋊C4—3142no—
244C42—1572no—
245C42—1571no—
246C42—1571no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—3141no—
252C23—3142no—
253C23—6284no—
254C23—6282no—
255C23D4×D15711yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—3141no—
259Q8—3141no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—3141no—
267D4—3142no—
268D4—3141no—
269D4—3142no—
270D4—6288no—
271D4—6282no—
272D4—6288no—
273D4—6284no—
274D4—6282no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—3144no—
278C2×C4—3141no—
279C2×C4—6282no—
280C2×C4—1572no—
281C2×C4—1572no—
282C2×C4D4×D15711yescharacteristic
283C2×C4D4×D15712yes—
284C2×C4C2×C22⋊D15712yes—
285C2×C4C23×D15711yescharacteristic
286C8—1574no—
287C8C2×C22⋊D15714yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—6284no—
295C22—6282no—
296C22—6281no—
297C22—6282no—
298C22—12564no—
299C22#1909776712yes—
300C22C2×D4×D15711yescharacteristic
301C4—41no—
302C4—3142no—
303C4—6281no—
304C4—1574no—
305C4#1909776712yes—
306C4C2×D4×D15711yescharacteristic
307C4C22×C22⋊D15711yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—12562no—
312C2—12561no—
313C2C2×#1909776711yescharacteristic
314C2AGL(1,ℤ/8)×D15712yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×AGL(1,ℤ/8)×C157C2
socleC22×C157C22×D4
hypercenterC2×AGL(1,ℤ/8)D157
hypocenterC157#3151
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#19097799C2semidirect
2. C2×#19097797C2semidirect
3. C2×#19097796C2semidirect
4. #23757819C2semidirect
5. C2×#19097767C2semidirect
6. #23757661C2semidirect
7. #23757659C2semidirect
8. #23757648C2semidirect
9. #23757644C2semidirect
10. M16×Dic157C2semidirect
11. C2×AGL(1,ℤ/8)×C157C2semidirect
18. C2×#14578943C22semidirect
19. C2×#14578942C22semidirect
20. C2×#14578941C22semidirect
21. C2×D1256C22semidirect
22. C2×#14578939C22semidirect
23. C2×#14578939C22semidirect
25. #19097769C22semidirect
29. #19097767C22semidirect
30. #19097760C22semidirect
31. #19097758C22semidirect
35. #19097681C22semidirect
36. C8×Dic157C22semidirect
40. C2×SD16×C157C22semidirect
41. C2×D8×C157C22semidirect
65. #14578938D4semidirect
87. M16×C157D4semidirect
109. #11929434C2×D4semidirect
129. C8×C157C2×D4semidirect
149. Dic157AGL(1,ℤ/8)semidirect
157. C157#3151semidirect
162. C2×AGL(1,ℤ/8)D157semidirect
182. C2×SD16C2×D157semidirect
186. C2×D8C2×D157semidirect
238. M16C22⋊D157semidirect
287. C8C2×C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977992
C2×#190977972
C2×#190977962
#237578192
C2×#190977672
#237576612
#237576592
#237576482
#237576442
M16×Dic1572
C2×AGL(1,ℤ/8)×C1572
#3151157