Finite Group Databasebeta

← #496×D157 #34110566

Subgroups — #496×D157 #34110566

Order 20096.

668 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 668 conjugacy classes of subgroups — too many to draw. Only its 145 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×SD16×D157C212yesmaximal, complemented
2C2×D8×D157C212yesmaximal, complemented
3#127×D157C211yesmaximal, characteristic, complemented
4#124×D157C211yesmaximal, characteristic, complemented
5#23757663C211yesmaximal, characteristic, complemented
6#23757662C211yesmaximal, characteristic, complemented
7#23757661C212yesmaximal, complemented
8#23757660C211yesmaximal, characteristic, complemented
9#23757659C212yesmaximal, complemented
10#97×D157C211yesmaximal, characteristic, complemented
11#496×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157C2211yescharacteristic
13C2×D4×D157—22no—
14C2×D4×D157C2211yescharacteristic
15C2×D4×D157C2212yes—
16C2×#14578943C2212yescomplemented
17C2×#14578942C2212yescomplemented
18C2×#14578941C2212yescomplemented
19SD16×D157—28no—
20C2×D1256C2212yescomplemented
21C2×#14578939C2212yescomplemented
22C2×#14578939C2212yescomplemented
23D8×D157—28no—
24C2×C8×D157C2212yes—
25#19097768C2211yescharacteristic, complemented
26#19097767—22no—
27#19097767C2212yescomplemented
28#19097760C2211yescharacteristic, complemented
29#19097759C2211yescharacteristic, complemented
30#19097758C2211yescharacteristic, complemented
31#19097757—22no—
32C22⋊C4×D157—22no—
33C42×D157C2211yescharacteristic
34#19097682C2211yescharacteristic, complemented
35#19097681C2212yescomplemented
36C2×SD16×C157C2212yescomplemented
37C2×D8×C157C2212yescomplemented
38#127×C157C2211yescharacteristic, complemented
39#124×C157C2211yescharacteristic, complemented
40#97×C157C2211yescharacteristic, complemented
41C23×D157—21no—
42C23×D157—22no—
43C2×Dic314—21no—
44C2×Dic314C2311yescharacteristic
45Q8×D157—24no—
46C2×D628—22no—
47C2×D628C2311yescharacteristic
48C2×D628C2312yes—
49C2×C22⋊D157—21no—
50C2×C22⋊D157—22no—
51C2×C22⋊D157—24no—
52D4×D157—24no—
53D4×D157—24no—
54D4×D157—24no—
55D4×D157—48no—
56C2×C4×D157—21no—
57C2×C4×D157D412yes—
58C2×C4×D157C2311yescharacteristic
59#14578943—24no—
60#14578942—24no—
61#14578941—24no—
62D1256—24no—
63#14578939—24no—
64#14578939—24no—
65C2×#11929434C2312yes—
66C8×D157D418yescomplemented
67C22⋊Dic157—22no—
68#14578931—22no—
69#14578931—22no—
70C4×Dic157D412yes—
71C4×Dic157C2311yescharacteristic
72C2×Q8×C157C2311yescharacteristic
73C2×D4×C157—22no—
74C2×D4×C157C2311yescharacteristic
75C2×D4×C157C2312yes—
76SD16×C157—24no—
77D8×C157—24no—
78C2×C8×C157C2312yes—
79C22⋊C4×C157—22no—
80C42×C157C2311yescharacteristic
81C22×D157—22no—
82C22×D157—21no—
83C22×D157—42no—
84C22×D157—44no—
85C22×D157—48no—
86C22×D157—44no—
87C22×D157C2×D411yescharacteristic
88Dic314—22no—
89Dic314—42no—
90D628—22no—
91D628—22no—
92D628—22no—
93D628—44no—
94C22⋊D157—48no—
95C22⋊D157—42no—
96C22⋊D157—44no—
97C2×Dic157—21no—
98C2×Dic157C2×D412yes—
99C2×Dic157C2×D411yescharacteristic
100C4×D157—24no—
101C4×D157—42no—
102C4×D157C2×D414yes—
103#11929434C2×D414yescomplemented
104C23×C157—22no—
105C23×C157—21no—
106Q8×C157—22no—
107D4×C157—22no—
108D4×C157—22no—
109D4×C157—22no—
110D4×C157—44no—
111C2×C4×C157—21no—
112C2×C4×C157C2×D412yes—
113C2×C4×C157C2411yescharacteristic
114C8×C157C2×D414yescomplemented
115C22×C157—41no—
116C22×C157—42no—
117C22×C157—44no—
118C22×C157—42no—
119C22×C157C22×D411yescharacteristic
120C2×D157—41no—
121C2×D157—42no—
122C2×D157—44no—
123C2×D157—42no—
124C2×D157—88no—
125C2×D157—84no—
126C2×D157#12712yes—
127C2×D157AGL(1,ℤ/8)14yes—
128C4×C157—22no—
129C4×C157—41no—
130C4×C157C22×D412yes—
131Dic157—22no—
132Dic157—41no—
133Dic157#12712yes—
134C2×C157—82no—
135C2×C157—81no—
136C2×C157C2×#12711yescharacteristic
137C2×C157C2×AGL(1,ℤ/8)12yes—
138D157—82no—
139D157—81no—
140D157#49614yesdirect product factor
141C157C2×#49611yescharacteristic, complemented, Sylow
142C2×#496—1571nomaximal, Sylow
143C22×SD16—1572no—
144C22×D8—1572no—
145C2×#127—1571no—
146C2×#124—1571no—
147#496—1572no—
148#496—1572no—
149#496—1571no—
150#496—1571no—
151#496—1571no—
152#496D15711yescharacteristic, direct product factor
153C2×#97—1571no—
154C22×Q8—1571no—
155C22×D4—3142no—
156C22×D4—1571no—
157C22×D4—1572no—
158C2×SD16—3148no—
159C2×SD16—1572no—
160C2×SD16—1572no—
161C2×SD16—1572no—
162C2×SD16C2×D15712yescomplemented
163C2×D8—3148no—
164C2×D8—1572no—
165C2×D8—1572no—
166C2×D8—1572no—
167C2×D8C2×D15712yescomplemented
168C22×C8—1572no—
169#127—3142no—
170#127—1572no—
171#127—1571no—
172#127C2×D15711yescharacteristic, complemented
173#124—3142no—
174#124—1571no—
175#124—1571no—
176#124—1571no—
177#124C2×D15711yescharacteristic, complemented
178C2×C22⋊C4—3142no—
179C2×C42—1571no—
180#97—1572no—
181#97—1571no—
182#97C2×D15711yescharacteristic, complemented
183C24—3141no—
184C24—3142no—
185C2×Q8—3141no—
186C2×Q8—3144no—
187C2×Q8—1571no—
188C2×Q8C22×D15711yescharacteristic
189C2×D4—22no—
190C2×D4—3144no—
191C2×D4—3144no—
192C2×D4—3144no—
193C2×D4—3141no—
194C2×D4—3142no—
195C2×D4—3144no—
196C2×D4—3142no—
197C2×D4—6288no—
198C2×D4—1571no—
199C2×D4—1572no—
200C2×D4C22×D15711yescharacteristic
201C2×D4C22×D15712yes—
202C22×C4—3141no—
203C22×C4—1572no—
204C22×C4—1571no—
205SD16—24no—
206SD16—3144no—
207SD16—3144no—
208SD16—3144no—
209D8—24no—
210D8—3144no—
211D8—3144no—
212D8—3144no—
213C2×C8—1572no—
214C2×C8—1578no—
215C2×C8C22×D15712yes—
216C22⋊C4—22no—
217C22⋊C4—3142no—
218C22⋊C4—3142no—
219C22⋊C4—3142no—
220C42—1572no—
221C42—1571no—
222C42C22×D15711yescharacteristic
223C23—21no—
224C23—22no—
225C23—3141no—
226C23—3142no—
227C23—6282no—
228C23—6284no—
229C23—6288no—
230C23—6284no—
231C23—1571no—
232Q8—22no—
233Q8—3142no—
234Q8—6282no—
235D4—22no—
236D4—22no—
237D4—22no—
238D4—44no—
239D4—3142no—
240D4—3142no—
241D4—3142no—
242D4—6284no—
243D4—6288no—
244D4—6282no—
245D4—6284no—
246C2×C4—21no—
247C2×C4—3144no—
248C2×C4—3141no—
249C2×C4—6282no—
250C2×C4—1572no—
251C2×C4—1571no—
252C2×C4—1574no—
253C2×C4D4×D15712yes—
254C2×C4C23×D15711yescharacteristic
255C8—1574no—
256C8D4×D15714yescomplemented
257C22—44no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—6284no—
262C22—6282no—
263C22—6281no—
264C22—6282no—
265C22—1574no—
266C22—1572no—
267C22—12568no—
268C22—12564no—
269C22C2×D4×D15711yescharacteristic
270C4—22no—
271C4—41no—
272C4—3142no—
273C4—6281no—
274C4—1572no—
275C4C2×D4×D15712yes—
276C2—82no—
277C2—81no—
278C2—1574no—
279C2—12562no—
280C2—12561no—
281C2#127×D15711yescharacteristic
282C2AGL(1,ℤ/8)×D15712yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#496×C157C2
socleC22×C157C22×D4
hypercenter#496D157
hypocenterC157C2×#496
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D157C2semidirect
2. C2×D8×D157C2semidirect
3. #127×D157C2semidirect
4. #124×D157C2semidirect
5. #23757663C2semidirect
6. #23757662C2semidirect
7. #23757661C2semidirect
8. #23757660C2semidirect
9. #23757659C2semidirect
10. #97×D157C2semidirect
11. #496×C157C2semidirect
16. C2×#14578943C22semidirect
17. C2×#14578942C22semidirect
18. C2×#14578941C22semidirect
20. C2×D1256C22semidirect
21. C2×#14578939C22semidirect
22. C2×#14578939C22semidirect
25. #19097768C22semidirect
27. #19097767C22semidirect
28. #19097760C22semidirect
29. #19097759C22semidirect
30. #19097758C22semidirect
34. #19097682C22semidirect
35. #19097681C22semidirect
36. C2×SD16×C157C22semidirect
37. C2×D8×C157C22semidirect
38. #127×C157C22semidirect
39. #124×C157C22semidirect
40. #97×C157C22semidirect
66. C8×D157D4semidirect
103. #11929434C2×D4semidirect
114. C8×C157C2×D4semidirect
140. D157#496direct (N × H)
141. C157C2×#496semidirect
152. #496D157direct (N × H)
162. C2×SD16C2×D157semidirect
167. C2×D8C2×D157semidirect
172. #127C2×D157semidirect
177. #124C2×D157semidirect
182. #97C2×D157semidirect
256. C8D4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1572
C2×D8×D1572
#127×D1572
#124×D1572
#237576632
#237576622
#237576612
#237576602
#237576592
#97×D1572
#496×C1572
C2×#496157