Finite Group Databasebeta

← #34110585

Subgroups — #34110585

Order 20096.

656 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 141 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×#19097798C212yesmaximal, complemented
2C2×#19097795C212yesmaximal, complemented
3#23757822C211yesmaximal, characteristic, complemented
4#127×D157C211yesmaximal, characteristic, complemented
5#23757663C211yesmaximal, characteristic, complemented
6#23757662C211yesmaximal, characteristic, complemented
7#23757660C211yesmaximal, characteristic, complemented
8#23757648C212yesmaximal, complemented
9#23757644C212yesmaximal, complemented
10#23757225C211yesmaximal, characteristic, complemented
11#496×C157C211yesmaximal, characteristic, complemented
12C2×#14578955C2211yescharacteristic
13C2×#14578953C2211yescharacteristic
14C2×D4×D157—22no—
15C2×D4×D157C2212yes—
16#19097798—28no—
17#19097795—28no—
18C2×#14578943C2212yescomplemented
19C2×#14578942C2212yescomplemented
20C2×#14578941C2212yescomplemented
21C2×D1256C2212yescomplemented
22C2×#14578939C2212yescomplemented
23C2×#14578939C2212yescomplemented
24C2×#14578937C2212yes—
25#19097769C2211yescharacteristic, complemented
26#19097768C2211yescharacteristic
27#19097767—22no—
28#19097767C2212yes—
29#19097767C2211yescharacteristic, complemented
30#19097759C2211yescharacteristic, complemented
31#19097744—22no—
32#19097731—22no—
33C42×D157C2211yescharacteristic
34#19097682C2211yescharacteristic, complemented
35C8×Dic157C2212yescomplemented
36C2×SD16×C157C2212yescomplemented
37C2×D8×C157C2212yescomplemented
38#127×C157C2211yescharacteristic
39#124×C157C2211yescharacteristic, complemented
40#97×C157C2211yescharacteristic, complemented
41C23×D157—22no—
42#14578955—24no—
43#14578953—24no—
44C2×Dic314C2311yescharacteristic
45C2×D628—21no—
46C2×D628—22no—
47C2×D628C2311yescharacteristic
48C2×D628C2312yes—
49C2×C22⋊D157—21no—
50C2×C22⋊D157—22no—
51C2×C22⋊D157—22no—
52C2×C22⋊D157—24no—
53D4×D157—24no—
54D4×D157—24no—
55D4×D157—48no—
56C22×Dic157—21no—
57C2×C4×D157—21no—
58C2×C4×D157D412yes—
59C2×C4×D157C2311yescharacteristic
60#14578943—24no—
61#14578942—24no—
62#14578941—24no—
63D1256—24no—
64#14578939—24no—
65#14578939—24no—
66#14578937D418yescomplemented
67C2×#11929434C2312yes—
68#14578933—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—21no—
82C22×D157—22no—
83C22×D157—48no—
84C22×D157—44no—
85C22×D157C2×D411yescharacteristic
86Dic314—22no—
87D628—22no—
88D628—22no—
89D628—22no—
90D628—44no—
91D628—42no—
92C22⋊D157—42no—
93C22⋊D157—44no—
94C22⋊D157—48no—
95C22⋊D157—44no—
96C2×Dic157—21no—
97C2×Dic157—42no—
98C2×Dic157C2×D411yescharacteristic
99C2×Dic157C2×D412yes—
100C4×D157—24no—
101C4×D157—42no—
102C4×D157C2×D414yes—
103#11929434C2×D414yescomplemented
104C23×C157—21no—
105C23×C157—22no—
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—22no—
121C2×D157—41no—
122C2×D157—42no—
123C2×D157—44no—
124C2×D157—42no—
125C2×D157—88no—
126C2×D157#12712yes—
127C4×C157—22no—
128C4×C157—41no—
129C4×C157C22×D412yes—
130Dic157—41no—
131Dic157#12712yes—
132Dic157AGL(1,ℤ/8)14yescomplemented
133C2×C157—82no—
134C2×C157—81no—
135C2×C157C2×#12711yescharacteristic
136C2×C157C2×AGL(1,ℤ/8)12yes—
137D157—22no—
138D157—81no—
139D157—82no—
140C157#315111yescharacteristic, complemented, Sylow
141#3151—1571nomaximal, Sylow
142C2×AGL(1,ℤ/8)—1572no—
143C2×AGL(1,ℤ/8)—1572no—
144#532—1571no—
145C2×#127—1571no—
146#496—1571no—
147#496—1571no—
148#496—1571no—
149#496D15711yescharacteristic, complemented
150#493—1572no—
151#492—1572no—
152C4×M16—1571no—
153C2×#41—1571no—
154C2×#41—1571no—
155C22×D4—3142no—
156C22×D4—1572no—
157AGL(1,ℤ/8)—3148no—
158AGL(1,ℤ/8)—3148no—
159C2×SD16—1572no—
160C2×SD16—1572no—
161C2×SD16—1572no—
162C2×SD16C2×D15712yescomplemented
163C2×D8—1572no—
164C2×D8—1572no—
165C2×D8—1572no—
166C2×D8C2×D15712yescomplemented
167C2×M16—1572no—
168C4⋊Q8—1571no—
169#127—3142no—
170#127—1572no—
171#127—1571no—
172#127—1571no—
173#127C2×D15711yescharacteristic
174#124—1571no—
175#124C2×D15711yescharacteristic, complemented
176#121—3142no—
177C4×D4—3142no—
178C2×C42—1571no—
179#97—1571no—
180#97C2×D15711yescharacteristic, complemented
181C4×C8—1572no—
182C24—3142no—
183#41—3144no—
184#41—3144no—
185C2×Q8—1571no—
186C2×Q8C22×D15711yescharacteristic
187C2×D4—22no—
188C2×D4—3141no—
189C2×D4—3141no—
190C2×D4—3144no—
191C2×D4—3144no—
192C2×D4—3142no—
193C2×D4—3142no—
194C2×D4—3144no—
195C2×D4—3142no—
196C2×D4—6288no—
197C2×D4—1571no—
198C2×D4—1572no—
199C2×D4C22×D15711yescharacteristic
200C2×D4C22×D15712yes—
201C22×C4—3141no—
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—
213M16—1578no—
214C2×C8—1572no—
215C2×C8C22×D15712yes—
216C4⋊C4—3142no—
217C22⋊C4—22no—
218C22⋊C4—3142no—
219C42—1572no—
220C42—1571no—
221C42C22×D15711yescharacteristic
222C23—21no—
223C23—22no—
224C23—3141no—
225C23—3142no—
226C23—6288no—
227C23—6284no—
228C23—1571no—
229Q8—22no—
230Q8—3142no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—3142no—
236D4—3142no—
237D4—3142no—
238D4—6284no—
239D4—6282no—
240D4—6282no—
241D4—6284no—
242D4—6288no—
243D4—6284no—
244C2×C4—21no—
245C2×C4—3144no—
246C2×C4—3141no—
247C2×C4—6282no—
248C2×C4—6282no—
249C2×C4—1572no—
250C2×C4—1571no—
251C2×C4—1574no—
252C2×C4D4×D15712yes—
253C2×C4C23×D15711yescharacteristic
254C8—1574no—
255C8D4×D15714yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—3142no—
261C22—6284no—
262C22—6282no—
263C22—6281no—
264C22—6282no—
265C22—1572no—
266C22—12568no—
267C22C2×D4×D15711yescharacteristic
268C4—22no—
269C4—41no—
270C4—6281no—
271C4—1574no—
272C4—1572no—
273C4C2×D4×D15712yes—
274C2—82no—
275C2—81no—
276C2—3142no—
277C2—12562no—
278C2—12561no—
279C2#127×D15711yescharacteristic
280C2AGL(1,ℤ/8)×D15712yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#496×C157C2
socleC22×C157C22×D4
hypercenter#496D157
hypocenterC157#3151
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×#19097798C2semidirect
2. C2×#19097795C2semidirect
3. #23757822C2semidirect
4. #127×D157C2semidirect
5. #23757663C2semidirect
6. #23757662C2semidirect
7. #23757660C2semidirect
8. #23757648C2semidirect
9. #23757644C2semidirect
10. #23757225C2semidirect
11. #496×C157C2semidirect
18. C2×#14578943C22semidirect
19. C2×#14578942C22semidirect
20. C2×#14578941C22semidirect
21. C2×D1256C22semidirect
22. C2×#14578939C22semidirect
23. C2×#14578939C22semidirect
25. #19097769C22semidirect
29. #19097767C22semidirect
30. #19097759C22semidirect
34. #19097682C22semidirect
35. C8×Dic157C22semidirect
36. C2×SD16×C157C22semidirect
37. C2×D8×C157C22semidirect
39. #124×C157C22semidirect
40. #97×C157C22semidirect
66. #14578937D4semidirect
103. #11929434C2×D4semidirect
114. C8×C157C2×D4semidirect
132. Dic157AGL(1,ℤ/8)semidirect
140. C157#3151semidirect
149. #496D157semidirect
162. C2×SD16C2×D157semidirect
166. C2×D8C2×D157semidirect
175. #124C2×D157semidirect
180. #97C2×D157semidirect
255. C8D4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
C2×#190977952
#237578222
#127×D1572
#237576632
#237576622
#237576602
#237576482
#237576442
#237572252
#496×C1572
#3151157