Finite Group Databasebeta

← #34110618

Subgroups — #34110618

Order 20096.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19097794C211yesmaximal, characteristic, complemented
3C2×D8×D157C211yesmaximal, characteristic, complemented
4#23757841C212yesmaximal, complemented
5#23757663C212yesmaximal, complemented
6#23757659C212yesmaximal, complemented
7#23757658C211yesmaximal, characteristic, complemented
8#23757650C211yesmaximal, characteristic, complemented
9#23757648C211yesmaximal, characteristic, complemented
10#23757227C211yesmaximal, characteristic, complemented
11#495×C157C211yesmaximal, characteristic, complemented
12C2×#14578955C2212yes—
13C2×D4×D157C2212yes—
14#19097798—28no—
15#19097794—24no—
16C2×#14578941C2212yescomplemented
17C2×#14578941C2212yescomplemented
18C2×D1256C2211yescharacteristic
19C2×D1256C2211yescharacteristic, complemented
20C2×D1256C2212yescomplemented
21C2×#14578939C2212yes—
22C2×#14578939C2212yescomplemented
23D8×D157—24no—
24C2×#14578937C2211yescharacteristic
25C2×C8×D157C2211yescharacteristic
26#19097768C2212yescomplemented
27#19097767C2212yescomplemented
28#19097760C2212yescomplemented
29#19097744—24no—
30#19097741—24no—
31#19097728C2211yescharacteristic
32#19097682C2211yescharacteristic, complemented
33#19097681C2211yescharacteristic, complemented
34C8×Dic157C2211yescharacteristic, complemented
35C2×Q16×C157C2211yescharacteristic, complemented
36C2×SD16×C157C2212yescomplemented
37C2×D8×C157C2211yescharacteristic
38#124×C157C2212yescomplemented
39C4×C8×C157C2211yescharacteristic, complemented
40C23×D157—22no—
41#14578955—24no—
42#14578955—24no—
43C2×D628—22no—
44C2×D628—24no—
45C2×D628C2312yes—
46C2×D628C2312yes—
47C2×C22⋊D157—22no—
48C2×C22⋊D157—24no—
49D4×D157—24no—
50D4×D157—24no—
51C2×C4×D157—22no—
52C2×C4×D157C2311yescharacteristic
53#14578941—24no—
54#14578941—24no—
55D1256—22no—
56D1256—22no—
57D1256—24no—
58#14578939—24no—
59#14578939—24no—
60#14578937D414yescomplemented
61C2×#11929434C2311yescharacteristic
62C2×#11929434C2311yescharacteristic
63C8×D157D414yescomplemented
64#14578933D412yes—
65#14578931—24no—
66#14578931D412yes—
67C4×Dic157C2311yescharacteristic
68C2×Q8×C157C2312yes—
69C2×D4×C157C2312yes—
70Q16×C157—22no—
71SD16×C157—24no—
72D8×C157—22no—
73C2×C8×C157C2311yescharacteristic
74C2×C8×C157C2311yescharacteristic
75C22⋊C4×C157—24no—
76C42×C157C2311yescharacteristic
77C22×D157—22no—
78C22×D157—22no—
79C22×D157—44no—
80C22×D157—84no—
81C22×D157C2×D411yescharacteristic
82D628—22no—
83D628—22no—
84D628—22no—
85D628—22no—
86D628—44no—
87D628—84no—
88C22⋊D157—44no—
89C22⋊D157—84no—
90C2×Dic157C2×D412yes—
91C2×Dic157C2×D411yescharacteristic
92C4×D157—44no—
93C4×D157C2×D412yes—
94C4×D157C2×D412yes—
95#11929434C2×D412yes—
96#11929434C2×D412yes—
97C23×C157—22no—
98Q8×C157—22no—
99Q8×C157—22no—
100D4×C157—22no—
101D4×C157—22no—
102C2×C4×C157—22no—
103C2×C4×C157C2×D412yes—
104C2×C4×C157C2411yescharacteristic
105C8×C157C2×D412yes—
106C8×C157C2×D412yes—
107C22×C157—42no—
108C22×C157—44no—
109C22×C157C22×D411yescharacteristic
110C2×D157—22no—
111C2×D157—42no—
112C2×D157—42no—
113C2×D157—44no—
114C2×D157—44no—
115C2×D157—88no—
116C2×D157#12712yes—
117C4×C157—22no—
118C4×C157—42no—
119C4×C157C22×D411yescharacteristic
120C4×C157C22×D411yescharacteristic
121Dic157—22no—
122Dic157#12712yes—
123C2×C157—82no—
124C2×C157C2×#12711yescharacteristic
125C2×C157#57612yes—
126D157—22no—
127D157—82no—
128D157—82no—
129C157#315511yescharacteristic, complemented, Sylow
130#3155—1571nomaximal, Sylow
131C2×AGL(1,ℤ/8)—1572no—
132C2×#135—1571no—
133C22×D8—1571no—
134#535—1572no—
135#496—1572no—
136#496—1572no—
137#495—1571no—
138#495D15711yescharacteristic, complemented
139#493—1571no—
140#493—1571no—
141#405—1571no—
142C2×#41—1572no—
143C22×D4—1572no—
144AGL(1,ℤ/8)—3148no—
145#135—3144no—
146C2×Q16C2×D15711yescharacteristic, complemented
147C2×SD16—1572no—
148C2×SD16—1572no—
149C2×SD16C2×D15712yescomplemented
150C2×D8—3144no—
151C2×D8—1572no—
152C2×D8—1572no—
153C2×D8—1571no—
154C2×D8—1571no—
155C2×D8—1572no—
156C2×D8C2×D15711yescharacteristic
157C2×M16—1571no—
158C22×C8—1571no—
159#127—1572no—
160#127—1572no—
161#124—1572no—
162#124C2×D15712yescomplemented
163#121—3144no—
164#120—3144no—
165#117—1571no—
166#97—1571no—
167#97—1571no—
168C4×C8—1571no—
169C4×C8C2×D15711yescharacteristic, complemented
170C24—3142no—
171#41—3144no—
172#41—3144no—
173C2×Q8C22×D15712yes—
174C2×D4—3142no—
175C2×D4—3144no—
176C2×D4—3144no—
177C2×D4—3142no—
178C2×D4—3144no—
179C2×D4—3144no—
180C2×D4—1572no—
181C2×D4—1572no—
182C2×D4C22×D15712yes—
183C22×C4—3142no—
184C22×C4—1571no—
185Q16—22no—
186SD16—24no—
187SD16—3144no—
188SD16—3144no—
189D8—22no—
190D8—3144no—
191D8—3144no—
192D8—3142no—
193D8—3142no—
194D8—3144no—
195M16—1574no—
196C2×C8—1571no—
197C2×C8—1571no—
198C2×C8—1574no—
199C2×C8C22×D15711yescharacteristic
200C2×C8C22×D15711yescharacteristic
201C4⋊C4—1572no—
202C22⋊C4—24no—
203C22⋊C4—3144no—
204C22⋊C4—1572no—
205C42—1571no—
206C42C22×D15711yescharacteristic
207C23—22no—
208C23—3142no—
209C23—3142no—
210C23—6284no—
211C23—1571no—
212C23—12564no—
213Q8—22no—
214Q8—22no—
215D4—22no—
216D4—22no—
217D4—3142no—
218D4—3142no—
219D4—3142no—
220D4—3142no—
221D4—6284no—
222D4—6284no—
223D4—12564no—
224D4—12564no—
225C2×C4—22no—
226C2×C4—6284no—
227C2×C4—1572no—
228C2×C4—1571no—
229C2×C4—1572no—
230C2×C4—1572no—
231C2×C4D4×D15712yes—
232C2×C4C23×D15711yescharacteristic
233C8—1572no—
234C8—1572no—
235C8D4×D15712yes—
236C8D4×D15712yes—
237C22—42no—
238C22—44no—
239C22—3142no—
240C22—6282no—
241C22—6282no—
242C22—6284no—
243C22—6284no—
244C22—1572no—
245C22—12568no—
246C22C2×D4×D15711yescharacteristic
247C4—22no—
248C4—42no—
249C4—3142no—
250C4—1572no—
251C4C2×D4×D15711yescharacteristic
252C4C2×D4×D15711yescharacteristic
253C2—82no—
254C2—3142no—
255C2—12562no—
256C2—12562no—
257C2#127×D15711yescharacteristic
258C2#2375807512yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#495×C157C2
socleC22×C157C22×D4
hypercenter#495D157
hypocenterC157#3155
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#19097798C2semidirect
2. C2×#19097794C2semidirect
3. C2×D8×D157C2semidirect
4. #23757841C2semidirect
5. #23757663C2semidirect
6. #23757659C2semidirect
7. #23757658C2semidirect
8. #23757650C2semidirect
9. #23757648C2semidirect
10. #23757227C2semidirect
11. #495×C157C2semidirect
16. C2×#14578941C22semidirect
17. C2×#14578941C22semidirect
19. C2×D1256C22semidirect
20. C2×D1256C22semidirect
22. C2×#14578939C22semidirect
26. #19097768C22semidirect
27. #19097767C22semidirect
28. #19097760C22semidirect
32. #19097682C22semidirect
33. #19097681C22semidirect
34. C8×Dic157C22semidirect
35. C2×Q16×C157C22semidirect
36. C2×SD16×C157C22semidirect
38. #124×C157C22semidirect
39. C4×C8×C157C22semidirect
60. #14578937D4semidirect
63. C8×D157D4semidirect
129. C157#3155semidirect
138. #495D157semidirect
146. C2×Q16C2×D157semidirect
149. C2×SD16C2×D157semidirect
162. #124C2×D157semidirect
169. C4×C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977982
C2×#190977942
C2×D8×D1572
#237578412
#237576632
#237576592
#237576582
#237576502
#237576482
#237572272
#495×C1572
#3155157