Finite Group Databasebeta

← #34111821

Subgroups — #34111821

Order 20096.

526 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19097790C211yesmaximal, characteristic, complemented
2#23757917C212yesmaximal, complemented
3#23757656C211yesmaximal, characteristic, complemented
4#23757514C211yesmaximal, characteristic, complemented
5#23757506C211yesmaximal, characteristic, complemented
6#23757486C212yesmaximal, complemented
7#23757424C212yesmaximal, complemented
8#23757415C212yesmaximal, complemented
9D8×Dic157C211yesmaximal, characteristic, complemented
10C8×C22⋊D157C211yesmaximal, characteristic, complemented
11C22×D8×C157C211yesmaximal, characteristic, complemented
12C2×#14578953C2212yescomplemented
13#19097790—24no—
14C2×Dic628C2211yescharacteristic, complemented
15C2×#14578942C2212yescomplemented
16C2×#14578942C2212yescomplemented
17C2×C8×D157C2211yescharacteristic, complemented
18#19097758C2212yescomplemented
19#19097752—22no—
20#19097750C2212yes—
21#19097743—22no—
22#19097742C2212yescomplemented
23#19097740—22no—
24D4×Dic157C2212yescomplemented
25C4×C22⋊D157C2211yescharacteristic
26C2×C22⋊Dic157—22no—
27#19097705C2211yescharacteristic, complemented
28#19097700C2211yescharacteristic, complemented
29#19097696C2212yescomplemented
30#19097692C2212yescomplemented
31#19097692C2212yescomplemented
32C22⋊#11929434C2211yescharacteristic
33#19097683C2211yescharacteristic, complemented
34C8×Dic157C2211yescharacteristic, complemented
35C22×D4×C157C2212yes—
36C2×D8×C157—24no—
37C2×D8×C157C2211yescharacteristic
38C2×D8×C157C2212yes—
39C2×D8×C157C2211yescharacteristic
40C22×C8×C157C2211yescharacteristic
41#14578953—24no—
42#14578953—42no—
43C2×Dic314C2312yes—
44C2×C22⋊D157—22no—
45C2×C22⋊D157—22no—
46C2×C22⋊D157D411yescharacteristic
47C22×Dic157—22no—
48C22×Dic157—22no—
49C2×C4×D157C2311yescharacteristic
50Dic628—22no—
51#14578942—24no—
52#14578942—42no—
53C2×#11929434C2311yescharacteristic
54C2×#11929434C2311yescharacteristic
55C8×D157—22no—
56C4⋊Dic157C2312yes—
57#14578933—22no—
58#14578933D411yescharacteristic
59C22⋊Dic157—22no—
60C22⋊Dic157—22no—
61C22⋊Dic157—22no—
62C22⋊Dic157—22no—
63C22⋊Dic157—22no—
64C22⋊Dic157D411yescharacteristic
65#14578931D411yescharacteristic
66C4×Dic157C2311yescharacteristic
67C24×C157—22no—
68C2×D4×C157—24no—
69C2×D4×C157—22no—
70C2×D4×C157—42no—
71C2×D4×C157C2312yes—
72C2×D4×C157C2312yes—
73C22×C4×C157C2311yescharacteristic
74D8×C157—24no—
75D8×C157—22no—
76D8×C157D414yescomplemented
77C2×C8×C157—22no—
78C2×C8×C157C2311yescharacteristic
79C2×C8×C157C2311yescharacteristic
80C22×D157C2×D411yescharacteristic
81Dic314—22no—
82Dic314—22no—
83C22⋊D157—22no—
84C22⋊D157—44no—
85C22⋊D157—82no—
86C2×Dic157—22no—
87C2×Dic157—22no—
88C2×Dic157—44no—
89C2×Dic157—82no—
90C2×Dic157C2×D411yescharacteristic
91C2×Dic157C2×D411yescharacteristic
92C2×Dic157C2×D411yescharacteristic
93C4×D157—21no—
94C4×D157—21no—
95#11929434—21no—
96#11929434—21no—
97C23×C157—22no—
98C23×C157—22no—
99C23×C157—44no—
100C23×C157—44no—
101C23×C157—82no—
102C23×C157C2×D411yescharacteristic
103D4×C157—22no—
104D4×C157—22no—
105D4×C157—22no—
106D4×C157—44no—
107D4×C157C2×D414yes—
108C2×C4×C157—21no—
109C2×C4×C157—21no—
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2411yescharacteristic
112C8×C157—21no—
113C8×C157C2×D412yes—
114C22×C157—21no—
115C22×C157—21no—
116C22×C157—24no—
117C22×C157—42no—
118C22×C157—42no—
119C22×C157—42no—
120C22×C157—42no—
121C22×C157—42no—
122C22×C157—88no—
123C22×C157#13512yescomplemented
124C22×C157C22×D411yescharacteristic
125C2×D157—21no—
126C2×D157—21no—
127C2×D157—21no—
128C4×C157—21no—
129C4×C157C22×D411yescharacteristic
130C4×C157H(5,𝔽2)11yescharacteristic
131Dic157—21no—
132Dic157—21no—
133Dic157—21no—
134Dic157—42no—
135Dic157—42no—
136C2×C157—22no—
137C2×C157—44no—
138C2×C157—82no—
139C2×C157D4211yescharacteristic
140C2×C157C2×#13511yescharacteristic
141C2×C157#57611yescharacteristic
142D157—41no—
143C157#328011yescharacteristic, complemented, Sylow
144#3280—1571nomaximal, Sylow
145C2×#135—1571no—
146C22×D8D15711yescharacteristic, complemented
147#546—1572no—
148#495—1571no—
149#467—1571no—
150#466—1571no—
151#463—1572no—
152#450—1572no—
153#449—1572no—
154C4×D8—1571no—
155C8×D4—1571no—
156C2×#41—1572no—
157C22×D4C2×D15712yes—
158#135—3144no—
159C2×Q16—1571no—
160C2×SD16—1572no—
161C2×SD16—1572no—
162C2×D8—24no—
163C2×D8C2×D15712yes—
164C2×D8C2×D15711yescharacteristic
165C2×D8C2×D15711yescharacteristic
166C22×C8—1571no—
167C22×C8C2×D15711yescharacteristic
168#124—1572no—
169#123—3142no—
170C22⋊Q8—1572no—
171#121—3142no—
172#121—1572no—
173#120—3142no—
174C4×D4—1572no—
175C4×D4—1571no—
176C2×C22⋊C4—3142no—
177#107—1571no—
178C4⋊C8—1571no—
179#103—1572no—
180#102—1572no—
181#102—1572no—
182C22⋊C8—1571no—
183C22⋊C8—1571no—
184C4×C8—1571no—
185C24—22no—
186#41—3144no—
187#41—6282no—
188C2×Q8—1572no—
189C2×D4—24no—
190C2×D4—22no—
191C2×D4—42no—
192C2×D4—3142no—
193C2×D4—3142no—
194C2×D4—1571no—
195C2×D4C22×D15712yes—
196C2×D4C22×D15712yes—
197C22×C4—3142no—
198C22×C4—3142no—
199C22×C4—1571no—
200C22×C4C22×D15711yescharacteristic
201Q16—3142no—
202SD16—3144no—
203SD16—6282no—
204D8—24no—
205D8—22no—
206D8C22⋊D15714yescomplemented
207C2×C8—22no—
208C2×C8—3142no—
209C2×C8—1571no—
210C2×C8—1571no—
211C2×C8C22×D15711yescharacteristic
212C2×C8C22×D15711yescharacteristic
213C4⋊C4—3142no—
214C4⋊C4—1572no—
215C4⋊C4—1571no—
216C22⋊C4—3142no—
217C22⋊C4—3142no—
218C22⋊C4—3142no—
219C22⋊C4—3142no—
220C22⋊C4—3142no—
221C22⋊C4—1571no—
222C22⋊C4—1571no—
223C42—1571no—
224C23—22no—
225C23—22no—
226C23—44no—
227C23—44no—
228C23—82no—
229C23—1571no—
230C23D4×D15711yescharacteristic
231Q8—3142no—
232Q8—3142no—
233D4—22no—
234D4—22no—
235D4—22no—
236D4—44no—
237D4—3142no—
238D4—6284no—
239D4—12562no—
240D4C2×C22⋊D15714yes—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—3141no—
244C2×C4—3141no—
245C2×C4—3142no—
246C2×C4—3142no—
247C2×C4—6284no—
248C2×C4—1571no—
249C2×C4—1571no—
250C2×C4—1571no—
251C2×C4—12562no—
252C2×C4D4×D15711yescharacteristic
253C2×C4C23×D15711yescharacteristic
254C8—21no—
255C8—3141no—
256C8—3141no—
257C8C2×C22⋊D15712yes—
258C22—21no—
259C22—21no—
260C22—24no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—88no—
267C22—3141no—
268C22—3141no—
269C22—3141no—
270C22#1909779012yescomplemented
271C22C2×D4×D15711yescharacteristic
272C4—21no—
273C4—3141no—
274C4—3141no—
275C4—3141no—
276C4—6282no—
277C4—6282no—
278C4C22×C22⋊D15711yescharacteristic
279C4#1909781611yescharacteristic
280C2—22no—
281C2—44no—
282C2—82no—
283C2—6281no—
284C2D4×C22⋊D15711yescharacteristic
285C2C2×#1909779011yescharacteristic
286C2#2375807411yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC22×D8×C157C2
socleC22×C157C22×D4
hypercenterC22×D8D157
hypocenterC157#3280
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C2×#19097790C2semidirect
2. #23757917C2semidirect
3. #23757656C2semidirect
4. #23757514C2semidirect
5. #23757506C2semidirect
6. #23757486C2semidirect
7. #23757424C2semidirect
8. #23757415C2semidirect
9. D8×Dic157C2semidirect
10. C8×C22⋊D157C2semidirect
11. C22×D8×C157C2semidirect
12. C2×#14578953C22semidirect
14. C2×Dic628C22semidirect
15. C2×#14578942C22semidirect
16. C2×#14578942C22semidirect
17. C2×C8×D157C22semidirect
18. #19097758C22semidirect
22. #19097742C22semidirect
24. D4×Dic157C22semidirect
27. #19097705C22semidirect
28. #19097700C22semidirect
29. #19097696C22semidirect
30. #19097692C22semidirect
31. #19097692C22semidirect
33. #19097683C22semidirect
34. C8×Dic157C22semidirect
76. D8×C157D4semidirect
123. C22×C157#135semidirect
143. C157#3280semidirect
146. C22×D8D157semidirect
206. D8C22⋊D157semidirect
270. C22#19097790semidirect

Maximal subgroups

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

subgroupindex
C2×#190977902
#237579172
#237576562
#237575142
#237575062
#237574862
#237574242
#237574152
D8×Dic1572
C8×C22⋊D1572
C22×D8×C1572
#3280157