Finite Group Databasebeta

← #34111824

Subgroups — #34111824

Order 20096.

526 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 123 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×#19097793C211yesmaximal, characteristic, complemented
2C2×D8×D157C211yesmaximal, characteristic, complemented
3#23757920C212yesmaximal, complemented
4#23757656C211yesmaximal, characteristic, complemented
5#23757515C211yesmaximal, characteristic, complemented
6#23757487C212yesmaximal, complemented
7#23757425C212yesmaximal, complemented
8#23757416C212yesmaximal, complemented
9#23757355C211yesmaximal, characteristic, complemented
10C8×C22⋊D157C211yesmaximal, characteristic, complemented
11#466×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2212yescomplemented
13C2×D4×D157C2212yes—
14#19097793—24no—
15C2×Dic628C2211yescharacteristic, complemented
16C2×#14578943C2212yescomplemented
17C2×#14578942C2212yescomplemented
18C2×D1256C2211yescharacteristic
19C2×#14578939C2212yes—
20D8×D157—24no—
21C2×C8×D157C2211yescharacteristic
22#19097758C2212yescomplemented
23#19097753—22no—
24#19097751C2212yes—
25#19097742—22no—
26#19097739—22no—
27#19097733C2212yescomplemented
28C4×C22⋊D157C2211yescharacteristic
29C22⋊C4×D157—22no—
30#19097700C2211yescharacteristic, complemented
31#19097694C2212yescomplemented
32#19097691C2212yescomplemented
33C22⋊#11929434C2211yescharacteristic, complemented
34#19097683C2211yescharacteristic
35C8×Dic157C2211yescharacteristic, complemented
36C2×D8×C157C2211yescharacteristic
37C22×C8×C157C2211yescharacteristic, complemented
38#121×C157C2212yescomplemented
39#107×C157C2211yescharacteristic, complemented
40#102×C157C2212yescomplemented
41C23×D157—22no—
42#14578954—24no—
43#14578954—42no—
44C2×Dic314C2312yes—
45C2×D628C2312yes—
46C2×C22⋊D157—22no—
47C2×C22⋊D157—22no—
48C2×C22⋊D157D411yescharacteristic
49D4×D157—24no—
50D4×D157—42no—
51C2×C4×D157—22no—
52C2×C4×D157—22no—
53C2×C4×D157C2311yescharacteristic
54Dic628—22no—
55#14578943—24no—
56#14578942—42no—
57D1256D414yescomplemented
58#14578939—24no—
59C2×#11929434C2311yescharacteristic
60C2×#11929434C2311yescharacteristic
61C8×D157—22no—
62C4⋊Dic157—22no—
63#14578933D411yescharacteristic
64C22⋊Dic157—22no—
65C22⋊Dic157—22no—
66C22⋊Dic157D411yescharacteristic
67#14578931—22no—
68#14578931—22no—
69#14578931D411yescharacteristic
70C4×Dic157C2311yescharacteristic
71C2×D4×C157—22no—
72C2×D4×C157C2312yes—
73C22×C4×C157C2311yescharacteristic
74D8×C157—22no—
75C2×C8×C157—22no—
76C2×C8×C157C2311yescharacteristic
77C2×C8×C157C2311yescharacteristic
78C4⋊C4×C157C2312yes—
79C22⋊C4×C157—22no—
80C22×D157—22no—
81C22×D157—44no—
82C22×D157—44no—
83C22×D157—82no—
84C22×D157C2×D411yescharacteristic
85Dic314—22no—
86Dic314—22no—
87D628—22no—
88D628C2×D414yes—
89C22⋊D157—22no—
90C22⋊D157—44no—
91C22⋊D157—44no—
92C2×Dic157—22no—
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96C4×D157—21no—
97C4×D157—21no—
98C4×D157—44no—
99C4×D157—82no—
100#11929434—21no—
101#11929434—21no—
102C23×C157—22no—
103C23×C157C2×D411yescharacteristic
104D4×C157—22no—
105D4×C157—22no—
106D4×C157—82no—
107C2×C4×C157—21no—
108C2×C4×C157—21no—
109C2×C4×C157—22no—
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2411yescharacteristic
112C8×C157—21no—
113C8×C157C2×D412yes—
114C22×C157—21no—
115C22×C157—21no—
116C22×C157—21no—
117C22×C157—42no—
118C22×C157—42no—
119C22×C157—42no—
120C22×C157C22×D411yescharacteristic
121C2×D157—21no—
122C2×D157—21no—
123C2×D157—24no—
124C2×D157—42no—
125C2×D157—42no—
126C2×D157—88no—
127C2×D157#13512yescomplemented
128C4×C157—21no—
129C4×C157—42no—
130C4×C157C22×D411yescharacteristic
131C4×C157H(5,𝔽2)11yescharacteristic
132Dic157—21no—
133Dic157—21no—
134Dic157—21no—
135Dic157—42no—
136C2×C157—41no—
137C2×C157—82no—
138C2×C157D4211yescharacteristic
139C2×C157C2×#13511yescharacteristic
140C2×C157#57611yescharacteristic
141D157—22no—
142D157—44no—
143C157#328011yescharacteristic, complemented, Sylow
144#3280—1571nomaximal, Sylow
145C2×#135—1571no—
146C22×D8—1571no—
147#546—1572no—
148#495—1571no—
149#467—1571no—
150#466D15711yescharacteristic, complemented
151#463—1572no—
152#450—1572no—
153#449—1572no—
154C4×D8—1571no—
155C8×D4—1571no—
156C2×#41—1572no—
157C22×D4—1572no—
158#135—3144no—
159C2×Q16—1571no—
160C2×SD16—1572no—
161C2×SD16—1572no—
162C2×D8—3144no—
163C2×D8—1572no—
164C2×D8—1571no—
165C2×D8C2×D15711yescharacteristic
166C22×C8—1571no—
167C22×C8C2×D15711yescharacteristic, complemented
168#124—1572no—
169#123—3142no—
170C22⋊Q8—1572no—
171#121—3142no—
172#121C2×D15712yescomplemented
173#120—3142no—
174C4×D4—1572no—
175C4×D4—1571no—
176C2×C22⋊C4—3142no—
177#107C2×D15711yescharacteristic, complemented
178C4⋊C8—1571no—
179#103—1572no—
180#102—1572no—
181#102C2×D15712yescomplemented
182C22⋊C8—1571no—
183C22⋊C8—1571no—
184C4×C8—1571no—
185C24—3142no—
186#41—3144no—
187#41—6282no—
188C2×Q8—1572no—
189C2×D4—22no—
190C2×D4—3144no—
191C2×D4—3142no—
192C2×D4—3142no—
193C2×D4—6282no—
194C2×D4—1571no—
195C2×D4—1572no—
196C2×D4C22×D15712yes—
197C22×C4—3142no—
198C22×C4—3142no—
199C22×C4—1571no—
200C22×C4C22×D15711yescharacteristic
201Q16—3142no—
202SD16—3144no—
203SD16—6282no—
204D8—22no—
205D8—3144no—
206D8—1574no—
207C2×C8—22no—
208C2×C8—3142no—
209C2×C8—1571no—
210C2×C8—1571no—
211C2×C8C22×D15711yescharacteristic
212C2×C8C22×D15711yescharacteristic
213C4⋊C4—3142no—
214C4⋊C4—1571no—
215C4⋊C4C22×D15712yes—
216C22⋊C4—22no—
217C22⋊C4—3142no—
218C22⋊C4—3142no—
219C22⋊C4—3142no—
220C22⋊C4—3142no—
221C22⋊C4—1571no—
222C22⋊C4—1571no—
223C42—1571no—
224C23—22no—
225C23—3142no—
226C23—6284no—
227C23—6284no—
228C23—1571no—
229C23—12562no—
230C23D4×D15711yescharacteristic
231Q8—3142no—
232Q8—3142no—
233D4—22no—
234D4—22no—
235D4—82no—
236D4—3142no—
237D4—3142no—
238D4—6284no—
239D4—6284no—
240D4—1574no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—3141no—
245C2×C4—3141no—
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—
257C8D4×D15712yes—
258C22—21no—
259C22—21no—
260C22—21no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—3141no—
265C22—3141no—
266C22—3144no—
267C22—6282no—
268C22—6282no—
269C22—1572no—
270C22—12568no—
271C22C2×D4×D15711yescharacteristic
272C4—21no—
273C4—42no—
274C4—3141no—
275C4—3141no—
276C4—3141no—
277C4—6282no—
278C4C2×D4×D15711yescharacteristic
279C4#1909781711yescharacteristic
280C2—41no—
281C2—82no—
282C2—3142no—
283C2—6284no—
284C2#2375790711yescharacteristic
285C2#135×D15711yescharacteristic
286C2#2375807411yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#466×C157C2
socleC22×C157C22×D4
hypercenter#466D157
hypocenterC157#3280
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#19097793C2semidirect
2. C2×D8×D157C2semidirect
3. #23757920C2semidirect
4. #23757656C2semidirect
5. #23757515C2semidirect
6. #23757487C2semidirect
7. #23757425C2semidirect
8. #23757416C2semidirect
9. #23757355C2semidirect
10. C8×C22⋊D157C2semidirect
11. #466×C157C2semidirect
12. C2×#14578954C22semidirect
15. C2×Dic628C22semidirect
16. C2×#14578943C22semidirect
17. C2×#14578942C22semidirect
22. #19097758C22semidirect
27. #19097733C22semidirect
30. #19097700C22semidirect
31. #19097694C22semidirect
32. #19097691C22semidirect
33. C22⋊#11929434C22semidirect
35. C8×Dic157C22semidirect
37. C22×C8×C157C22semidirect
38. #121×C157C22semidirect
39. #107×C157C22semidirect
40. #102×C157C22semidirect
57. D1256D4semidirect
127. C2×D157#135semidirect
143. C157#3280semidirect
150. #466D157semidirect
167. C22×C8C2×D157semidirect
172. #121C2×D157semidirect
177. #107C2×D157semidirect
181. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190977932
C2×D8×D1572
#237579202
#237576562
#237575152
#237574872
#237574252
#237574162
#237573552
C8×C22⋊D1572
#466×C1572
#3280157