Finite Group Databasebeta

← #34111961

Subgroups — #34111961

Order 20096.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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
1#23757980C211yesmaximal, characteristic, complemented
2D4×D628C211yesmaximal, characteristic
3#23757620C211yesmaximal, characteristic, complemented
4#23757592C212yesmaximal, complemented
5#23757564C211yesmaximal, characteristic, complemented
6#23757502C212yesmaximal, complemented
7#23757474C211yesmaximal, characteristic, complemented
8#23757426C212yesmaximal, complemented
9C4×#14578943C211yesmaximal, characteristic, complemented
10C2×#19097693C212yesmaximal, complemented
11C8×D4×C157C211yesmaximal, characteristic, complemented
12C22×D628C2212yes—
13C2×D4×D157—21no—
14C2×#14578943C2211yescharacteristic
15C2×#14578943C2212yescomplemented
16#19097771C2211yescharacteristic, complemented
17#19097768C2211yescharacteristic
18#19097762—21no—
19#19097750C2212yescomplemented
20#19097748—22no—
21#19097746—21no—
22#19097745C2212yes—
23#19097741—22no—
24C4×Dic314C2211yescharacteristic, complemented
25C4×D628C2211yescharacteristic
26D4×Dic157—21no—
27C2×C4⋊Dic157C2212yescomplemented
28#19097703C2211yescharacteristic, complemented
29#19097703C2212yescomplemented
30#19097696C2211yescharacteristic, complemented
31#19097693C2211yescharacteristic
32#19097693C2212yes—
33#19097693C2212yescomplemented
34#19097693C2212yescomplemented
35#19097693C2212yescomplemented
36C22×C8×C157C2212yescomplemented
37C4×D4×C157C2211yescharacteristic
38C4⋊C8×C157C2211yescharacteristic, complemented
39C22⋊C8×C157C2212yescomplemented
40C4×C8×C157C2211yescharacteristic, complemented
41C23×D157—22no—
42C2×Dic314—21no—
43C2×Dic314C2311yescharacteristic
44C2×D628—24no—
45C2×D628—22no—
46C2×D628—21no—
47C2×D628—21no—
48C2×D628—42no—
49C2×D628C2312yes—
50C2×D628C2311yescharacteristic
51C2×C22⋊D157—22no—
52D4×D157—44no—
53C22×Dic157—22no—
54C2×C4×D157—21no—
55#14578943—22no—
56#14578943—42no—
57C4⋊Dic157—22no—
58C4⋊Dic157—21no—
59C4⋊Dic157C2311yescharacteristic
60C4⋊Dic157C2312yes—
61C4⋊Dic157C2311yescharacteristic
62#14578933—21no—
63#14578933—22no—
64C22⋊Dic157—22no—
65#14578931—21no—
66#14578931—22no—
67C4×Dic157—21no—
68C2×D4×C157D411yescharacteristic
69C22×C4×C157C2312yes—
70C2×C8×C157—42no—
71C2×C8×C157C2311yescharacteristic
72C2×C8×C157C2311yescharacteristic
73C2×C8×C157C2312yes—
74C4⋊C4×C157D411yescharacteristic
75C22⋊C4×C157D412yes—
76C42×C157C2311yescharacteristic
77C22×D157—21no—
78C22×D157—22no—
79C22×D157—44no—
80C22×D157—44no—
81C22×D157—82no—
82Dic314—21no—
83Dic314—21no—
84Dic314—81no—
85D628—22no—
86D628—21no—
87D628—22no—
88D628—44no—
89D628—42no—
90D628—81no—
91D628#4112yes—
92C22⋊D157—44no—
93C2×Dic157—21no—
94C2×Dic157—21no—
95C2×Dic157—22no—
96C2×Dic157—21no—
97C2×Dic157—82no—
98C4×D157—42no—
99C23×C157C2×D412yes—
100D4×C157SD1614yescomplemented
101C2×C4×C157—22no—
102C2×C4×C157—22no—
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157C2×D412yes—
105C2×C4×C157C2×D411yescharacteristic
106C2×C4×C157C2411yescharacteristic
107C8×C157—21no—
108C8×C157—21no—
109C8×C157—22no—
110C22×C157—22no—
111C22×C157—22no—
112C22×C157C2×SD1614yes—
113C22×C157C22×D411yescharacteristic
114C2×D157—22no—
115C2×D157—41no—
116C2×D157—41no—
117C2×D157—42no—
118C2×D157—42no—
119C2×D157—42no—
120C2×D157—88no—
121C4×C157—22no—
122C4×C157—21no—
123C4×C157C2×SD1612yes—
124C4×C157C2×#4111yescharacteristic
125C4×C157H(5,𝔽2)11yescharacteristic
126Dic157—41no—
127Dic157—41no—
128Dic157—41no—
129Dic157—42no—
130C2×C157—24no—
131C2×C157#54611yescharacteristic
132C2×C157C22×SD1611yescharacteristic
133C2×C157#57611yescharacteristic
134D157—42no—
135D157—82no—
136C157#329511yescharacteristic, complemented, Sylow
137#3295—1571nomaximal, Sylow
138#554—1571no—
139D42—1571no—
140#486—1571no—
141#481—1572no—
142#476—1571no—
143#465—1572no—
144#460—1571no—
145#450—1572no—
146C4×SD16—1571no—
147C8×D4D15711yescharacteristic, complemented
148C2×#102—1572no—
149C22×D4—3141no—
150C22×D4—1572no—
151C2×SD16—1571no—
152C2×SD16—1572no—
153C22×C8C2×D15712yescomplemented
154C4⋊Q8—1571no—
155#127—1571no—
156#125—3141no—
157C22⋊Q8—3142no—
158C22⋊Q8—1572no—
159#121—3141no—
160#121—1572no—
161#120—3142no—
162C4×Q8—1571no—
163C4×D4—3141no—
164C4×D4—1571no—
165C4×D4C2×D15711yescharacteristic
166C2×C4⋊C4—1572no—
167#106—1571no—
168#106—1572no—
169C4⋊C8C2×D15711yescharacteristic, complemented
170#103—1571no—
171#102—1571no—
172#102—1572no—
173#102—1572no—
174#102—1572no—
175#102—1572no—
176C22⋊C8C2×D15712yescomplemented
177C4×C8C2×D15711yescharacteristic, complemented
178C24—3142no—
179C2×Q8—3141no—
180C2×Q8—1571no—
181C2×D4—3144no—
182C2×D4—3142no—
183C2×D4—3142no—
184C2×D4—3141no—
185C2×D4—3141no—
186C2×D4—6282no—
187C2×D4—6284no—
188C2×D4—1572no—
189C2×D4—1571no—
190C2×D4D62811yescharacteristic
191C22×C4—3142no—
192C22×C4—3141no—
193C22×C4C22×D15712yes—
194SD16—3142no—
195SD16—6282no—
196C2×C8—42no—
197C2×C8C22×D15711yescharacteristic
198C2×C8C22×D15711yescharacteristic
199C2×C8C22×D15712yes—
200C4⋊C4—3142no—
201C4⋊C4—3141no—
202C4⋊C4—3141no—
203C4⋊C4—3142no—
204C4⋊C4—1571no—
205C4⋊C4—1572no—
206C4⋊C4—1571no—
207C4⋊C4D62811yescharacteristic
208C22⋊C4—3142no—
209C22⋊C4—3141no—
210C22⋊C4—3142no—
211C22⋊C4D62812yes—
212C42—3141no—
213C42C22×D15711yescharacteristic
214C23—3141no—
215C23—3142no—
216C23—6284no—
217C23—6284no—
218C23—12562no—
219C23C2×D62812yes—
220Q8—3141no—
221Q8—3141no—
222Q8—12561no—
223D4—3142no—
224D4—3141no—
225D4—3142no—
226D4—6284no—
227D4—6284no—
228D4—6282no—
229D4—1572no—
230D4—12561no—
231D4#1457894314yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—3141no—
235C2×C4—3141no—
236C2×C4—3141no—
237C2×C4—3142no—
238C2×C4—6282no—
239C2×C4—12562no—
240C2×C4C2×D62811yescharacteristic
241C2×C4C2×D62812yes—
242C2×C4C2×D62811yescharacteristic
243C2×C4C23×D15711yescharacteristic
244C8—21no—
245C8—21no—
246C8—22no—
247C22—22no—
248C22—22no—
249C22—3142no—
250C22—6281no—
251C22—6281no—
252C22—6282no—
253C22—6282no—
254C22—6282no—
255C22—12568no—
256C22C2×#1457894314yes—
257C22C22×D62811yescharacteristic
258C4—22no—
259C4—21no—
260C4—6281no—
261C4—6281no—
262C4—6281no—
263C4—6282no—
264C4C2×#1457894312yes—
265C4#41×D15711yescharacteristic
266C4#1909781611yescharacteristic
267C2—24no—
268C2—6282no—
269C2—12562no—
270C2#2375791811yescharacteristic
271C2C22×#1457894311yescharacteristic
272C2#2375807811yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC8×D4×C157C2
socleC22×C157C22×D4
prime radicalD4×D628C2
hypercenterC8×D4D157
hypocenterC157#3295
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #23757980C2semidirect
3. #23757620C2semidirect
4. #23757592C2semidirect
5. #23757564C2semidirect
6. #23757502C2semidirect
7. #23757474C2semidirect
8. #23757426C2semidirect
9. C4×#14578943C2semidirect
10. C2×#19097693C2semidirect
11. C8×D4×C157C2semidirect
15. C2×#14578943C22semidirect
16. #19097771C22semidirect
19. #19097750C22semidirect
24. C4×Dic314C22semidirect
27. C2×C4⋊Dic157C22semidirect
28. #19097703C22semidirect
29. #19097703C22semidirect
30. #19097696C22semidirect
33. #19097693C22semidirect
34. #19097693C22semidirect
35. #19097693C22semidirect
36. C22×C8×C157C22semidirect
38. C4⋊C8×C157C22semidirect
39. C22⋊C8×C157C22semidirect
40. C4×C8×C157C22semidirect
100. D4×C157SD16semidirect
136. C157#3295semidirect
147. C8×D4D157semidirect
153. C22×C8C2×D157semidirect
169. C4⋊C8C2×D157semidirect
176. C22⋊C8C2×D157semidirect
177. C4×C8C2×D157semidirect
231. D4#14578943semidirect

Maximal subgroups

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

subgroupindex
#237579802
D4×D6282
#237576202
#237575922
#237575642
#237575022
#237574742
#237574262
C4×#145789432
C2×#190976932
C8×D4×C1572
#3295157