Finite Group Databasebeta

← #34111957

Subgroups — #34111957

Order 20096.

462 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 125 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#23757984C211yesmaximal, characteristic, complemented
2D4×D628C211yesmaximal, characteristic
3#23757621C211yesmaximal, characteristic, complemented
4#23757593C212yesmaximal, complemented
5#23757503C212yesmaximal, complemented
6#23757474C211yesmaximal, characteristic, complemented
7#23757425C212yesmaximal, complemented
8C4×#14578942C211yesmaximal, characteristic, complemented
9#23757338C211yesmaximal, characteristic, complemented
10#102×D157C212yesmaximal, complemented
11#476×C157C211yesmaximal, characteristic, complemented
12C22×D628—21no—
13C2×D4×D157C2212yes—
14C2×#14578943C2212yescomplemented
15C2×#14578942C2211yescharacteristic
16C2×C8×D157C2212yescomplemented
17#19097768C2211yescharacteristic
18#19097762—21no—
19#19097751C2212yescomplemented
20#19097747—22no—
21#19097746C2212yes—
22#19097745—21no—
23#19097741—22no—
24C4×Dic314C2211yescharacteristic, complemented
25C4×D628C2211yescharacteristic
26#19097733—21no—
27C4⋊C4×D157C2212yescomplemented
28#19097702C2211yescharacteristic, complemented
29#19097694C2211yescharacteristic, complemented
30#19097693C2212yescomplemented
31#19097692C2211yescharacteristic
32#19097691C2212yescomplemented
33#19097691C2212yescomplemented
34#19097683C2212yescomplemented
35C4×#11929434C2211yescharacteristic, complemented
36C4⋊Q8×C157C2211yescharacteristic, complemented
37C4×D4×C157C2211yescharacteristic
38#106×C157C2212yescomplemented
39C4⋊C8×C157C2211yescharacteristic, complemented
40#102×C157C2212yes—
41C23×D157—22no—
42C2×Dic314C2311yescharacteristic
43C2×D628—22no—
44C2×D628—21no—
45C2×D628—21no—
46C2×D628—44no—
47C2×D628D411yescharacteristic
48C2×D628C2312yes—
49C2×C22⋊D157—22no—
50D4×D157—24no—
51D4×D157—42no—
52C2×C4×D157—22no—
53C2×C4×D157C2312yes—
54#14578943—42no—
55#14578942—22no—
56C2×#11929434C2311yescharacteristic
57C2×#11929434C2311yescharacteristic
58C8×D157—42no—
59C4⋊Dic157—22no—
60C4⋊Dic157D411yescharacteristic
61C4⋊Dic157C2311yescharacteristic
62#14578933—22no—
63#14578933—21no—
64#14578931—22no—
65#14578931—22no—
66#14578931D412yes—
67C4×Dic157—21no—
68C2×Q8×C157—21no—
69C2×D4×C157C2311yescharacteristic
70C22×C4×C157—21no—
71C2×C8×C157C2312yes—
72C4⋊C4×C157—21no—
73C4⋊C4×C157C2312yes—
74C4⋊C4×C157C2311yescharacteristic
75C22⋊C4×C157—21no—
76C42×C157C2311yescharacteristic
77C22×D157—22no—
78C22×D157—44no—
79C22×D157—44no—
80C22×D157—82no—
81C22×D157C2×D412yes—
82Dic314—21no—
83Dic314—21no—
84D628—22no—
85D628—22no—
86D628—44no—
87D628—42no—
88D628—81no—
89D628SD1614yescomplemented
90C22⋊D157—44no—
91C2×Dic157—21no—
92C2×Dic157—21no—
93C2×Dic157C2×D412yes—
94C4×D157—22no—
95C4×D157—22no—
96C4×D157—82no—
97#11929434—21no—
98#11929434—21no—
99C23×C157—21no—
100Q8×C157—81no—
101D4×C157—21no—
102D4×C157#4112yes—
103C2×C4×C157—22no—
104C2×C4×C157—21no—
105C2×C4×C157—42no—
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2411yescharacteristic
109C8×C157—22no—
110C22×C157—22no—
111C22×C157—41no—
112C22×C157—41no—
113C22×C157C22×D411yescharacteristic
114C2×D157—22no—
115C2×D157—22no—
116C2×D157—42no—
117C2×D157—42no—
118C2×D157—42no—
119C2×D157—88no—
120C2×D157C2×SD1614yes—
121C4×C157—21no—
122C4×C157—41no—
123C4×C157—42no—
124C4×C157C2×SD1612yes—
125C4×C157C2×#4111yescharacteristic
126C4×C157H(5,𝔽2)11yescharacteristic
127Dic157—22no—
128Dic157—41no—
129Dic157—41no—
130C2×C157—42no—
131C2×C157#54611yescharacteristic
132C2×C157C22×SD1611yescharacteristic
133C2×C157#57611yescharacteristic
134D157—24no—
135D157—82no—
136C157#329511yescharacteristic, complemented, Sylow
137#3295—1571nomaximal, Sylow
138#554—1571no—
139D42—1571no—
140#486—1571no—
141#481—1572no—
142#476D15711yescharacteristic, complemented
143#465—1572no—
144#460—1571no—
145#450—1572no—
146C4×SD16—1571no—
147C8×D4—1571no—
148C2×#102—1572no—
149C22×D4—3141no—
150C22×D4—1572no—
151C2×SD16—1571no—
152C2×SD16—1572no—
153C22×C8—1572no—
154C4⋊Q8C2×D15711yescharacteristic, complemented
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#106C2×D15712yescomplemented
169C4⋊C8C2×D15711yescharacteristic, complemented
170#103—1571no—
171#102—1571no—
172#102—1572no—
173#102—1572no—
174#102—1572no—
175#102C2×D15712yes—
176C22⋊C8—1572no—
177C4×C8—1571no—
178C24—3142no—
179C2×Q8—21no—
180C2×Q8—1571no—
181C2×D4—3142no—
182C2×D4—3144no—
183C2×D4—3142no—
184C2×D4—3141no—
185C2×D4—3141no—
186C2×D4—6284no—
187C2×D4—6282no—
188C2×D4—1571no—
189C2×D4—1572no—
190C2×D4C22×D15711yescharacteristic
191C22×C4—21no—
192C22×C4—3142no—
193C22×C4—1572no—
194SD16—3142no—
195SD16—6282no—
196C2×C8—6282no—
197C2×C8—1571no—
198C2×C8—1571no—
199C2×C8C22×D15712yes—
200C4⋊C4—21no—
201C4⋊C4—3142no—
202C4⋊C4—3142no—
203C4⋊C4—3141no—
204C4⋊C4—1571no—
205C4⋊C4—1571no—
206C4⋊C4C22×D15712yes—
207C4⋊C4C22×D15711yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—3142no—
210C22⋊C4—3142no—
211C22⋊C4—1572no—
212C42—3141no—
213C42C22×D15711yescharacteristic
214C23—21no—
215C23—3142no—
216C23—6284no—
217C23—6284no—
218C23—1572no—
219C23—12562no—
220Q8—81no—
221Q8—3141no—
222Q8—3141no—
223D4—21no—
224D4—3142no—
225D4—3142no—
226D4—6284no—
227D4—6284no—
228D4—6282no—
229D4—1574no—
230D4—12561no—
231D4#1457895512yes—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—3142no—
236C2×C4—3142no—
237C2×C4—3141no—
238C2×C4—3141no—
239C2×C4—1572no—
240C2×C4—12562no—
241C2×C4D4×D15711yescharacteristic
242C2×C4D4×D15711yescharacteristic
243C2×C4C23×D15711yescharacteristic
244C8—22no—
245C8—3141no—
246C8—3141no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—3142no—
251C22—3142no—
252C22—6282no—
253C22—6282no—
254C22—6282no—
255C22—1574no—
256C22—12568no—
257C22C2×D4×D15711yescharacteristic
258C4—21no—
259C4—41no—
260C4—42no—
261C4—3142no—
262C4—6281no—
263C4—6281no—
264C4SD16×D15712yes—
265C4C2×#1457895511yescharacteristic
266C4#1909781711yescharacteristic
267C2—42no—
268C2—3144no—
269C2—12562no—
270C2#2375791411yescharacteristic
271C2C2×SD16×D15711yescharacteristic
272C2#2375807711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#476×C157C2
socleC22×C157C22×D4
prime radicalD4×D628C2
hypercenter#476D157
hypocenterC157#3295
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #23757984C2semidirect
3. #23757621C2semidirect
4. #23757593C2semidirect
5. #23757503C2semidirect
6. #23757474C2semidirect
7. #23757425C2semidirect
8. C4×#14578942C2semidirect
9. #23757338C2semidirect
10. #102×D157C2semidirect
11. #476×C157C2semidirect
14. C2×#14578943C22semidirect
16. C2×C8×D157C22semidirect
19. #19097751C22semidirect
24. C4×Dic314C22semidirect
27. C4⋊C4×D157C22semidirect
28. #19097702C22semidirect
29. #19097694C22semidirect
30. #19097693C22semidirect
32. #19097691C22semidirect
33. #19097691C22semidirect
34. #19097683C22semidirect
35. C4×#11929434C22semidirect
36. C4⋊Q8×C157C22semidirect
38. #106×C157C22semidirect
39. C4⋊C8×C157C22semidirect
89. D628SD16semidirect
136. C157#3295semidirect
142. #476D157semidirect
154. C4⋊Q8C2×D157semidirect
168. #106C2×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
#237579842
D4×D6282
#237576212
#237575932
#237575032
#237574742
#237574252
C4×#145789422
#237573382
#102×D1572
#476×C1572
#3295157