Finite Group Databasebeta

← #34111950

Subgroups — #34111950

Order 20096.

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

normal subgroups only

This group has 494 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#23757927C211yesmaximal, characteristic, complemented
2D4×D628C211yesmaximal, characteristic, complemented
3#23757620C211yesmaximal, characteristic, complemented
4#23757587C212yesmaximal, complemented
5#23757554C211yesmaximal, characteristic, complemented
6#23757509C212yesmaximal, complemented
7#23757469C211yesmaximal, characteristic, complemented
8#23757409C212yesmaximal, complemented
9C4×D1256C211yesmaximal, characteristic, complemented
10C2×#19097693C212yesmaximal, complemented
11C8×D4×C157C211yesmaximal, characteristic, complemented
12C2×#14578953—21no—
13C22×D628C2212yes—
14C2×D4×D157—21no—
15C2×D1256C2211yescharacteristic
16C2×D1256C2212yescomplemented
17#19097771C2211yescharacteristic, complemented
18#19097768C2211yescharacteristic
19#19097755—22no—
20#19097751—21no—
21#19097746—21no—
22#19097745C2212yes—
23#19097745C2212yescomplemented
24#19097741—22no—
25C4×D628C2211yescharacteristic
26C4×D628C2211yescharacteristic, complemented
27C2×C4⋊Dic157C2212yescomplemented
28#19097705C2211yescharacteristic, complemented
29#19097705C2212yescomplemented
30#19097693C2211yescharacteristic
31#19097693C2212yes—
32#19097693C2212yescomplemented
33#19097693C2212yescomplemented
34#19097693C2211yescharacteristic, complemented
35#19097693C2212yescomplemented
36C22×C8×C157C2212yescomplemented
37C4×D4×C157C2211yescharacteristic
38C4⋊C8×C157C2211yescharacteristic, complemented
39C22⋊C8×C157C2212yescomplemented
40C4×C8×C157C2211yescharacteristic, complemented
41C23×D157—22no—
42#14578953—44no—
43C2×Dic314—21no—
44C2×D628—24no—
45C2×D628—22no—
46C2×D628—21no—
47C2×D628—21no—
48C2×D628—42no—
49C2×D628C2312yes—
50C2×D628C2311yescharacteristic
51C2×D628C2311yescharacteristic
52C2×C22⋊D157—22no—
53C2×C22⋊D157—22no—
54D4×D157—44no—
55C22×Dic157—22no—
56C2×C4×D157—21no—
57C2×C4×D157—21no—
58D1256—22no—
59D1256—42no—
60C4⋊Dic157—22no—
61C4⋊Dic157—21no—
62C4⋊Dic157C2311yescharacteristic
63C4⋊Dic157C2312yes—
64C4⋊Dic157C2311yescharacteristic
65#14578931—21no—
66#14578931—21no—
67#14578931—22no—
68#14578931—22no—
69C2×D4×C157D411yescharacteristic
70C22×C4×C157C2312yes—
71C2×C8×C157—42no—
72C2×C8×C157C2311yescharacteristic
73C2×C8×C157C2311yescharacteristic
74C2×C8×C157C2312yes—
75C4⋊C4×C157D411yescharacteristic
76C22⋊C4×C157D412yes—
77C42×C157C2311yescharacteristic
78C22×D157—21no—
79C22×D157—21no—
80C22×D157—22no—
81C22×D157—44no—
82C22×D157—44no—
83C22×D157—82no—
84Dic314—42no—
85D628—22no—
86D628—21no—
87D628—21no—
88D628—22no—
89D628—21no—
90D628—44no—
91D628—42no—
92D628—81no—
93D628#4112yescomplemented
94C22⋊D157—44no—
95C22⋊D157—44no—
96C2×Dic157—21no—
97C2×Dic157—21no—
98C2×Dic157—22no—
99C2×Dic157—44no—
100C4×D157—42no—
101C4×D157—42no—
102C23×C157C2×D412yes—
103D4×C157D814yescomplemented
104C2×C4×C157—22no—
105C2×C4×C157—22no—
106C2×C4×C157C2×D411yescharacteristic
107C2×C4×C157C2×D412yes—
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2411yescharacteristic
110C8×C157—21no—
111C8×C157—21no—
112C8×C157—22no—
113C22×C157—22no—
114C22×C157—22no—
115C22×C157C2×D814yes—
116C22×C157C22×D411yescharacteristic
117C2×D157—22no—
118C2×D157—41no—
119C2×D157—41no—
120C2×D157—42no—
121C2×D157—41no—
122C2×D157—41no—
123C2×D157—42no—
124C2×D157—42no—
125C2×D157—41no—
126C2×D157—88no—
127C4×C157—22no—
128C4×C157—21no—
129C4×C157C2×D812yes—
130C4×C157C2×#4111yescharacteristic
131C4×C157H(5,𝔽2)11yescharacteristic
132Dic157—41no—
133Dic157—41no—
134Dic157—42no—
135C2×C157—24no—
136C2×C157#54611yescharacteristic
137C2×C157C22×D811yescharacteristic
138C2×C157#57711yescharacteristic
139D157—42no—
140D157—81no—
141D157—82no—
142C157#329411yescharacteristic, complemented, Sylow
143#3294—1571nomaximal, Sylow
144#547—1571no—
145D42—1571no—
146#486—1571no—
147#480—1572no—
148#474—1571no—
149#466—1572no—
150#459—1571no—
151#447—1572no—
152C4×D8—1571no—
153C8×D4D15711yescharacteristic, complemented
154C2×#102—1572no—
155C2×#41—3141no—
156C22×D4—3141no—
157C22×D4—1572no—
158C2×D8—1571no—
159C2×D8—1572no—
160C22×C8C2×D15712yescomplemented
161C4⋊Q8—1571no—
162#127—1571no—
163#123—3142no—
164C22⋊Q8—3141no—
165#121—3141no—
166#121—1572no—
167#121—1572no—
168#120—3142no—
169C4×D4—1571no—
170C4×D4—1571no—
171C4×D4C2×D15711yescharacteristic
172C2×C4⋊C4—1572no—
173#107—1571no—
174#107—1572no—
175C4⋊C8C2×D15711yescharacteristic, complemented
176#102—1571no—
177#102—1572no—
178#102—1572no—
179#102—1572no—
180#102—1571no—
181#102—1572no—
182C22⋊C8C2×D15712yescomplemented
183C4×C8C2×D15711yescharacteristic, complemented
184C24—3142no—
185#41—6284no—
186C2×Q8—3141no—
187C2×D4—3144no—
188C2×D4—3142no—
189C2×D4—3142no—
190C2×D4—3142no—
191C2×D4—3141no—
192C2×D4—3141no—
193C2×D4—6282no—
194C2×D4—6284no—
195C2×D4—1572no—
196C2×D4—1571no—
197C2×D4—1571no—
198C2×D4D62811yescharacteristic
199C22×C4—3142no—
200C22×C4—3141no—
201C22×C4—3141no—
202C22×C4C22×D15712yes—
203D8—3142no—
204D8—6282no—
205C2×C8—42no—
206C2×C8C22×D15711yescharacteristic
207C2×C8C22×D15711yescharacteristic
208C2×C8C22×D15712yes—
209C4⋊C4—3142no—
210C4⋊C4—3141no—
211C4⋊C4—1571no—
212C4⋊C4—1572no—
213C4⋊C4—1571no—
214C4⋊C4D62811yescharacteristic
215C22⋊C4—3141no—
216C22⋊C4—3141no—
217C22⋊C4—3142no—
218C22⋊C4—3142no—
219C22⋊C4D62812yes—
220C42C22×D15711yescharacteristic
221C23—3141no—
222C23—3141no—
223C23—3142no—
224C23—6284no—
225C23—6284no—
226C23—12562no—
227C23C2×D62812yes—
228Q8—6282no—
229D4—3141no—
230D4—3142no—
231D4—3141no—
232D4—3141no—
233D4—3142no—
234D4—6284no—
235D4—6284no—
236D4—6284no—
237D4—6282no—
238D4—1572no—
239D4—12561no—
240D4D125614yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—3141no—
244C2×C4—3141no—
245C2×C4—3142no—
246C2×C4—6284no—
247C2×C4—6282no—
248C2×C4—6282no—
249C2×C4C2×D62811yescharacteristic
250C2×C4C2×D62812yes—
251C2×C4C2×D62811yescharacteristic
252C2×C4C23×D15711yescharacteristic
253C8—21no—
254C8—21no—
255C8—22no—
256C22—22no—
257C22—22no—
258C22—3142no—
259C22—6281no—
260C22—6281no—
261C22—6282no—
262C22—6281no—
263C22—6282no—
264C22—6281no—
265C22—6281no—
266C22—6282no—
267C22—12568no—
268C22C2×D125614yes—
269C22C22×D62811yescharacteristic
270C4—21no—
271C4—22no—
272C4—6281no—
273C4—6281no—
274C4—6282no—
275C4C2×D125612yes—
276C4#41×D15711yescharacteristic
277C4#1909781611yescharacteristic
278C2—24no—
279C2—6282no—
280C2—12562no—
281C2—12561no—
282C2#2375791811yescharacteristic
283C2C22×D125611yescharacteristic
284C2#2375808511yescharacteristic
subgroupG/H
centerC22C22×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC8×D4×C157C2
socleC22×C157C22×D4
hypercenterC8×D4D157
hypocenterC157#3294
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. #23757927C2semidirect
2. D4×D628C2semidirect
3. #23757620C2semidirect
4. #23757587C2semidirect
5. #23757554C2semidirect
6. #23757509C2semidirect
7. #23757469C2semidirect
8. #23757409C2semidirect
9. C4×D1256C2semidirect
10. C2×#19097693C2semidirect
11. C8×D4×C157C2semidirect
16. C2×D1256C22semidirect
17. #19097771C22semidirect
23. #19097745C22semidirect
26. C4×D628C22semidirect
27. C2×C4⋊Dic157C22semidirect
28. #19097705C22semidirect
29. #19097705C22semidirect
32. #19097693C22semidirect
33. #19097693C22semidirect
34. #19097693C22semidirect
35. #19097693C22semidirect
36. C22×C8×C157C22semidirect
38. C4⋊C8×C157C22semidirect
39. C22⋊C8×C157C22semidirect
40. C4×C8×C157C22semidirect
93. D628#41semidirect
103. D4×C157D8semidirect
142. C157#3294semidirect
153. C8×D4D157semidirect
160. C22×C8C2×D157semidirect
175. C4⋊C8C2×D157semidirect
182. C22⋊C8C2×D157semidirect
183. C4×C8C2×D157semidirect
240. D4D1256semidirect

Maximal subgroups

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

subgroupindex
#237579272
D4×D6282
#237576202
#237575872
#237575542
#237575092
#237574692
#237574092
C4×D12562
C2×#190976932
C8×D4×C1572
#3294157