Finite Group Databasebeta

← #34111943

Subgroups — #34111943

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
1D4×D628C211yesmaximal, characteristic, complemented
2#23757621C211yesmaximal, characteristic, complemented
3#23757586C212yesmaximal, complemented
4#23757550C211yesmaximal, characteristic, complemented
5#23757507C212yesmaximal, complemented
6#23757468C211yesmaximal, characteristic, complemented
7#23757406C212yesmaximal, complemented
8C4×#14578939C211yesmaximal, characteristic, complemented
9D4×#11929434C211yesmaximal, characteristic, complemented
10C2×#19097691C212yesmaximal, complemented
11#547×C157C211yesmaximal, characteristic, complemented
12C22×D628C2212yes—
13C2×D4×D157—21no—
14C2×#14578939C2211yescharacteristic
15C2×#14578939C2212yescomplemented
16C22×#11929434C2212yescomplemented
17#19097768C2211yescharacteristic
18#19097746—21no—
19#19097745C2212yes—
20#19097741—22no—
21C4×D628C2211yescharacteristic
22#19097704C2211yescharacteristic, complemented
23#19097704C2212yescomplemented
24C4⋊#11929434C2211yescharacteristic, complemented
25#19097692C2211yescharacteristic, complemented
26#19097691C2211yescharacteristic
27#19097691C2212yes—
28#19097691C2212yescomplemented
29#19097691C2212yescomplemented
30#19097691C2212yescomplemented
31C22⋊#11929434C2212yescomplemented
32C4×#11929434C2211yescharacteristic, complemented
33C2×#41×C157—21no—
34C4⋊Q8×C157C2211yescharacteristic, complemented
35#123×C157—22no—
36C22⋊Q8×C157—21no—
37#121×C157C2212yescomplemented
38C4×D4×C157C2211yescharacteristic
39C4×D4×C157C2211yescharacteristic, complemented
40C2×C4⋊C4×C157C2212yescomplemented
41C23×D157—22no—
42C2×D628—24no—
43C2×D628—22no—
44C2×D628—21no—
45C2×D628—21no—
46C2×D628—42no—
47C2×D628C2312yes—
48C2×D628C2311yescharacteristic
49C2×C22⋊D157—22no—
50D4×D157—44no—
51C2×C4×D157—21no—
52#14578939—22no—
53#14578939—42no—
54C2×#11929434—42no—
55C2×#11929434C2311yescharacteristic
56C2×#11929434C2311yescharacteristic
57C2×#11929434C2312yes—
58C4⋊Dic157C2311yescharacteristic
59#14578931—21no—
60#14578931—22no—
61#41×C157—44no—
62C2×Q8×C157—21no—
63C2×D4×C157—22no—
64C2×D4×C157D411yescharacteristic
65C2×D4×C157C2311yescharacteristic
66C22×C4×C157—22no—
67C22×C4×C157—21no—
68C22×C4×C157C2312yes—
69C4⋊C4×C157—22no—
70C4⋊C4×C157—21no—
71C4⋊C4×C157D411yescharacteristic
72C4⋊C4×C157C2311yescharacteristic
73C4⋊C4×C157C2312yes—
74C22⋊C4×C157—21no—
75C22⋊C4×C157—22no—
76C22⋊C4×C157D412yes—
77C42×C157C2311yescharacteristic
78C22×D157—21no—
79C22×D157—22no—
80C22×D157—44no—
81C22×D157—44no—
82C22×D157—82no—
83D628—22no—
84D628—21no—
85D628—22no—
86D628—44no—
87D628—42no—
88D628—81no—
89D628#4112yescomplemented
90C22⋊D157—44no—
91C2×Dic157—21no—
92C4×D157—42no—
93#11929434—21no—
94#11929434—21no—
95#11929434—22no—
96C23×C157—21no—
97C23×C157C2×D412yes—
98Q8×C157—42no—
99D4×C157—21no—
100D4×C157—21no—
101D4×C157—44no—
102D4×C157D814yescomplemented
103C2×C4×C157—22no—
104C2×C4×C157—22no—
105C2×C4×C157—21no—
106C2×C4×C157—22no—
107C2×C4×C157—44no—
108C2×C4×C157—42no—
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2×D412yes—
111C2×C4×C157C2×D411yescharacteristic
112C2×C4×C157C2411yescharacteristic
113C22×C157—22no—
114C22×C157—22no—
115C22×C157—41no—
116C22×C157—41no—
117C22×C157—41no—
118C22×C157C2×D814yes—
119C22×C157C22×D411yescharacteristic
120C2×D157—22no—
121C2×D157—41no—
122C2×D157—41no—
123C2×D157—42no—
124C2×D157—42no—
125C2×D157—42no—
126C2×D157—88no—
127C4×C157—22no—
128C4×C157—21no—
129C4×C157—41no—
130C4×C157—42no—
131C4×C157C2×D812yes—
132C4×C157C2×#4111yescharacteristic
133C4×C157H(5,𝔽2)11yescharacteristic
134Dic157—41no—
135C2×C157—24no—
136C2×C157—81no—
137C2×C157#54611yescharacteristic
138C2×C157C22×D811yescharacteristic
139C2×C157#57711yescharacteristic
140D157—42no—
141D157—82no—
142C157#329411yescharacteristic, complemented, Sylow
143#3294—1571nomaximal, Sylow
144#547D15711yescharacteristic, complemented
145D42—1571no—
146#486—1571no—
147#480—1572no—
148#474—1571no—
149#466—1572no—
150#459—1571no—
151#447—1572no—
152C4×D8—1571no—
153C8×D4—1571no—
154C2×#102—1572no—
155C2×#41—21no—
156C22×D4—3141no—
157C22×D4—1572no—
158C2×D8—1571no—
159C2×D8—1572no—
160C22×C8—1572no—
161C4⋊Q8C2×D15711yescharacteristic, complemented
162#127—1571no—
163#123—22no—
164C22⋊Q8—21no—
165#121—3141no—
166#121—1572no—
167#121C2×D15712yescomplemented
168#120—3142no—
169C4×D4—1571no—
170C4×D4C2×D15711yescharacteristic
171C4×D4C2×D15711yescharacteristic, complemented
172C2×C4⋊C4C2×D15712yescomplemented
173#107—1571no—
174#107—1572no—
175C4⋊C8—1571no—
176#102—1571no—
177#102—1571no—
178#102—1572no—
179#102—1572no—
180#102—1572no—
181#102—1572no—
182C22⋊C8—1572no—
183C4×C8—1571no—
184C24—3142no—
185#41—44no—
186C2×Q8—21no—
187C2×D4—22no—
188C2×D4—3144no—
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×D4C22⋊D15711yescharacteristic
198C2×D4C22×D15711yescharacteristic
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—3141no—
202C22×C4C22×D15712yes—
203D8—3142no—
204D8—6282no—
205C2×C8—6282no—
206C2×C8—1571no—
207C2×C8—1571no—
208C2×C8—1572no—
209C4⋊C4—22no—
210C4⋊C4—21no—
211C4⋊C4—1571no—
212C4⋊C4C22⋊D15711yescharacteristic
213C4⋊C4C22×D15711yescharacteristic
214C4⋊C4C22×D15712yes—
215C22⋊C4—21no—
216C22⋊C4—22no—
217C22⋊C4—3141no—
218C22⋊C4—3142no—
219C22⋊C4C22⋊D15712yes—
220C42C22×D15711yescharacteristic
221C23—21no—
222C23—3141no—
223C23—3142no—
224C23—6284no—
225C23—6284no—
226C23—12562no—
227C23C2×C22⋊D15712yes—
228Q8—42no—
229D4—21no—
230D4—21no—
231D4—44no—
232D4—3142no—
233D4—3141no—
234D4—3142no—
235D4—6284no—
236D4—6284no—
237D4—6282no—
238D4—1572no—
239D4—12561no—
240D4#1457893914yescomplemented
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—22no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—3141no—
248C2×C4—6282no—
249C2×C4C2×C22⋊D15711yescharacteristic
250C2×C4C2×C22⋊D15712yes—
251C2×C4C2×C22⋊D15711yescharacteristic
252C2×C4C23×D15711yescharacteristic
253C8—3141no—
254C8—3141no—
255C8—3142no—
256C22—22no—
257C22—22no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—3142no—
262C22—6281no—
263C22—6281no—
264C22—6282no—
265C22—6282no—
266C22—6282no—
267C22—12568no—
268C22C2×#1457893914yes—
269C22C22×C22⋊D15711yescharacteristic
270C4—21no—
271C4—22no—
272C4—41no—
273C4—42no—
274C4—6281no—
275C4C2×#1457893912yes—
276C4#41×D15711yescharacteristic
277C4#1909781711yescharacteristic
278C2—24no—
279C2—81no—
280C2—6282no—
281C2—12562no—
282C2#2375791111yescharacteristic
283C2C22×#1457893911yescharacteristic
284C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#547×C157C2
socleC22×C157C22×D4
hypercenter#547D157
hypocenterC157#3294
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×D628C2semidirect
2. #23757621C2semidirect
3. #23757586C2semidirect
4. #23757550C2semidirect
5. #23757507C2semidirect
6. #23757468C2semidirect
7. #23757406C2semidirect
8. C4×#14578939C2semidirect
9. D4×#11929434C2semidirect
10. C2×#19097691C2semidirect
11. #547×C157C2semidirect
15. C2×#14578939C22semidirect
16. C22×#11929434C22semidirect
22. #19097704C22semidirect
23. #19097704C22semidirect
24. C4⋊#11929434C22semidirect
25. #19097692C22semidirect
28. #19097691C22semidirect
29. #19097691C22semidirect
30. #19097691C22semidirect
31. C22⋊#11929434C22semidirect
32. C4×#11929434C22semidirect
34. C4⋊Q8×C157C22semidirect
37. #121×C157C22semidirect
39. C4×D4×C157C22semidirect
40. C2×C4⋊C4×C157C22semidirect
89. D628#41semidirect
102. D4×C157D8semidirect
142. C157#3294semidirect
144. #547D157semidirect
161. C4⋊Q8C2×D157semidirect
167. #121C2×D157semidirect
171. C4×D4C2×D157semidirect
172. C2×C4⋊C4C2×D157semidirect
240. D4#14578939semidirect

Maximal subgroups

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

subgroupindex
D4×D6282
#237576212
#237575862
#237575502
#237575072
#237574682
#237574062
C4×#145789392
D4×#119294342
C2×#190976912
#547×C1572
#3294157