Finite Group Databasebeta

← #34111952

Subgroups — #34111952

Order 20096.

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

normal subgroups only

This group has 494 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
1#23757928C211yesmaximal, characteristic, complemented
2#23757908C211yesmaximal, characteristic, complemented
3#23757619C211yesmaximal, characteristic, complemented
4#23757585C212yesmaximal, complemented
5#23757554C211yesmaximal, characteristic, complemented
6#23757506C212yesmaximal, complemented
7#23757405C212yesmaximal, complemented
8C4×#14578939C211yesmaximal, characteristic, complemented
9#23757338C211yesmaximal, characteristic, complemented
10#102×D157C212yesmaximal, complemented
11#459×C157C211yesmaximal, characteristic, complemented
12C2×#14578954—21no—
13C2×D4×D157—21no—
14C2×D4×D157C2212yes—
15C2×#14578939C2211yescharacteristic
16C2×C8×D157C2212yescomplemented
17#19097771C2211yescharacteristic, complemented
18#19097753—22no—
19#19097750—21no—
20#19097742—21no—
21#19097742C2212yes—
22#19097742C2212yescomplemented
23#19097739—22no—
24C4×D628C2211yescharacteristic
25C4×D628C2211yescharacteristic
26C4⋊C4×D157C2212yescomplemented
27#19097705C2212yescomplemented
28#19097704C2211yescharacteristic, complemented
29#19097693C2212yes—
30#19097692C2212yescomplemented
31#19097692C2211yescharacteristic, complemented
32#19097692C2212yescomplemented
33#19097691C2211yescharacteristic
34#19097683C2212yescomplemented
35C4×#11929434C2211yescharacteristic, complemented
36C2×D8×C157C2212yescomplemented
37#127×C157C2211yescharacteristic
38C4×D4×C157C2211yescharacteristic, complemented
39C4⋊C8×C157C2211yescharacteristic, complemented
40#102×C157C2212yescomplemented
41C23×D157—22no—
42#14578954—44no—
43C2×Dic314—21no—
44C2×D628D411yescharacteristic
45C2×D628C2311yescharacteristic
46C2×C22⋊D157—22no—
47C2×C22⋊D157—22no—
48C2×C22⋊D157—22no—
49D4×D157—24no—
50D4×D157—42no—
51D4×D157—44no—
52C2×C4×D157—22no—
53C2×C4×D157—21no—
54C2×C4×D157C2312yes—
55#14578939—22no—
56C2×#11929434C2311yescharacteristic
57C2×#11929434C2311yescharacteristic
58C8×D157—42no—
59C4⋊Dic157—21no—
60C4⋊Dic157D411yescharacteristic
61C4⋊Dic157C2312yes—
62C4⋊Dic157C2311yescharacteristic
63#14578933—22no—
64C22⋊Dic157—22no—
65C22⋊Dic157—22no—
66#14578931—21no—
67#14578931D412yes—
68C2×D4×C157—21no—
69C2×D4×C157—21no—
70C2×D4×C157C2312yes—
71C2×D4×C157C2311yescharacteristic
72C22×C4×C157—21no—
73D8×C157—42no—
74C2×C8×C157C2312yes—
75C4⋊C4×C157C2311yescharacteristic
76C22⋊C4×C157—21no—
77C42×C157C2311yescharacteristic
78C22×D157—21no—
79C22×D157—44no—
80C22×D157—44no—
81C22×D157—82no—
82C22×D157C2×D412yes—
83Dic314—42no—
84D628—21no—
85D628D814yescomplemented
86D628#4112yescomplemented
87C22⋊D157—44no—
88C22⋊D157—44no—
89C22⋊D157—44no—
90C2×Dic157—21no—
91C2×Dic157—22no—
92C2×Dic157C2×D412yes—
93C4×D157—22no—
94C4×D157—22no—
95C4×D157—44no—
96C4×D157—42no—
97#11929434—21no—
98#11929434—21no—
99C23×C157—22no—
100C23×C157—21no—
101D4×C157—22no—
102D4×C157—21no—
103D4×C157—22no—
104D4×C157—21no—
105D4×C157—42no—
106D4×C157—81no—
107C2×C4×C157—21no—
108C2×C4×C157—42no—
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2411yescharacteristic
112C8×C157—22no—
113C22×C157—41no—
114C22×C157—42no—
115C22×C157—41no—
116C22×C157—42no—
117C22×C157—42no—
118C22×C157—41no—
119C22×C157C22×D411yescharacteristic
120C2×D157—22no—
121C2×D157—22no—
122C2×D157—22no—
123C2×D157—41no—
124C2×D157—41no—
125C2×D157—88no—
126C2×D157C2×D814yes—
127C4×C157—21no—
128C4×C157—41no—
129C4×C157C2×D812yes—
130C4×C157C2×#4111yescharacteristic
131C4×C157H(5,𝔽2)11yescharacteristic
132Dic157—22no—
133Dic157—41no—
134Dic157—42no—
135C2×C157—82no—
136C2×C157—81no—
137C2×C157#54611yescharacteristic
138C2×C157C22×D811yescharacteristic
139C2×C157#57711yescharacteristic
140D157—24no—
141D157—42no—
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#459D15711yescharacteristic, complemented
151#447—1572no—
152C4×D8—1571no—
153C8×D4—1571no—
154C2×#102—1572no—
155C2×#41—3141no—
156C22×D4—3141no—
157C22×D4—1572no—
158C2×D8—1571no—
159C2×D8C2×D15712yescomplemented
160C22×C8—1572no—
161C4⋊Q8—1571no—
162#127C2×D15711yescharacteristic
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, complemented
172C2×C4⋊C4—1572no—
173#107—1572no—
174#107—1571no—
175C4⋊C8C2×D15711yescharacteristic, complemented
176#102—1572no—
177#102—1571no—
178#102—1572no—
179#102—1572no—
180#102—1571no—
181#102C2×D15712yescomplemented
182C22⋊C8—1572no—
183C4×C8—1571no—
184C24—3142no—
185#41—6284no—
186C2×Q8—3141no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—3144no—
190C2×D4—3142no—
191C2×D4—3142no—
192C2×D4—3142no—
193C2×D4—6282no—
194C2×D4—6284no—
195C2×D4—1571no—
196C2×D4—1571no—
197C2×D4C22×D15712yes—
198C2×D4C22×D15711yescharacteristic
199C22×C4—21no—
200C22×C4—3142no—
201C22×C4—3141no—
202C22×C4—1572no—
203D8—42no—
204D8—3142no—
205C2×C8—6282no—
206C2×C8—1571no—
207C2×C8—1571no—
208C2×C8C22×D15712yes—
209C4⋊C4—3142no—
210C4⋊C4—3141no—
211C4⋊C4—1572no—
212C4⋊C4—1571no—
213C4⋊C4—1571no—
214C4⋊C4C22×D15711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—3142no—
217C22⋊C4—3142no—
218C22⋊C4—3141no—
219C22⋊C4—1572no—
220C42C22×D15711yescharacteristic
221C23—21no—
222C23—22no—
223C23—3141no—
224C23—6284no—
225C23—6284no—
226C23—1572no—
227C23—12562no—
228Q8—6282no—
229D4—21no—
230D4—22no—
231D4—21no—
232D4—22no—
233D4—42no—
234D4—81no—
235D4—3141no—
236D4—6284no—
237D4—6284no—
238D4—6284no—
239D4—1574no—
240D4—1572no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—3142no—
244C2×C4—3142no—
245C2×C4—3141no—
246C2×C4—3142no—
247C2×C4—6284no—
248C2×C4—6282no—
249C2×C4—1572no—
250C2×C4D4×D15711yescharacteristic
251C2×C4D4×D15711yescharacteristic
252C2×C4C23×D15711yescharacteristic
253C8—22no—
254C8—3141no—
255C8—3141no—
256C22—42no—
257C22—41no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—3142no—
263C22—3142no—
264C22—3142no—
265C22—6281no—
266C22—6281no—
267C22—1574no—
268C22—12568no—
269C22C2×D4×D15711yescharacteristic
270C4—21no—
271C4—41no—
272C4—3142no—
273C4—6281no—
274C4—6282no—
275C4D8×D15712yes—
276C4#41×D15711yescharacteristic
277C4#1909781611yescharacteristic
278C2—82no—
279C2—81no—
280C2—3144no—
281C2—6282no—
282C2#2375792011yescharacteristic
283C2C2×D8×D15711yescharacteristic
284C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#459×C157C2
socleC22×C157C22×D4
hypercenter#459D157
hypocenterC157#3294
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #23757928C2semidirect
2. #23757908C2semidirect
3. #23757619C2semidirect
4. #23757585C2semidirect
5. #23757554C2semidirect
6. #23757506C2semidirect
7. #23757405C2semidirect
8. C4×#14578939C2semidirect
9. #23757338C2semidirect
10. #102×D157C2semidirect
11. #459×C157C2semidirect
16. C2×C8×D157C22semidirect
17. #19097771C22semidirect
22. #19097742C22semidirect
26. C4⋊C4×D157C22semidirect
27. #19097705C22semidirect
28. #19097704C22semidirect
30. #19097692C22semidirect
31. #19097692C22semidirect
32. #19097692C22semidirect
34. #19097683C22semidirect
35. C4×#11929434C22semidirect
36. C2×D8×C157C22semidirect
38. C4×D4×C157C22semidirect
39. C4⋊C8×C157C22semidirect
40. #102×C157C22semidirect
85. D628D8semidirect
86. D628#41semidirect
142. C157#3294semidirect
150. #459D157semidirect
159. C2×D8C2×D157semidirect
171. C4×D4C2×D157semidirect
175. C4⋊C8C2×D157semidirect
181. #102C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237579282
#237579082
#237576192
#237575852
#237575542
#237575062
#237574052
C4×#145789392
#237573382
#102×D1572
#459×C1572
#3294157