Finite Group Databasebeta

← #34111752

Subgroups — #34111752

Order 20096.

556 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C2×#19097797C212yesmaximal, complemented
2#23757944C211yesmaximal, characteristic, complemented
3#23757649C211yesmaximal, characteristic, complemented
4#23757524C212yesmaximal, complemented
5#23757470C211yesmaximal, characteristic, complemented
6#23757468C211yesmaximal, characteristic, complemented
7#23757416C212yesmaximal, complemented
8#23757407C212yesmaximal, complemented
9C4×#14578942C211yesmaximal, characteristic, complemented
10#23757351C211yesmaximal, characteristic, complemented
11D42×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2212yescomplemented
13#19097797—28no—
14C2×#14578942C2211yescharacteristic
15C2×#14578942C2212yescomplemented
16C2×#14578939C2212yes—
17C2×#14578939C2212yescomplemented
18C2×#14578939C2212yescomplemented
19C2×#14578938C2212yescomplemented
20#19097768C2211yescharacteristic, complemented
21#19097767—21no—
22#19097745C2212yescomplemented
23#19097744—22no—
24C4×Dic314C2211yescharacteristic, complemented
25#19097733—21no—
26#19097702C2211yescharacteristic, complemented
27C4⋊#11929434C2211yescharacteristic, complemented
28#19097694C2211yescharacteristic, complemented
29#19097692C2211yescharacteristic
30#19097692C2212yescomplemented
31#19097691C2212yescomplemented
32C22⋊#11929434C2212yescomplemented
33C4×#11929434C2211yescharacteristic, complemented
34C22×D4×C157—21no—
35C22×D4×C157C2212yes—
36#127×C157C2211yescharacteristic
37#121×C157—21no—
38#121×C157C2212yes—
39#120×C157—22no—
40C4×D4×C157C2211yescharacteristic
41C4×D4×C157C2211yescharacteristic
42#14578954—24no—
43#14578954—42no—
44C2×Dic314C2311yescharacteristic
45C2×D628—21no—
46C2×D628—21no—
47C2×D628C2312yes—
48C2×C22⋊D157—22no—
49C2×C22⋊D157—22no—
50C2×C4×D157—22no—
51#14578942—24no—
52#14578942D414yescomplemented
53#14578939—24no—
54#14578939—24no—
55#14578939—42no—
56#14578938—24no—
57C2×#11929434C2311yescharacteristic
58C2×#11929434C2312yes—
59C2×#11929434C2311yescharacteristic
60C4⋊Dic157C2311yescharacteristic
61#14578933—21no—
62#14578931—22no—
63C4×Dic157—21no—
64C24×C157—22no—
65C2×D4×C157—24no—
66C2×D4×C157—22no—
67C2×D4×C157—22no—
68C2×D4×C157—21no—
69C2×D4×C157—21no—
70C2×D4×C157—42no—
71C2×D4×C157—44no—
72C2×D4×C157D411yescharacteristic
73C2×D4×C157C2312yes—
74C2×D4×C157C2311yescharacteristic
75C22×C4×C157—21no—
76C22×C4×C157C2312yes—
77C4⋊C4×C157D411yescharacteristic
78C4⋊C4×C157C2311yescharacteristic
79C22⋊C4×C157—21no—
80C22⋊C4×C157—22no—
81C22⋊C4×C157D412yes—
82C42×C157C2311yescharacteristic
83C22×D157—22no—
84Dic314—21no—
85Dic314C2×D412yescomplemented
86D628—22no—
87D628—22no—
88D628—42no—
89D628—81no—
90C22⋊D157—44no—
91C22⋊D157—82no—
92C2×Dic157—21no—
93C2×Dic157—21no—
94C4×D157—44no—
95#11929434—22no—
96#11929434—21no—
97#11929434C2×D412yescomplemented
98C23×C157—21no—
99C23×C157—22no—
100C23×C157—44no—
101C23×C157—44no—
102C23×C157—82no—
103C23×C157C2×D412yes—
104D4×C157—22no—
105D4×C157—22no—
106D4×C157—21no—
107D4×C157—22no—
108D4×C157—44no—
109D4×C157—44no—
110D4×C157—42no—
111D4×C157—81no—
112D4×C157C2×D412yes—
113C2×C4×C157—22no—
114C2×C4×C157—22no—
115C2×C4×C157—21no—
116C2×C4×C157—42no—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2×D411yescharacteristic
119C2×C4×C157C2×D411yescharacteristic
120C2×C4×C157C2411yescharacteristic
121C22×C157—22no—
122C22×C157—22no—
123C22×C157—22no—
124C22×C157—22no—
125C22×C157—41no—
126C22×C157—41no—
127C22×C157—42no—
128C22×C157—42no—
129C22×C157—42no—
130C22×C157—88no—
131C22×C157C22×D411yescharacteristic
132C2×D157—42no—
133C2×D157—42no—
134C2×D157—42no—
135C4×C157—22no—
136C4×C157—21no—
137C4×C157—41no—
138C4×C157AGL(1,ℤ/8)12yescomplemented
139C4×C157C22×D411yescharacteristic
140C4×C157H(5,𝔽2)11yescharacteristic
141Dic157—22no—
142Dic157—41no—
143C2×C157—42no—
144C2×C157—42no—
145C2×C157—82no—
146C2×C157D4211yescharacteristic
147C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
148C2×C157#57611yescharacteristic
149D157—82no—
150C157#327411yescharacteristic, complemented, Sylow
151#3274—1571nomaximal, Sylow
152C2×AGL(1,ℤ/8)—1572no—
153#550—1571no—
154D42D15711yescharacteristic, complemented
155#493—1571no—
156#469—1572no—
157#460—1571no—
158#459—1571no—
159#449—1572no—
160#447—1572no—
161C4×SD16—1571no—
162#436—1571no—
163C2×#41—1572no—
164C22×D4—21no—
165C22×D4C2×D15712yes—
166AGL(1,ℤ/8)—3148no—
167C2×SD16—1572no—
168C2×SD16—1571no—
169C2×D8—1572no—
170C2×D8—1572no—
171C2×D8—1572no—
172C2×M16—1572no—
173#127—3141no—
174#127—1571no—
175#127C2×D15711yescharacteristic
176#121—21no—
177#121—3142no—
178#121—1572no—
179#121C2×D15712yes—
180#120—22no—
181C4×Q8—1571no—
182C4×D4—3141no—
183C4×D4C2×D15711yescharacteristic
184C4×D4C2×D15711yescharacteristic
185#106—1571no—
186C4⋊C8—1571no—
187#103—1571no—
188#102—1571no—
189#102—1572no—
190#102—1572no—
191C22⋊C8—1572no—
192C4×C8—1571no—
193C24—22no—
194#41—3144no—
195#41—6282no—
196C2×Q8—1571no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—42no—
203C2×D4—44no—
204C2×D4—3142no—
205C2×D4—3142no—
206C2×D4—3141no—
207C2×D4—3141no—
208C2×D4—1572no—
209C2×D4C22⋊D15711yescharacteristic
210C2×D4C22×D15712yes—
211C2×D4C22×D15711yescharacteristic
212C22×C4—21no—
213C22×C4—3142no—
214C22×C4C22×D15712yes—
215SD16—3144no—
216SD16—1574no—
217D8—3144no—
218D8—3144no—
219D8—6282no—
220M16—3144no—
221C2×C8—1571no—
222C2×C8—1572no—
223C2×C8—1571no—
224C4⋊C4—3141no—
225C4⋊C4—1571no—
226C4⋊C4C22⋊D15711yescharacteristic
227C4⋊C4C22×D15711yescharacteristic
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—3142no—
231C22⋊C4C22⋊D15712yes—
232C42—3141no—
233C42C22×D15711yescharacteristic
234C23—21no—
235C23—22no—
236C23—44no—
237C23—44no—
238C23—82no—
239C23—3142no—
240C23C2×C22⋊D15712yes—
241Q8—3141no—
242Q8—1572no—
243D4—22no—
244D4—22no—
245D4—21no—
246D4—22no—
247D4—44no—
248D4—44no—
249D4—42no—
250D4—81no—
251D4—3142no—
252D4—3142no—
253D4—6284no—
254D4—6282no—
255D4—12561no—
256D4—12562no—
257D4D4×D15712yes—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—42no—
262C2×C4—3141no—
263C2×C4—3141no—
264C2×C4—6284no—
265C2×C4C2×C22⋊D15711yescharacteristic
266C2×C4C2×C22⋊D15712yes—
267C2×C4C2×C22⋊D15711yescharacteristic
268C2×C4C23×D15711yescharacteristic
269C8—3142no—
270C8—3141no—
271C8—1572no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—88no—
282C22—6282no—
283C22—6282no—
284C22—6282no—
285C22C22×C22⋊D15711yescharacteristic
286C4—22no—
287C4—21no—
288C4—41no—
289C4—3142no—
290C4—6281no—
291C4#1909779712yescomplemented
292C4C2×D4×D15711yescharacteristic
293C4#1909781611yescharacteristic
294C2—42no—
295C2—42no—
296C2—82no—
297C2—12562no—
298C2D4×C22⋊D15711yescharacteristic
299C2C2×#1909779711yescharacteristic
300C2#2375807611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingD42×C157C2
socleC22×C157C22×D4
hypercenterD42D157
hypocenterC157#3274
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#19097797C2semidirect
2. #23757944C2semidirect
3. #23757649C2semidirect
4. #23757524C2semidirect
5. #23757470C2semidirect
6. #23757468C2semidirect
7. #23757416C2semidirect
8. #23757407C2semidirect
9. C4×#14578942C2semidirect
10. #23757351C2semidirect
11. D42×C157C2semidirect
12. C2×#14578954C22semidirect
15. C2×#14578942C22semidirect
17. C2×#14578939C22semidirect
18. C2×#14578939C22semidirect
19. C2×#14578938C22semidirect
20. #19097768C22semidirect
22. #19097745C22semidirect
24. C4×Dic314C22semidirect
26. #19097702C22semidirect
27. C4⋊#11929434C22semidirect
28. #19097694C22semidirect
30. #19097692C22semidirect
31. #19097691C22semidirect
32. C22⋊#11929434C22semidirect
33. C4×#11929434C22semidirect
52. #14578942D4semidirect
85. Dic314C2×D4semidirect
97. #11929434C2×D4semidirect
138. C4×C157AGL(1,ℤ/8)semidirect
150. C157#3274semidirect
154. D42D157semidirect
291. C4#19097797semidirect

Maximal subgroups

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

subgroupindex
C2×#190977972
#237579442
#237576492
#237575242
#237574702
#237574682
#237574162
#237574072
C4×#145789422
#237573512
D42×C1572
#3274157