Finite Group Databasebeta

← #34111732

Subgroups — #34111732

Order 20096.

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

normal subgroups only

This group has 546 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#23757928C211yesmaximal, characteristic, complemented
3#23757645C211yesmaximal, characteristic, complemented
4#23757517C212yesmaximal, complemented
5#23757478C211yesmaximal, characteristic, complemented
6#23757465C211yesmaximal, characteristic, complemented
7#23757424C212yesmaximal, complemented
8#23757416C212yesmaximal, complemented
9C4×#14578939C211yesmaximal, characteristic, complemented
10#23757351C211yesmaximal, characteristic, complemented
11D42×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2212yescomplemented
13#19097797—28no—
14C2×#14578942C2212yes—
15C2×#14578942C2212yescomplemented
16C2×#14578942C2212yescomplemented
17C2×#14578939C2211yescharacteristic
18C2×#14578939C2212yescomplemented
19C2×#14578938C2212yescomplemented
20#19097771C2211yescharacteristic, complemented
21#19097753—22no—
22#19097750C2212yescomplemented
23#19097742—21no—
24C4×D628C2211yescharacteristic, complemented
25C4⋊C4×D157—21no—
26#19097704C2211yescharacteristic, complemented
27C4⋊#11929434C2211yescharacteristic, complemented
28#19097694C2212yescomplemented
29#19097692C2211yescharacteristic
30#19097692C2212yescomplemented
31#19097691C2211yescharacteristic, complemented
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×Dic314C2312yes—
45C2×D628C2311yescharacteristic
46C2×C22⋊D157—22no—
47C2×C4×D157—21no—
48C2×C4×D157—22no—
49#14578942—24no—
50#14578942—24no—
51#14578942—42no—
52#14578939—24no—
53#14578939D414yescomplemented
54#14578938—24no—
55C2×#11929434C2311yescharacteristic
56C2×#11929434C2312yes—
57C2×#11929434C2311yescharacteristic
58C4⋊Dic157—21no—
59C4⋊Dic157—21no—
60C4⋊Dic157C2311yescharacteristic
61#14578933—22no—
62C22⋊Dic157—22no—
63#14578931—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—21no—
84Dic314—22no—
85Dic314—22no—
86D628—21no—
87D628C2×D412yescomplemented
88C22⋊D157—44no—
89C2×Dic157—22no—
90C2×Dic157—21no—
91C4×D157—44no—
92C4×D157—42no—
93#11929434—22no—
94#11929434—21no—
95#11929434C2×D412yescomplemented
96C23×C157—21no—
97C23×C157—22no—
98C23×C157—44no—
99C23×C157—44no—
100C23×C157—82no—
101C23×C157C2×D412yes—
102D4×C157—22no—
103D4×C157—22no—
104D4×C157—21no—
105D4×C157—22no—
106D4×C157—44no—
107D4×C157—44no—
108D4×C157—42no—
109D4×C157—81no—
110D4×C157C2×D412yes—
111C2×C4×C157—22no—
112C2×C4×C157—22no—
113C2×C4×C157—21no—
114C2×C4×C157—42no—
115C2×C4×C157C2×D412yes—
116C2×C4×C157C2×D411yescharacteristic
117C2×C4×C157C2×D411yescharacteristic
118C2×C4×C157C2411yescharacteristic
119C22×C157—22no—
120C22×C157—22no—
121C22×C157—22no—
122C22×C157—22no—
123C22×C157—41no—
124C22×C157—41no—
125C22×C157—42no—
126C22×C157—42no—
127C22×C157—42no—
128C22×C157—88no—
129C22×C157C22×D411yescharacteristic
130C2×D157—22no—
131C2×D157—41no—
132C2×D157—41no—
133C4×C157—22no—
134C4×C157—21no—
135C4×C157—41no—
136C4×C157AGL(1,ℤ/8)12yescomplemented
137C4×C157C22×D411yescharacteristic
138C4×C157H(5,𝔽2)11yescharacteristic
139Dic157—42no—
140Dic157—41no—
141C2×C157—42no—
142C2×C157—42no—
143C2×C157—82no—
144C2×C157D4211yescharacteristic
145C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
146C2×C157#57711yescharacteristic
147D157—42no—
148C157#327211yescharacteristic, complemented, Sylow
149#3272—1571nomaximal, Sylow
150C2×AGL(1,ℤ/8)—1572no—
151#547—1571no—
152D42D15711yescharacteristic, complemented
153#492—1571no—
154#468—1572no—
155#461—1571no—
156#459—1571no—
157#450—1572no—
158#449—1572no—
159C4×D8—1571no—
160#436—1571no—
161C2×#41—1572no—
162C22×D4—21no—
163C22×D4C2×D15712yes—
164AGL(1,ℤ/8)—3148no—
165C2×SD16—1572no—
166C2×SD16—1572no—
167C2×SD16—1572no—
168C2×D8—1572no—
169C2×D8—1571no—
170C2×M16—1572no—
171C4⋊Q8—1571no—
172#127C2×D15711yescharacteristic
173#123—3142no—
174C22⋊Q8—1572no—
175#121—21no—
176#121—3141no—
177#121C2×D15712yes—
178#120—22no—
179C4×D4—1571no—
180C4×D4C2×D15711yescharacteristic
181C4×D4C2×D15711yescharacteristic
182C2×C4⋊C4—3141no—
183#107—1571no—
184C4⋊C8—1571no—
185#103—1572no—
186#102—1571no—
187#102—1572no—
188#102—1571no—
189C22⋊C8—1572no—
190C4×C8—1571no—
191C24—22no—
192#41—3144no—
193#41—6282no—
194C2×Q8—1572no—
195C2×D4—24no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C2×D4—3142no—
203C2×D4—1571no—
204C2×D4C22⋊D15711yescharacteristic
205C2×D4C22×D15712yes—
206C2×D4C22×D15711yescharacteristic
207C22×C4—21no—
208C22×C4—3141no—
209C22×C4—3142no—
210C22×C4C22×D15712yes—
211SD16—3144no—
212SD16—3144no—
213SD16—6282no—
214D8—3144no—
215D8—1574no—
216M16—3144no—
217C2×C8—1571no—
218C2×C8—1572no—
219C2×C8—1571no—
220C4⋊C4—3141no—
221C4⋊C4—3142no—
222C4⋊C4—3141no—
223C4⋊C4—1571no—
224C4⋊C4C22⋊D15711yescharacteristic
225C4⋊C4C22×D15711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—22no—
228C22⋊C4—3142no—
229C22⋊C4—3141no—
230C22⋊C4C22⋊D15712yes—
231C42C22×D15711yescharacteristic
232C23—21no—
233C23—22no—
234C23—44no—
235C23—44no—
236C23—82no—
237C23—3141no—
238C23C2×C22⋊D15712yes—
239Q8—3142no—
240Q8—3142no—
241D4—22no—
242D4—22no—
243D4—21no—
244D4—22no—
245D4—44no—
246D4—44no—
247D4—42no—
248D4—81no—
249D4—3141no—
250D4—6284no—
251D4—1572no—
252D4D4×D15712yes—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—3141no—
258C2×C4—3142no—
259C2×C4—6284no—
260C2×C4—6282no—
261C2×C4C2×C22⋊D15711yescharacteristic
262C2×C4C2×C22⋊D15712yes—
263C2×C4C2×C22⋊D15711yescharacteristic
264C2×C4C23×D15711yescharacteristic
265C8—3142no—
266C8—3141no—
267C8—1572no—
268C22—22no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—88no—
278C22—3142no—
279C22—6281no—
280C22—6281no—
281C22C22×C22⋊D15711yescharacteristic
282C4—21no—
283C4—22no—
284C4—41no—
285C4—6282no—
286C4—6281no—
287C4#1909779712yescomplemented
288C4C2×D4×D15711yescharacteristic
289C4#1909781611yescharacteristic
290C2—42no—
291C2—42no—
292C2—82no—
293C2—6282no—
294C2D4×C22⋊D15711yescharacteristic
295C2C2×#1909779711yescharacteristic
296C2#2375808411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingD42×C157C2
socleC22×C157C22×D4
hypercenterD42D157
hypocenterC157#3272
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. #23757928C2semidirect
3. #23757645C2semidirect
4. #23757517C2semidirect
5. #23757478C2semidirect
6. #23757465C2semidirect
7. #23757424C2semidirect
8. #23757416C2semidirect
9. C4×#14578939C2semidirect
10. #23757351C2semidirect
11. D42×C157C2semidirect
12. C2×#14578954C22semidirect
15. C2×#14578942C22semidirect
16. C2×#14578942C22semidirect
18. C2×#14578939C22semidirect
19. C2×#14578938C22semidirect
20. #19097771C22semidirect
22. #19097750C22semidirect
24. C4×D628C22semidirect
26. #19097704C22semidirect
27. C4⋊#11929434C22semidirect
28. #19097694C22semidirect
30. #19097692C22semidirect
31. #19097691C22semidirect
32. C22⋊#11929434C22semidirect
33. C4×#11929434C22semidirect
53. #14578939D4semidirect
87. D628C2×D4semidirect
95. #11929434C2×D4semidirect
136. C4×C157AGL(1,ℤ/8)semidirect
148. C157#3272semidirect
152. D42D157semidirect
287. C4#19097797semidirect

Maximal subgroups

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

subgroupindex
C2×#190977972
#237579282
#237576452
#237575172
#237574782
#237574652
#237574242
#237574162
C4×#145789392
#237573512
D42×C1572
#3272157