Finite Group Databasebeta

← #34109382

Subgroups — #34109382

Order 20096.

758 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 758 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C22×#14578955C211yesmaximal, characteristic, complemented
2C2×#19097802C211yesmaximal, characteristic, complemented
3C2×#19097799C211yesmaximal, characteristic, complemented
4#23757785C211yesmaximal, characteristic, complemented
5#23757438C214yesmaximal, complemented
6#23757421C214yesmaximal, complemented
7#23757277C211yesmaximal, characteristic, complemented
8#23757236C211yesmaximal, characteristic, complemented
9#419×C157C211yesmaximal, characteristic, complemented
10C2×#14578955—24no—
11C2×#14578955—24no—
12C2×#14578955—22no—
13C2×#14578955C2214yes—
14C2×#14578954C2211yescharacteristic
15C22×D628—21no—
16C22×D628C2211yescharacteristic
17C22×C4×D157—21no—
18C22×C4×D157C2211yescharacteristic
19#19097802—24no—
20#19097799—24no—
21C2×#14578944C2212yescomplemented
22C2×#14578943C2212yescomplemented
23C2×#14578941C2212yescomplemented
24C2×D1256C2212yescomplemented
25C2×#14578938C2211yescharacteristic
26#19097755—21no—
27#19097751C2212yescomplemented
28#19097746C2212yescomplemented
29#19097741—21no—
30C4×D628—22no—
31#19097691C2214yescomplemented
32#19097683C2214yescomplemented
33C22×Q8×C157C2211yescharacteristic
34C2×M16×C157C2211yescharacteristic
35#117×C157C2211yescharacteristic
36#103×C157C2214yescomplemented
37C23×D157—21no—
38C23×D157D411yescharacteristic
39#14578955—216no—
40#14578955—28no—
41#14578955—28no—
42#14578954—22no—
43#14578954—22no—
44C2×Dic314C2311yescharacteristic
45C2×D628—28no—
46C2×D628—24no—
47C2×D628—22no—
48C2×D628—21no—
49C2×D628—22no—
50C2×D628D414yescomplemented
51C2×D628C2311yescharacteristic
52C2×D628C2312yes—
53C2×C22⋊D157—21no—
54C22×Dic157D411yescharacteristic
55C2×C4×D157—28no—
56C2×C4×D157—24no—
57C2×C4×D157—22no—
58C2×C4×D157—22no—
59C2×C4×D157—21no—
60C2×C4×D157D412yes—
61C2×C4×D157C2312yes—
62#14578944—24no—
63#14578943—24no—
64#14578941—24no—
65D1256—24no—
66#14578938—22no—
67C2×#11929434C2312yes—
68C4⋊Dic157—22no—
69#14578931—22no—
70#14578931—22no—
71C2×Q8×C157—22no—
72C2×Q8×C157—21no—
73C2×Q8×C157D414yescomplemented
74C2×Q8×C157C2312yes—
75C22×C4×C157—21no—
76C22×C4×C157C2311yescharacteristic
77M16×C157—22no—
78C2×C8×C157C2312yes—
79C4⋊C4×C157C2312yes—
80C22⋊C4×C157—21no—
81C42×C157—21no—
82C22×D157—22no—
83C22×D157—24no—
84C22×D157—22no—
85C22×D157—21no—
86C22×D157—42no—
87C22×D157—44no—
88C22×D157C2×D412yes—
89Dic314—21no—
90Dic314—21no—
91D628—216no—
92D628—24no—
93D628—22no—
94D628—21no—
95D628—21no—
96D628—44no—
97D628C2×D414yes—
98C22⋊D157—42no—
99C2×Dic157—22no—
100C2×Dic157—21no—
101C2×Dic157C2×D412yes—
102C4×D157—216no—
103C4×D157—24no—
104C4×D157—22no—
105C4×D157—22no—
106C4×D157—42no—
107#11929434—22no—
108C23×C157C2×D411yescharacteristic
109Q8×C157—24no—
110Q8×C157—22no—
111Q8×C157C2×D414yes—
112C2×C4×C157—24no—
113C2×C4×C157—22no—
114C2×C4×C157—22no—
115C2×C4×C157C2×D411yescharacteristic
116C2×C4×C157C2×D412yes—
117C2×C4×C157C2×D412yes—
118C2×C4×C157C2411yescharacteristic
119C8×C157—22no—
120C22×C157—22no—
121C22×C157#12012yes—
122C22×C157C22×D411yescharacteristic
123C2×D157—22no—
124C2×D157—24no—
125C2×D157—22no—
126C2×D157—22no—
127C2×D157—44no—
128C2×D157—48no—
129C2×D157—44no—
130C2×D157—41no—
131C2×D157—42no—
132C4×C157—24no—
133C4×C157—42no—
134C4×C157#12012yes—
135C4×C157C22×D411yescharacteristic
136C4×C157C22×D411yescharacteristic
137Dic157—22no—
138Dic157—41no—
139C2×C157—22no—
140C2×C157C2×#12011yescharacteristic
141C2×C157#57512yes—
142D157—42no—
143D157—44no—
144D157—81no—
145C157#300511yescharacteristic, complemented, Sylow
146#3005—1571nomaximal, Sylow
147C22×#41—1571no—
148C2×#137—1571no—
149C2×AGL(1,ℤ/8)—1571no—
150#525—1571no—
151#452—1574no—
152#449—1574no—
153#419D15711yescharacteristic, complemented
154#418—1571no—
155#407—1571no—
156C2×#41—3144no—
157C2×#41—3144no—
158C2×#41—3142no—
159C2×#41—1574no—
160C2×#41—1571no—
161C22×Q8C2×D15711yescharacteristic
162C22×D4—3141no—
163C22×D4—1571no—
164C23×C4—3141no—
165C23×C4—1571no—
166#137—3144no—
167AGL(1,ℤ/8)—3144no—
168C2×Q16—1572no—
169C2×SD16—1572no—
170C2×SD16—1572no—
171C2×D8—1572no—
172C2×M16—1571no—
173C2×M16C2×D15711yescharacteristic
174#123—3141no—
175C22⋊Q8—1572no—
176#121—1572no—
177#120—3141no—
178C4×D4—3142no—
179#117C2×D15711yescharacteristic
180#103C2×D15714yescomplemented
181#102—1574no—
182C22⋊C8—1574no—
183C24—3141no—
184C24—1571no—
185#41—31416no—
186#41—3148no—
187#41—3148no—
188#41—3142no—
189#41—3142no—
190C2×Q8—22no—
191C2×Q8—21no—
192C2×Q8—1571no—
193C2×Q8D62814yescomplemented
194C2×Q8C22×D15712yes—
195C2×D4—3148no—
196C2×D4—3144no—
197C2×D4—3142no—
198C2×D4—3141no—
199C2×D4—3141no—
200C2×D4—3142no—
201C2×D4—1574no—
202C2×D4—1571no—
203C2×D4—1572no—
204C22×C4—21no—
205C22×C4—3148no—
206C22×C4—3144no—
207C22×C4—3142no—
208C22×C4—3142no—
209C22×C4—3141no—
210C22×C4—1571no—
211C22×C4—1572no—
212C22×C4—1572no—
213C22×C4C22×D15711yescharacteristic
214Q16—3144no—
215SD16—3144no—
216SD16—3144no—
217D8—3144no—
218M16—22no—
219M16—3142no—
220C2×C8—1572no—
221C2×C8C22×D15712yes—
222C4⋊C4—3142no—
223C4⋊C4C22×D15712yes—
224C22⋊C4—21no—
225C22⋊C4—3142no—
226C22⋊C4—3142no—
227C42—21no—
228C23—3142no—
229C23—3144no—
230C23—3142no—
231C23—3141no—
232C23—6282no—
233C23—6284no—
234C23—1572no—
235C23D4×D15711yescharacteristic
236Q8—24no—
237Q8—22no—
238Q8—3141no—
239Q8—3141no—
240Q8C2×D62814yes—
241D4—31416no—
242D4—3144no—
243D4—3141no—
244D4—3142no—
245D4—3141no—
246D4—6284no—
247D4—6282no—
248D4—1574no—
249C2×C4—24no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—31416no—
253C2×C4—3142no—
254C2×C4—3144no—
255C2×C4—3142no—
256C2×C4—3142no—
257C2×C4—3141no—
258C2×C4—6282no—
259C2×C4—1572no—
260C2×C4D4×D15711yescharacteristic
261C2×C4D4×D15712yes—
262C2×C4C2×D62812yes—
263C2×C4C23×D15711yescharacteristic
264C8—22no—
265C8—3142no—
266C22—22no—
267C22—3142no—
268C22—3144no—
269C22—3142no—
270C22—3142no—
271C22—6284no—
272C22—6288no—
273C22—6284no—
274C22—6281no—
275C22—6282no—
276C22#1909774112yes—
277C22C2×D4×D15711yescharacteristic
278C4—24no—
279C4—42no—
280C4—3142no—
281C4—6281no—
282C4#1909774112yes—
283C4C2×D4×D15711yescharacteristic
284C4C22×D62811yescharacteristic
285C2—22no—
286C2—6282no—
287C2—6284no—
288C2—12561no—
289C2C2×#1909774111yescharacteristic
290C2#2375807212yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#419×C157C2
socleC22×C157C22×D4
hypercenter#419D157
hypocenterC157#3005
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14578955C2semidirect
2. C2×#19097802C2semidirect
3. C2×#19097799C2semidirect
4. #23757785C2semidirect
5. #23757438C2semidirect
6. #23757421C2semidirect
7. #23757277C2semidirect
8. #23757236C2semidirect
9. #419×C157C2semidirect
21. C2×#14578944C22semidirect
22. C2×#14578943C22semidirect
23. C2×#14578941C22semidirect
24. C2×D1256C22semidirect
27. #19097751C22semidirect
28. #19097746C22semidirect
31. #19097691C22semidirect
32. #19097683C22semidirect
36. #103×C157C22semidirect
50. C2×D628D4semidirect
73. C2×Q8×C157D4semidirect
145. C157#3005semidirect
153. #419D157semidirect
180. #103C2×D157semidirect
193. C2×Q8D628semidirect

Maximal subgroups

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

subgroupindex
C22×#145789552
C2×#190978022
C2×#190977992
#237577852
#237574382
#237574212
#237572772
#237572362
#419×C1572
#3005157