Finite Group Databasebeta

← #469×D157 #34109728

Subgroups — #469×D157 #34109728

Order 20096.

596 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 135 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×D8×D157C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic, complemented
3#121×D157C212yesmaximal, complemented
4#23757527C211yesmaximal, characteristic, complemented
5#23757526C211yesmaximal, characteristic, complemented
6#23757525C212yesmaximal, complemented
7#23757524C212yesmaximal, complemented
8#23757523C211yesmaximal, characteristic, complemented
9#106×D157C211yesmaximal, characteristic, complemented
10#102×D157C212yesmaximal, complemented
11#469×C157C211yesmaximal, characteristic, complemented
12C2×D4×D157—22no—
13C2×D4×D157C2212yes—
14C22×C4×D157C2211yescharacteristic
15C2×D1256C2211yescharacteristic, complemented
16C2×#14578939C2212yescomplemented
17D8×D157—24no—
18C2×#14578938C2211yescharacteristic, complemented
19C2×#14578937C2211yescharacteristic, complemented
20C2×#14578937C2211yescharacteristic, complemented
21M16×D157—24no—
22C2×C8×D157C2211yescharacteristic
23C2×C8×D157C2211yescharacteristic
24#19097746C2212yescomplemented
25#19097745C2212yescomplemented
26#19097744—22no—
27#19097743—22no—
28#19097742C2212yescomplemented
29C4⋊C4×D157C2212yes—
30C22⋊C4×D157—22no—
31#19097703C2211yescharacteristic, complemented
32#19097702C2212yescomplemented
33#19097693C2212yescomplemented
34#19097692C2212yescomplemented
35#19097691C2212yescomplemented
36C2×D8×C157C2211yescharacteristic, complemented
37C2×M16×C157C2211yescharacteristic, complemented
38#121×C157C2212yescomplemented
39#106×C157C2211yescharacteristic, complemented
40#102×C157C2212yescomplemented
41C23×D157—22no—
42C23×D157D411yescharacteristic
43C2×D628—22no—
44C2×D628C2312yes—
45C2×C22⋊D157—22no—
46C2×C22⋊D157—22no—
47C2×C22⋊D157—22no—
48D4×D157—24no—
49D4×D157—24no—
50D4×D157—48no—
51C22×Dic157D411yescharacteristic
52C2×C4×D157—22no—
53C2×C4×D157—22no—
54C2×C4×D157—22no—
55C2×C4×D157D411yescharacteristic
56C2×C4×D157D411yescharacteristic
57C2×C4×D157C2311yescharacteristic
58C2×C4×D157C2311yescharacteristic
59D1256—22no—
60#14578939—24no—
61#14578938—22no—
62#14578937—22no—
63#14578937—22no—
64C2×#11929434C2311yescharacteristic
65C2×#11929434C2311yescharacteristic
66C8×D157—22no—
67C8×D157D414yescomplemented
68C4⋊Dic157C2312yes—
69#14578933—22no—
70C22⋊Dic157—22no—
71#14578931—22no—
72#14578931—22no—
73C2×D4×C157—22no—
74C2×D4×C157C2312yes—
75C22×C4×C157C2311yescharacteristic
76D8×C157—22no—
77M16×C157—22no—
78C2×C8×C157C2311yescharacteristic
79C2×C8×C157C2311yescharacteristic
80C4⋊C4×C157C2312yes—
81C22⋊C4×C157—22no—
82C22×D157—22no—
83C22×D157—24no—
84C22×D157—22no—
85C22×D157—44no—
86C22×D157—44no—
87C22×D157—44no—
88C22×D157C2×D411yescharacteristic
89C22×D157C2×D411yescharacteristic
90D628—22no—
91D628—22no—
92D628—44no—
93C22⋊D157—44no—
94C22⋊D157—44no—
95C22⋊D157—44no—
96C2×Dic157—22no—
97C2×Dic157—22no—
98C2×Dic157C2×D411yescharacteristic
99C2×Dic157C2×D411yescharacteristic
100C4×D157—22no—
101C4×D157—21no—
102C4×D157—21no—
103C4×D157—22no—
104C4×D157—44no—
105C4×D157C2×D412yes—
106C4×D157#4112yes—
107#11929434—21no—
108#11929434C2×D412yescomplemented
109C23×C157—22no—
110C23×C157C2×D411yescharacteristic
111D4×C157—22no—
112D4×C157—22no—
113D4×C157—44no—
114C2×C4×C157—21no—
115C2×C4×C157—21no—
116C2×C4×C157—22no—
117C2×C4×C157C2×D411yescharacteristic
118C2×C4×C157C2411yescharacteristic
119C8×C157—21no—
120C8×C157C2×D412yescomplemented
121C22×C157—21no—
122C22×C157—22no—
123C22×C157—42no—
124C22×C157—42no—
125C22×C157—42no—
126C22×C157C22×D411yescharacteristic
127C2×D157—21no—
128C2×D157—22no—
129C2×D157—44no—
130C2×D157—42no—
131C2×D157—42no—
132C2×D157—42no—
133C2×D157—88no—
134C2×D157#12112yes—
135C2×D157AGL(1,ℤ/8)14yes—
136C4×C157—21no—
137C4×C157—42no—
138C4×C157C22×D411yescharacteristic
139C4×C157C2×#4111yescharacteristic
140Dic157—21no—
141Dic157—42no—
142Dic157#12112yes—
143C2×C157—41no—
144C2×C157—82no—
145C2×C157C2×#12111yescharacteristic
146C2×C157C2×AGL(1,ℤ/8)12yes—
147D157—41no—
148D157—82no—
149D157#46914yesdirect product factor
150C157C2×#46911yescharacteristic, complemented, Sylow
151C2×#469—1571nomaximal, Sylow
152C22×D8—1571no—
153C22×M16—1571no—
154C2×#121—1572no—
155#469—1571no—
156#469—1571no—
157#469—1572no—
158#469—1571no—
159#469—1572no—
160#469D15711yescharacteristic, direct product factor
161C2×#106—1571no—
162C2×#102—1572no—
163C22×D4—3142no—
164C22×D4—1572no—
165C23×C4—1571no—
166C2×D8—3144no—
167C2×D8—1572no—
168C2×D8—1571no—
169C2×D8C2×D15711yescharacteristic, complemented
170C2×M16—3144no—
171C2×M16—1571no—
172C2×M16—1571no—
173C2×M16—1571no—
174C2×M16C2×D15711yescharacteristic, complemented
175C22×C8—1571no—
176C22×C8—1571no—
177#121—3142no—
178#121—3142no—
179#121—1572no—
180#121—1572no—
181#121—1572no—
182#121C2×D15712yescomplemented
183C2×C4⋊C4—1572no—
184C2×C22⋊C4—3142no—
185#106—1571no—
186#106—1572no—
187#106C2×D15711yescharacteristic, complemented
188#102—1572no—
189#102—1572no—
190#102—1572no—
191#102C2×D15712yescomplemented
192C24—3142no—
193C24—1571no—
194C2×D4—22no—
195C2×D4—3142no—
196C2×D4—3142no—
197C2×D4—3142no—
198C2×D4—3144no—
199C2×D4—3144no—
200C2×D4—3142no—
201C2×D4—6288no—
202C2×D4—1572no—
203C2×D4C22×D15712yes—
204C22×C4—3142no—
205C22×C4—3142no—
206C22×C4—3142no—
207C22×C4—1571no—
208C22×C4—1571no—
209C22×C4—1571no—
210C22×C4—1571no—
211C22×C4—1571no—
212C22×C4C22×D15711yescharacteristic
213D8—22no—
214D8—3144no—
215D8—3142no—
216M16—22no—
217M16—3142no—
218M16—3142no—
219M16—3142no—
220C2×C8—3142no—
221C2×C8—1571no—
222C2×C8—1571no—
223C2×C8—1574no—
224C2×C8C22×D15711yescharacteristic
225C2×C8C22×D15711yescharacteristic
226C4⋊C4—3142no—
227C4⋊C4—1572no—
228C4⋊C4C22×D15712yes—
229C22⋊C4—22no—
230C22⋊C4—3142no—
231C22⋊C4—3142no—
232C22⋊C4—3142no—
233C23—22no—
234C23—3142no—
235C23—3144no—
236C23—3142no—
237C23—6284no—
238C23—6284no—
239C23—6284no—
240C23—1571no—
241C23—1571no—
242C23D4×D15711yescharacteristic
243D4—22no—
244D4—22no—
245D4—44no—
246D4—3142no—
247D4—3142no—
248D4—6284no—
249D4—6284no—
250D4—6284no—
251D4—6284no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—3142no—
256C2×C4—3142no—
257C2×C4—3142no—
258C2×C4—3141no—
259C2×C4—3141no—
260C2×C4—3142no—
261C2×C4—6284no—
262C2×C4—1571no—
263C2×C4—1571no—
264C2×C4—1572no—
265C2×C4—1572no—
266C2×C4D4×D15711yescharacteristic
267C2×C4C23×D15711yescharacteristic
268C8—21no—
269C8—3141no—
270C8—1572no—
271C8D4×D15712yescomplemented
272C22—21no—
273C22—22no—
274C22—42no—
275C22—42no—
276C22—42no—
277C22—3141no—
278C22—3142no—
279C22—6284no—
280C22—6282no—
281C22—6282no—
282C22—6282no—
283C22—1572no—
284C22—1574no—
285C22—12568no—
286C22C2×D4×D15711yescharacteristic
287C4—21no—
288C4—42no—
289C4—3141no—
290C4—6282no—
291C4—1572no—
292C4C2×D4×D15711yescharacteristic
293C4#41×D15711yescharacteristic
294C2—41no—
295C2—82no—
296C2—6281no—
297C2—1574no—
298C2—12562no—
299C2#121×D15711yescharacteristic
300C2AGL(1,ℤ/8)×D15712yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#469×C157C2
socleC22×C157C22×D4
hypercenter#469D157
hypocenterC157C2×#469
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D157C2semidirect
2. C2×M16×D157C2semidirect
3. #121×D157C2semidirect
4. #23757527C2semidirect
5. #23757526C2semidirect
6. #23757525C2semidirect
7. #23757524C2semidirect
8. #23757523C2semidirect
9. #106×D157C2semidirect
10. #102×D157C2semidirect
11. #469×C157C2semidirect
15. C2×D1256C22semidirect
16. C2×#14578939C22semidirect
18. C2×#14578938C22semidirect
19. C2×#14578937C22semidirect
20. C2×#14578937C22semidirect
24. #19097746C22semidirect
25. #19097745C22semidirect
28. #19097742C22semidirect
31. #19097703C22semidirect
32. #19097702C22semidirect
33. #19097693C22semidirect
34. #19097692C22semidirect
35. #19097691C22semidirect
36. C2×D8×C157C22semidirect
37. C2×M16×C157C22semidirect
38. #121×C157C22semidirect
39. #106×C157C22semidirect
40. #102×C157C22semidirect
67. C8×D157D4semidirect
108. #11929434C2×D4semidirect
120. C8×C157C2×D4semidirect
149. D157#469direct (N × H)
150. C157C2×#469semidirect
160. #469D157direct (N × H)
169. C2×D8C2×D157semidirect
174. C2×M16C2×D157semidirect
182. #121C2×D157semidirect
187. #106C2×D157semidirect
191. #102C2×D157semidirect
271. C8D4×D157semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1572
C2×M16×D1572
#121×D1572
#237575272
#237575262
#237575252
#237575242
#237575232
#106×D1572
#102×D1572
#469×C1572
C2×#469157