Finite Group Databasebeta

← #470×D157 #34109734

Subgroups — #470×D157 #34109734

Order 20096.

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

normal subgroups only

This group has 492 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×Q16×D157C211yesmaximal, characteristic, complemented
2C2×M16×D157C211yesmaximal, characteristic
3C22⋊Q8×D157C212yesmaximal
4#23757532C211yesmaximal, characteristic, complemented
5#23757531C211yesmaximal, characteristic, complemented
6#23757530C212yesmaximal, complemented
7#23757529C212yesmaximal, complemented
8#23757528C211yesmaximal, characteristic, complemented
9#106×D157C211yesmaximal, characteristic, complemented
10#103×D157C212yesmaximal, complemented
11#470×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157C2212yes—
13C22×C4×D157C2211yescharacteristic
14C2×Dic628C2211yescharacteristic, complemented
15C2×#14578944C2212yescomplemented
16Q16×D157—24no—
17C2×#14578938C2211yescharacteristic
18C2×#14578937C2211yescharacteristic
19C2×#14578937C2211yescharacteristic
20M16×D157—24no—
21C2×C8×D157C2211yescharacteristic
22C2×C8×D157C2211yescharacteristic
23#19097751C2212yes—
24#19097750C2212yes—
25#19097749—22no—
26#19097748—22no—
27#19097747C2212yes—
28C4⋊C4×D157—22no—
29C4⋊C4×D157C2212yes—
30C22⋊C4×D157—22no—
31#19097703C2211yescharacteristic, complemented
32#19097702C2212yescomplemented
33#19097696C2212yescomplemented
34#19097695C2212yescomplemented
35#19097694C2212yescomplemented
36C2×Q16×C157C2211yescharacteristic, complemented
37C2×M16×C157C2211yescharacteristic
38C22⋊Q8×C157C2212yes—
39#106×C157C2211yescharacteristic, complemented
40#103×C157C2212yescomplemented
41C23×D157D411yescharacteristic
42C2×Dic314—22no—
43C2×Dic314C2312yes—
44Q8×D157—24no—
45Q8×D157—24no—
46C22×Dic157D411yescharacteristic
47C2×C4×D157—22no—
48C2×C4×D157—22no—
49C2×C4×D157—22no—
50C2×C4×D157—22no—
51C2×C4×D157D411yescharacteristic
52C2×C4×D157D411yescharacteristic
53C2×C4×D157C2311yescharacteristic
54C2×C4×D157C2311yescharacteristic
55Dic628—22no—
56#14578944—24no—
57#14578938—22no—
58#14578937—22no—
59#14578937—22no—
60C2×#11929434C2311yescharacteristic
61C2×#11929434C2311yescharacteristic
62C8×D157—22no—
63C8×D157D414yes—
64C4⋊Dic157—22no—
65C4⋊Dic157C2312yes—
66#14578933—22no—
67#14578933—22no—
68#14578933—22no—
69C22⋊Dic157—22no—
70#14578931—22no—
71#14578931—22no—
72C2×Q8×C157C2312yes—
73C22×C4×C157C2311yescharacteristic
74Q16×C157—22no—
75M16×C157—22no—
76C2×C8×C157C2311yescharacteristic
77C2×C8×C157C2311yescharacteristic
78C4⋊C4×C157—22no—
79C4⋊C4×C157C2312yes—
80C22⋊C4×C157—22no—
81C22×D157—22no—
82C22×D157—24no—
83C22×D157C2×D411yescharacteristic
84C22×D157C2×D411yescharacteristic
85Dic314—22no—
86Dic314—22no—
87Dic314—44no—
88C2×Dic157—22no—
89C2×Dic157—22no—
90C2×Dic157—22no—
91C2×Dic157C2×D411yescharacteristic
92C2×Dic157C2×D411yescharacteristic
93C4×D157—22no—
94C4×D157—21no—
95C4×D157—21no—
96C4×D157—22no—
97C4×D157—44no—
98C4×D157—44no—
99C4×D157C2×D412yes—
100C4×D157#4112yes—
101#11929434—21no—
102#11929434C2×D412yes—
103C23×C157C2×D411yescharacteristic
104Q8×C157—22no—
105Q8×C157—22no—
106C2×C4×C157—21no—
107C2×C4×C157—21no—
108C2×C4×C157—22no—
109C2×C4×C157—22no—
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2411yescharacteristic
112C8×C157—21no—
113C8×C157C2×D412yes—
114C22×C157—21no—
115C22×C157—22no—
116C22×C157C22×D411yescharacteristic
117C2×D157—21no—
118C2×D157—22no—
119C2×D157—44no—
120C2×D157#12112yes—
121C2×D157#13714yes—
122C4×C157—21no—
123C4×C157—42no—
124C4×C157—42no—
125C4×C157C22×D411yescharacteristic
126C4×C157C2×#4111yescharacteristic
127Dic157—21no—
128Dic157—42no—
129Dic157—42no—
130Dic157#12112yes—
131C2×C157—41no—
132C2×C157C2×#12111yescharacteristic
133C2×C157C2×#13712yes—
134D157—41no—
135D157#47014yesdirect product factor
136C157C2×#47011yescharacteristic, complemented, Sylow
137C2×#470—1571nomaximal, Sylow
138C22×Q16—1571no—
139C22×M16—1571no—
140C2×C22⋊Q8—1572no—
141#470—1571no—
142#470—1571no—
143#470—1572no—
144#470—1571no—
145#470—1572no—
146#470D15711yescharacteristic, direct product factor
147C2×#106—1571no—
148C2×#103—1572no—
149C22×Q8—1572no—
150C23×C4—1571no—
151C2×Q16—3144no—
152C2×Q16—1572no—
153C2×Q16—1571no—
154C2×Q16C2×D15711yescharacteristic, complemented
155C2×M16—3144no—
156C2×M16—1571no—
157C2×M16—1571no—
158C2×M16—1571no—
159C2×M16C2×D15711yescharacteristic
160C22×C8—1571no—
161C22×C8—1571no—
162C22⋊Q8—3142no—
163C22⋊Q8—3142no—
164C22⋊Q8—1572no—
165C22⋊Q8—1572no—
166C22⋊Q8—1572no—
167C22⋊Q8C2×D15712yes—
168C2×C4⋊C4—3142no—
169C2×C4⋊C4—1572no—
170C2×C22⋊C4—3142no—
171#106—1571no—
172#106—1572no—
173#106C2×D15711yescharacteristic, complemented
174#103—1572no—
175#103—1572no—
176#103—1572no—
177#103C2×D15712yescomplemented
178C24—1571no—
179C2×Q8—3142no—
180C2×Q8—3144no—
181C2×Q8—3144no—
182C2×Q8—1572no—
183C2×Q8C22×D15712yes—
184C22×C4—3142no—
185C22×C4—3142no—
186C22×C4—3142no—
187C22×C4—3142no—
188C22×C4—1571no—
189C22×C4—1571no—
190C22×C4—1571no—
191C22×C4—1571no—
192C22×C4—1571no—
193C22×C4C22×D15711yescharacteristic
194Q16—22no—
195Q16—3144no—
196Q16—3142no—
197M16—22no—
198M16—3142no—
199M16—3142no—
200M16—3142no—
201C2×C8—3142no—
202C2×C8—1571no—
203C2×C8—1571no—
204C2×C8—1574no—
205C2×C8C22×D15711yescharacteristic
206C2×C8C22×D15711yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—3142no—
209C4⋊C4—3142no—
210C4⋊C4—3142no—
211C4⋊C4—3142no—
212C4⋊C4—1572no—
213C4⋊C4C22×D15712yes—
214C22⋊C4—22no—
215C22⋊C4—3142no—
216C22⋊C4—3142no—
217C22⋊C4—3142no—
218C23—3142no—
219C23—3144no—
220C23—1571no—
221C23—1571no—
222C23D4×D15711yescharacteristic
223Q8—22no—
224Q8—22no—
225Q8—3142no—
226Q8—3142no—
227Q8—6284no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—3142no—
233C2×C4—3142no—
234C2×C4—3142no—
235C2×C4—3141no—
236C2×C4—3141no—
237C2×C4—3142no—
238C2×C4—3142no—
239C2×C4—6284no—
240C2×C4—6284no—
241C2×C4—1571no—
242C2×C4—1571no—
243C2×C4—1572no—
244C2×C4—1572no—
245C2×C4D4×D15711yescharacteristic
246C2×C4C23×D15711yescharacteristic
247C8—21no—
248C8—3141no—
249C8—1572no—
250C8D4×D15712yes—
251C22—21no—
252C22—22no—
253C22—3141no—
254C22—3142no—
255C22—6284no—
256C22—1572no—
257C22—1574no—
258C22C2×D4×D15711yescharacteristic
259C4—21no—
260C4—42no—
261C4—42no—
262C4—3141no—
263C4—6282no—
264C4—6282no—
265C4—1572no—
266C4C2×D4×D15711yescharacteristic
267C4#41×D15711yescharacteristic
268C2—41no—
269C2—6281no—
270C2—1574no—
271C2#121×D15711yescharacteristic
272C2#137×D15712yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#470×C157C2
socleC22×C157C22×D4
prime radicalC23×D157D4
hypercenter#470D157
hypocenterC157C2×#470
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×Q16×D157C2semidirect
4. #23757532C2semidirect
5. #23757531C2semidirect
6. #23757530C2semidirect
7. #23757529C2semidirect
8. #23757528C2semidirect
9. #106×D157C2semidirect
10. #103×D157C2semidirect
11. #470×C157C2semidirect
14. C2×Dic628C22semidirect
15. C2×#14578944C22semidirect
31. #19097703C22semidirect
32. #19097702C22semidirect
33. #19097696C22semidirect
34. #19097695C22semidirect
35. #19097694C22semidirect
36. C2×Q16×C157C22semidirect
39. #106×C157C22semidirect
40. #103×C157C22semidirect
135. D157#470direct (N × H)
136. C157C2×#470semidirect
146. #470D157direct (N × H)
154. C2×Q16C2×D157semidirect
173. #106C2×D157semidirect
177. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1572
C2×M16×D1572
C22⋊Q8×D1572
#237575322
#237575312
#237575302
#237575292
#237575282
#106×D1572
#103×D1572
#470×C1572
C2×#470157