Finite Group Databasebeta

← #489×D157 #34110462

Subgroups — #489×D157 #34110462

Order 20096.

436 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C4⋊Q8×D157C211yesmaximal, characteristic, complemented
2#124×D157C211yesmaximal, characteristic
3#23757635C211yesmaximal, characteristic, complemented
4#23757634C212yesmaximal, complemented
5#23757633C211yesmaximal, characteristic, complemented
6#23757632C211yesmaximal, characteristic, complemented
7#23757631C212yesmaximal, complemented
8#103×D157C212yesmaximal, complemented
9#102×D157C212yesmaximal
10#97×D157C211yesmaximal, characteristic, complemented
11#489×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157—21no—
13C2×Q8×D157C2211yescharacteristic
14C2×D4×D157C2211yescharacteristic
15C2×C8×D157C2212yes—
16#19097771C2211yescharacteristic, complemented
17#19097770—21no—
18#19097770C2212yescomplemented
19#19097769—21no—
20#19097760C2211yescharacteristic
21#19097759C2211yescharacteristic
22#19097758C2211yescharacteristic
23#19097757—21no—
24#19097757—21no—
25C4⋊C4×D157—21no—
26C4⋊C4×D157C2212yes—
27C22⋊C4×D157—21no—
28C22⋊C4×D157—21no—
29C42×D157C2211yescharacteristic
30#19097696C2212yescomplemented
31#19097695C2212yescomplemented
32#19097694C2212yescomplemented
33#19097693C2212yes—
34#19097692C2212yes—
35#19097691C2212yes—
36#19097682C2211yescharacteristic, complemented
37#19097681C2212yescomplemented
38C4⋊Q8×C157C2211yescharacteristic, complemented
39#124×C157C2211yescharacteristic
40#103×C157C2212yescomplemented
41#102×C157C2212yes—
42#97×C157C2211yescharacteristic, complemented
43C23×D157—21no—
44C2×Dic314—21no—
45C2×Dic314—22no—
46C2×Dic314—21no—
47C2×Dic314C2311yescharacteristic
48Q8×D157—24no—
49Q8×D157—44no—
50C2×D628C2311yescharacteristic
51C2×C22⋊D157—21no—
52D4×D157—24no—
53C2×C4×D157—21no—
54C2×C4×D157—22no—
55C2×C4×D157D411yescharacteristic
56C2×C4×D157D411yescharacteristic
57C2×C4×D157C2311yescharacteristic
58C2×#11929434C2312yes—
59C8×D157—24no—
60C4⋊Dic157—21no—
61C4⋊Dic157C2312yes—
62#14578933—22no—
63#14578933—22no—
64C22⋊Dic157—21no—
65C22⋊Dic157—21no—
66#14578931—21no—
67#14578931—21no—
68#14578931—21no—
69#14578931—21no—
70C4×Dic157D411yescharacteristic
71C4×Dic157D411yescharacteristic
72C4×Dic157C2311yescharacteristic
73C2×Q8×C157—21no—
74C2×Q8×C157C2311yescharacteristic
75C2×D4×C157C2311yescharacteristic
76C2×C8×C157C2312yes—
77C4⋊C4×C157—21no—
78C4⋊C4×C157C2312yes—
79C22⋊C4×C157—21no—
80C22⋊C4×C157—21no—
81C42×C157C2311yescharacteristic
82C22×D157—21no—
83C22×D157—42no—
84C22×D157—42no—
85C22×D157—42no—
86C22×D157C2×D411yescharacteristic
87Dic314—22no—
88Dic314—44no—
89Dic314—42no—
90Dic314—42no—
91D628—22no—
92C22⋊D157—42no—
93C2×Dic157—22no—
94C2×Dic157—21no—
95C2×Dic157C2×D411yescharacteristic
96C2×Dic157C2×D411yescharacteristic
97C2×Dic157C2×D411yescharacteristic
98C4×D157—22no—
99C4×D157—22no—
100C4×D157—42no—
101C4×D157—44no—
102C4×D157#4114yes—
103#11929434—22no—
104C23×C157—21no—
105Q8×C157—22no—
106Q8×C157—42no—
107D4×C157—22no—
108C2×C4×C157—21no—
109C2×C4×C157—22no—
110C2×C4×C157C2×D411yescharacteristic
111C2×C4×C157C2×D411yescharacteristic
112C2×C4×C157C2411yescharacteristic
113C8×C157—22no—
114C22×C157—41no—
115C22×C157—41no—
116C22×C157—41no—
117C22×C157C22×D411yescharacteristic
118C2×D157—41no—
119C2×D157—41no—
120C2×D157—41no—
121C2×D157—84no—
122C2×D157#12412yes—
123C2×D157AGL(1,ℤ/8)12yes—
124C2×D157#13712yes—
125C4×C157—21no—
126C4×C157—21no—
127C4×C157—41no—
128C4×C157—42no—
129C4×C157C2×#4112yes—
130Dic157—21no—
131Dic157—21no—
132Dic157—41no—
133Dic157—42no—
134Dic157#12412yes—
135C2×C157—81no—
136C2×C157C2×#12411yescharacteristic
137C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
138C2×C157C2×#13711yescharacteristic
139D157—81no—
140D157#48914yesdirect product factor
141C157C2×#48911yescharacteristic, complemented, Sylow
142C2×#489—1571nomaximal, Sylow
143C2×C4⋊Q8—1571no—
144C2×#124—1571no—
145#489—1571no—
146#489—1571no—
147#489—1572no—
148#489—1572no—
149#489—1571no—
150#489D15711yescharacteristic, direct product factor
151C2×#103—1572no—
152C2×#102—1572no—
153C2×#97—1571no—
154C22×Q8—3141no—
155C22×Q8—1571no—
156C22×D4—1571no—
157C22×C8—1572no—
158C4⋊Q8—3141no—
159C4⋊Q8—3141no—
160C4⋊Q8—1572no—
161C4⋊Q8—1571no—
162C4⋊Q8C2×D15711yescharacteristic, complemented
163#124—3141no—
164#124—3141no—
165#124—1571no—
166#124—1571no—
167#124—1571no—
168#124C2×D15711yescharacteristic
169C2×C4⋊C4—3141no—
170C2×C4⋊C4—1572no—
171C2×C22⋊C4—3141no—
172C2×C22⋊C4—3141no—
173C2×C42—1571no—
174#103—1572no—
175#103—1572no—
176#103—1572no—
177#103C2×D15712yescomplemented
178#102—1572no—
179#102—1572no—
180#102—1572no—
181#102C2×D15712yes—
182#97—1572no—
183#97—1571no—
184#97C2×D15711yescharacteristic, complemented
185C24—3141no—
186C2×Q8—21no—
187C2×Q8—3141no—
188C2×Q8—3142no—
189C2×Q8—3141no—
190C2×Q8—3144no—
191C2×Q8—6284no—
192C2×Q8—1571no—
193C2×Q8C22×D15711yescharacteristic
194C2×D4—3144no—
195C2×D4—3141no—
196C2×D4—1571no—
197C2×D4C22×D15711yescharacteristic
198C22×C4—3141no—
199C22×C4—3142no—
200C22×C4—1571no—
201C22×C4—1571no—
202C22×C4—1571no—
203C2×C8—3144no—
204C2×C8—1572no—
205C2×C8C22×D15712yes—
206C4⋊C4—21no—
207C4⋊C4—3142no—
208C4⋊C4—3142no—
209C4⋊C4—3141no—
210C4⋊C4—1572no—
211C4⋊C4C22×D15712yes—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—3141no—
215C22⋊C4—3141no—
216C22⋊C4—3141no—
217C22⋊C4—3141no—
218C22⋊C4—3141no—
219C22⋊C4—3141no—
220C42—1571no—
221C42—1571no—
222C42—1571no—
223C42C22×D15711yescharacteristic
224C23—21no—
225C23—3141no—
226C23—6282no—
227C23—6282no—
228C23—6282no—
229C23—1571no—
230Q8—22no—
231Q8—42no—
232Q8—3142no—
233Q8—6282no—
234Q8—6284no—
235Q8—6282no—
236D4—22no—
237D4—3142no—
238D4—6282no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—3142no—
242C2×C4—3142no—
243C2×C4—3141no—
244C2×C4—3142no—
245C2×C4—6282no—
246C2×C4—6284no—
247C2×C4—1571no—
248C2×C4—1571no—
249C2×C4—1571no—
250C2×C4—1574no—
251C2×C4D4×D15711yescharacteristic
252C2×C4D4×D15711yescharacteristic
253C2×C4C23×D15711yescharacteristic
254C8—22no—
255C8—3142no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—6281no—
260C22—6281no—
261C22—6281no—
262C22—1572no—
263C22—1572no—
264C22—1572no—
265C22—12564no—
266C22C2×D4×D15711yescharacteristic
267C4—21no—
268C4—21no—
269C4—41no—
270C4—42no—
271C4—3141no—
272C4—3141no—
273C4—6281no—
274C4—6282no—
275C4—1572no—
276C4#41×D15712yes—
277C2—81no—
278C2—1574no—
279C2—12561no—
280C2#124×D15711yescharacteristic
281C2AGL(1,ℤ/8)×D15711yescharacteristic
282C2#137×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#489×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
hypercenter#489D157
hypocenterC157C2×#489
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D157C2semidirect
3. #23757635C2semidirect
4. #23757634C2semidirect
5. #23757633C2semidirect
6. #23757632C2semidirect
7. #23757631C2semidirect
8. #103×D157C2semidirect
10. #97×D157C2semidirect
11. #489×C157C2semidirect
16. #19097771C22semidirect
18. #19097770C22semidirect
30. #19097696C22semidirect
31. #19097695C22semidirect
32. #19097694C22semidirect
36. #19097682C22semidirect
37. #19097681C22semidirect
38. C4⋊Q8×C157C22semidirect
40. #103×C157C22semidirect
42. #97×C157C22semidirect
140. D157#489direct (N × H)
141. C157C2×#489semidirect
150. #489D157direct (N × H)
162. C4⋊Q8C2×D157semidirect
177. #103C2×D157semidirect
184. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1572
#124×D1572
#237576352
#237576342
#237576332
#237576322
#237576312
#103×D1572
#102×D1572
#97×D1572
#489×C1572
C2×#489157