Finite Group Databasebeta

← #484×D157 #34110038

Subgroups — #484×D157 #34110038

Order 20096.

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

normal subgroups only

This group has 460 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
1C22⋊Q8×D157C211yesmaximal, characteristic
2C2×C4⋊C4×D157C211yesmaximal, characteristic
3#23757610C212yesmaximal, complemented
4#23757609C211yesmaximal, characteristic, complemented
5#23757608C211yesmaximal, characteristic, complemented
6#23757607C212yesmaximal, complemented
7#23757606C211yesmaximal, characteristic, complemented
8#107×D157C212yesmaximal, complemented
9#103×D157C212yesmaximal, complemented
10C22⋊C8×D157C211yesmaximal, characteristic
11#484×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157C2211yescharacteristic
13C22×C4×D157—21no—
14C22×C4×D157C2211yescharacteristic
15C2×C8×D157C2212yes—
16#19097751C2211yescharacteristic
17#19097750C2211yescharacteristic
18#19097749—21no—
19#19097748—21no—
20#19097747C2211yescharacteristic
21C2×C4⋊Dic157C2211yescharacteristic
22C2×#14578933—21no—
23C4⋊C4×D157—21no—
24C4⋊C4×D157—21no—
25C4⋊C4×D157—21no—
26C4⋊C4×D157C2212yes—
27C4⋊C4×D157C2212yes—
28C4⋊C4×D157C2211yescharacteristic
29C22⋊C4×D157—21no—
30#19097705C2212yescomplemented
31#19097704C2212yescomplemented
32#19097704C2212yescomplemented
33#19097696C2212yescomplemented
34#19097695C2212yescomplemented
35#19097694C2212yescomplemented
36C22⋊#11929434C2211yescharacteristic
37#19097683C2212yes—
38C22⋊Q8×C157C2211yescharacteristic
39C2×C4⋊C4×C157C2211yescharacteristic
40#107×C157C2212yescomplemented
41#103×C157C2212yescomplemented
42C22⋊C8×C157C2211yescharacteristic
43C23×D157D411yescharacteristic
44C2×Dic314—21no—
45C2×Dic314C2311yescharacteristic
46Q8×D157—24no—
47C22×Dic157—21no—
48C22×Dic157D411yescharacteristic
49C2×C4×D157—24no—
50C2×C4×D157—21no—
51C2×C4×D157—21no—
52C2×C4×D157—22no—
53C2×C4×D157—22no—
54C2×C4×D157—44no—
55C2×C4×D157D411yescharacteristic
56C2×C4×D157D411yescharacteristic
57C2×C4×D157C2311yescharacteristic
58C2×C4×D157C2311yescharacteristic
59C2×#11929434C2312yes—
60C8×D157—24no—
61C4⋊Dic157—21no—
62C4⋊Dic157—21no—
63C4⋊Dic157C2312yes—
64C4⋊Dic157C2311yescharacteristic
65#14578933—22no—
66#14578933—22no—
67#14578933—21no—
68#14578933—21no—
69#14578933—21no—
70C22⋊Dic157—21no—
71#14578931—21no—
72#14578931—21no—
73C2×Q8×C157C2311yescharacteristic
74C22×C4×C157—21no—
75C22×C4×C157C2311yescharacteristic
76C2×C8×C157C2312yes—
77C4⋊C4×C157—21no—
78C4⋊C4×C157—21no—
79C4⋊C4×C157C2312yes—
80C4⋊C4×C157C2311yescharacteristic
81C22⋊C4×C157—21no—
82C22×D157—22no—
83C22×D157—22no—
84C22×D157Q1614yes—
85C22×D157C2×D411yescharacteristic
86C22×D157C2×D411yescharacteristic
87Dic314—22no—
88Dic314—42no—
89C2×Dic157—22no—
90C2×Dic157—22no—
91C2×Dic157—21no—
92C2×Dic157—21no—
93C2×Dic157—42no—
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96C4×D157—22no—
97C4×D157—22no—
98C4×D157—22no—
99C4×D157—44no—
100C4×D157—42no—
101C4×D157—42no—
102C4×D157—44no—
103C4×D157#4114yes—
104#11929434—22no—
105C23×C157C2×D411yescharacteristic
106Q8×C157—22no—
107C2×C4×C157—22no—
108C2×C4×C157—22no—
109C2×C4×C157—21no—
110C2×C4×C157—21no—
111C2×C4×C157—42no—
112C2×C4×C157C2×D411yescharacteristic
113C2×C4×C157C2411yescharacteristic
114C8×C157—22no—
115C22×C157—21no—
116C22×C157—21no—
117C22×C157C2×Q1612yes—
118C22×C157C22×D411yescharacteristic
119C2×D157—21no—
120C2×D157—21no—
121C2×D157—28no—
122C2×D157#12312yes—
123C2×D157C2×Q1612yes—
124C2×D157C2×Q1612yes—
125C2×D157AGL(1,ℤ/8)12yes—
126C4×C157—21no—
127C4×C157—41no—
128C4×C157—42no—
129C4×C157—41no—
130C4×C157C2×#4112yes—
131Dic157—21no—
132Dic157—41no—
133Dic157—42no—
134Dic157—41no—
135Dic157#12312yes—
136C2×C157—22no—
137C2×C157C2×#12311yescharacteristic
138C2×C157C22×Q1611yescharacteristic
139C2×C157C2×AGL(1,ℤ/8)11yescharacteristic
140D157—22no—
141D157#48414yesdirect product factor
142C157C2×#48411yescharacteristic, complemented, Sylow
143C2×#484—1571nomaximal, Sylow
144C2×C22⋊Q8—1571no—
145C22×C4⋊C4—1571no—
146#484—1571no—
147#484—1571no—
148#484—1572no—
149#484—1572no—
150#484—1571no—
151#484D15711yescharacteristic, direct product factor
152C2×#107—1572no—
153C2×#103—1572no—
154C2×C22⋊C8—1571no—
155C22×Q8—1571no—
156C23×C4—3141no—
157C23×C4—1571no—
158C22×C8—1572no—
159C22⋊Q8—3141no—
160C22⋊Q8—3141no—
161C22⋊Q8—1571no—
162C22⋊Q8—1571no—
163C22⋊Q8—1571no—
164C22⋊Q8C2×D15711yescharacteristic
165C2×C4⋊C4—3141no—
166C2×C4⋊C4—3141no—
167C2×C4⋊C4—3141no—
168C2×C4⋊C4—3141no—
169C2×C4⋊C4—1571no—
170C2×C4⋊C4—1572no—
171C2×C4⋊C4—1572no—
172C2×C4⋊C4—1571no—
173C2×C4⋊C4C2×D15711yescharacteristic
174C2×C22⋊C4—3141no—
175#107—1572no—
176#107—1572no—
177#107—1572no—
178#107C2×D15712yescomplemented
179#103—1572no—
180#103—1572no—
181#103—1572no—
182#103C2×D15712yescomplemented
183C22⋊C8—1571no—
184C22⋊C8—1572no—
185C22⋊C8C2×D15711yescharacteristic
186C24—1571no—
187C2×Q8—3141no—
188C2×Q8—3144no—
189C2×Q8—1571no—
190C2×Q8C22×D15711yescharacteristic
191C22×C4—21no—
192C22×C4—3144no—
193C22×C4—3141no—
194C22×C4—3141no—
195C22×C4—3142no—
196C22×C4—3142no—
197C22×C4—3141no—
198C22×C4—6284no—
199C22×C4—1571no—
200C22×C4—1571no—
201C22×C4—1571no—
202C22×C4—1571no—
203C22×C4—1571no—
204C22×C4C22×D15711yescharacteristic
205C2×C8—3144no—
206C2×C8—1572no—
207C2×C8C22×D15712yes—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—3142no—
211C4⋊C4—3142no—
212C4⋊C4—3141no—
213C4⋊C4—3141no—
214C4⋊C4—3141no—
215C4⋊C4—3141no—
216C4⋊C4—3141no—
217C4⋊C4—1572no—
218C4⋊C4—1571no—
219C4⋊C4C22×D15712yes—
220C4⋊C4C22×D15711yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—3141no—
223C22⋊C4—3141no—
224C22⋊C4—3141no—
225C23—3142no—
226C23—3142no—
227C23—1574no—
228C23—1571no—
229C23—1571no—
230C23D4×D15711yescharacteristic
231Q8—22no—
232Q8—3142no—
233Q8—6282no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—42no—
239C2×C4—3142no—
240C2×C4—3142no—
241C2×C4—3142no—
242C2×C4—3142no—
243C2×C4—3141no—
244C2×C4—3141no—
245C2×C4—3142no—
246C2×C4—6284no—
247C2×C4—6284no—
248C2×C4—6282no—
249C2×C4—6282no—
250C2×C4—6282no—
251C2×C4—1571no—
252C2×C4—1571no—
253C2×C4—1574no—
254C2×C4D4×D15711yescharacteristic
255C2×C4C23×D15711yescharacteristic
256C8—22no—
257C8—3142no—
258C22—21no—
259C22—21no—
260C22—3148no—
261C22—3141no—
262C22—3141no—
263C22—1572no—
264C22—1572no—
265C22—1572no—
266C22—1572no—
267C22Q16×D15712yes—
268C22C2×D4×D15711yescharacteristic
269C4—21no—
270C4—41no—
271C4—42no—
272C4—41no—
273C4—3141no—
274C4—6281no—
275C4—6282no—
276C4—6281no—
277C4—1572no—
278C4#41×D15712yes—
279C2—22no—
280C2—3142no—
281C2—1574no—
282C2#123×D15711yescharacteristic
283C2C2×Q16×D15711yescharacteristic
284C2AGL(1,ℤ/8)×D15711yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#484×C157C2
socleC22×C157C22×D4
prime radicalC23×D157D4
hypercenter#484D157
hypocenterC157C2×#484
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #23757610C2semidirect
4. #23757609C2semidirect
5. #23757608C2semidirect
6. #23757607C2semidirect
7. #23757606C2semidirect
8. #107×D157C2semidirect
9. #103×D157C2semidirect
11. #484×C157C2semidirect
30. #19097705C22semidirect
31. #19097704C22semidirect
32. #19097704C22semidirect
33. #19097696C22semidirect
34. #19097695C22semidirect
35. #19097694C22semidirect
40. #107×C157C22semidirect
41. #103×C157C22semidirect
141. D157#484direct (N × H)
142. C157C2×#484semidirect
151. #484D157direct (N × H)
178. #107C2×D157semidirect
182. #103C2×D157semidirect

Maximal subgroups

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

subgroupindex
C22⋊Q8×D1572
C2×C4⋊C4×D1572
#237576102
#237576092
#237576082
#237576072
#237576062
#107×D1572
#103×D1572
C22⋊C8×D1572
#484×C1572
C2×#484157