Finite Group Databasebeta

← #34110081

Subgroups — #34110081

Order 20096.

410 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 121 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×#19097729C211yesmaximal, characteristic
3#23757616C212yesmaximal, complemented
4#23757612C212yesmaximal, complemented
5#23757608C211yesmaximal, characteristic, complemented
6#23757604C211yesmaximal, characteristic, complemented
7#23757601C211yesmaximal, characteristic, complemented
8#23757314C212yesmaximal, complemented
9#23757280C212yesmaximal, complemented
10#23757235C211yesmaximal, characteristic
11#484×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157C2211yescharacteristic
13C22×C4×D157C2211yescharacteristic
14C2×#14578937C2212yes—
15#19097751C2211yescharacteristic
16#19097750C2211yescharacteristic
17#19097749—21no—
18#19097748—21no—
19#19097747C2211yescharacteristic
20#19097729—21no—
21#19097729—21no—
22#19097729C2212yes—
23#19097729C2212yes—
24C2×C4⋊Dic157C2211yescharacteristic
25C4⋊C4×D157—21no—
26C4⋊C4×D157C2211yescharacteristic
27C2×#14578931—21no—
28C22⋊C4×D157—21no—
29C2×C4×Dic157—21no—
30#19097704C2212yescomplemented
31#19097703C2212yescomplemented
32#19097702C2212yescomplemented
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×D157D411yescharacteristic
53C2×C4×D157D411yescharacteristic
54C2×C4×D157C2311yescharacteristic
55C2×C4×D157C2311yescharacteristic
56#14578937—24no—
57C2×#11929434C2312yes—
58C4⋊Dic157—21no—
59C4⋊Dic157—21no—
60C4⋊Dic157C2312yes—
61C4⋊Dic157C2311yescharacteristic
62#14578933—21no—
63#14578933—21no—
64#14578933—21no—
65C22⋊Dic157—21no—
66#14578931—21no—
67#14578931—21no—
68#14578931—22no—
69#14578931—22no—
70C4×Dic157—22no—
71C4×Dic157—22no—
72C2×Q8×C157C2311yescharacteristic
73C22×C4×C157—21no—
74C22×C4×C157C2311yescharacteristic
75C2×C8×C157C2312yes—
76C4⋊C4×C157—21no—
77C4⋊C4×C157—21no—
78C4⋊C4×C157C2312yes—
79C4⋊C4×C157C2311yescharacteristic
80C22⋊C4×C157—21no—
81C22×D157—22no—
82C22×D157—22no—
83C22×D157—41no—
84C22×D157C2×D411yescharacteristic
85C22×D157C2×D411yescharacteristic
86Dic314—22no—
87Dic314—42no—
88C2×Dic157—22no—
89C2×Dic157—21no—
90C2×Dic157—21no—
91C2×Dic157—22no—
92C2×Dic157—42no—
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C4×D157—22no—
96C4×D157—22no—
97C4×D157—22no—
98C4×D157—42no—
99C4×D157—42no—
100C4×D157#4114yes—
101#11929434—22no—
102C23×C157C2×D411yescharacteristic
103Q8×C157—22no—
104C2×C4×C157—22no—
105C2×C4×C157—21no—
106C2×C4×C157—21no—
107C2×C4×C157—22no—
108C2×C4×C157—42no—
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2411yescharacteristic
111C8×C157—22no—
112C22×C157—21no—
113C22×C157—21no—
114C22×C157#13712yes—
115C22×C157C22×D411yescharacteristic
116C2×D157—21no—
117C2×D157—21no—
118C2×D157—21no—
119C2×D157—21no—
120C2×D157—21no—
121C2×D157—44no—
122C2×D157#12312yes—
123C4×C157—21no—
124C4×C157—41no—
125C4×C157—42no—
126C4×C157—41no—
127C4×C157C2×#4112yes—
128Dic157—21no—
129Dic157—41no—
130Dic157—42no—
131Dic157—41no—
132Dic157#12312yes—
133C2×C157—22no—
134C2×C157C2×#12311yescharacteristic
135C2×C157C2×#13711yescharacteristic
136C2×C157#57511yescharacteristic
137D157—22no—
138D157—41no—
139C157#309511yescharacteristic, complemented, Sylow
140#3095—1571nomaximal, Sylow
141C2×C22⋊Q8—1571no—
142C2×#117—1571no—
143#485—1572no—
144#485—1572no—
145#484—1571no—
146#484D15711yescharacteristic, complemented
147#483—1571no—
148#483—1571no—
149#428—1572no—
150#419—1572no—
151#407—1571no—
152C22×Q8—1571no—
153C23×C4—1571no—
154C2×M16—1572no—
155C22⋊Q8—3141no—
156C22⋊Q8—3141no—
157C22⋊Q8—1571no—
158C22⋊Q8—1571no—
159C22⋊Q8—1571no—
160C22⋊Q8C2×D15711yescharacteristic
161#117—3141no—
162#117—3141no—
163#117—1572no—
164#117—1572no—
165C2×C4⋊C4—3141no—
166C2×C4⋊C4—1571no—
167C2×C4⋊C4—1571no—
168C2×C4⋊C4C2×D15711yescharacteristic
169C2×C22⋊C4—3141no—
170C2×C22⋊C4—3141no—
171C2×C42—3141no—
172#107—1572no—
173#107C2×D15712yescomplemented
174#106—1572no—
175#106—1572no—
176#103—1572no—
177#103—1572no—
178#103—1572no—
179#103C2×D15712yescomplemented
180C22⋊C8—1571no—
181C22⋊C8—1572no—
182C22⋊C8C2×D15711yescharacteristic
183C24—1571no—
184C2×Q8—3141no—
185C2×Q8—3144no—
186C2×Q8—1571no—
187C2×Q8C22×D15711yescharacteristic
188C22×C4—21no—
189C22×C4—3144no—
190C22×C4—3141no—
191C22×C4—3141no—
192C22×C4—3141no—
193C22×C4—1571no—
194C22×C4—1571no—
195C22×C4—1571no—
196C22×C4—1571no—
197C22×C4—1571no—
198C22×C4C22×D15711yescharacteristic
199M16—3144no—
200C2×C8—1572no—
201C2×C8C22×D15712yes—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—3141no—
205C4⋊C4—3141no—
206C4⋊C4—3141no—
207C4⋊C4—3141no—
208C4⋊C4—3141no—
209C4⋊C4—1572no—
210C4⋊C4—1571no—
211C4⋊C4C22×D15712yes—
212C4⋊C4C22×D15711yescharacteristic
213C22⋊C4—21no—
214C22⋊C4—3142no—
215C22⋊C4—3142no—
216C22⋊C4—3141no—
217C22⋊C4—3141no—
218C22⋊C4—3141no—
219C42—3142no—
220C42—3142no—
221C23—3142no—
222C23—3142no—
223C23—6281no—
224C23—1571no—
225C23—1571no—
226C23D4×D15711yescharacteristic
227Q8—22no—
228Q8—3142no—
229Q8—6282no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—3142no—
236C2×C4—3142no—
237C2×C4—3142no—
238C2×C4—3142no—
239C2×C4—3142no—
240C2×C4—3141no—
241C2×C4—3141no—
242C2×C4—6282no—
243C2×C4—6282no—
244C2×C4—6282no—
245C2×C4—1571no—
246C2×C4—1571no—
247C2×C4—1574no—
248C2×C4D4×D15711yescharacteristic
249C2×C4C23×D15711yescharacteristic
250C8—22no—
251C8—3142no—
252C22—21no—
253C22—21no—
254C22—3141no—
255C22—3141no—
256C22—3141no—
257C22—3141no—
258C22—3141no—
259C22—6284no—
260C22—1572no—
261C22#1909780312yes—
262C22C2×D4×D15711yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—41no—
267C4—3141no—
268C4—6281no—
269C4—6282no—
270C4—6281no—
271C4—1572no—
272C4#41×D15712yes—
273C2—22no—
274C2—3142no—
275C2—6281no—
276C2#123×D15711yescharacteristic
277C2C2×#1909780311yescharacteristic
278C2#2375806911yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#484×C157C2
socleC22×C157C22×D4
prime radicalC23×D157D4
hypercenter#484D157
hypocenterC157#3095
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. #23757616C2semidirect
4. #23757612C2semidirect
5. #23757608C2semidirect
6. #23757604C2semidirect
7. #23757601C2semidirect
8. #23757314C2semidirect
9. #23757280C2semidirect
11. #484×C157C2semidirect
30. #19097704C22semidirect
31. #19097703C22semidirect
32. #19097702C22semidirect
33. #19097696C22semidirect
34. #19097695C22semidirect
35. #19097694C22semidirect
40. #107×C157C22semidirect
41. #103×C157C22semidirect
139. C157#3095semidirect
146. #484D157semidirect
173. #107C2×D157semidirect
179. #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×#190977292
#237576162
#237576122
#237576082
#237576042
#237576012
#237573142
#237572802
#237572352
#484×C1572
#3095157