Finite Group Databasebeta

← #34110521

Subgroups — #34110521

Order 20096.

400 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 117 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
1#23757828C211yesmaximal, characteristic, complemented
2#124×D157C211yesmaximal, characteristic
3#23757635C211yesmaximal, characteristic, complemented
4#23757633C211yesmaximal, characteristic, complemented
5#23757632C211yesmaximal, characteristic, complemented
6#23757629C212yesmaximal, complemented
7#23757626C212yesmaximal, complemented
8#23757280C212yesmaximal, complemented
9#23757276C212yesmaximal
10#23757225C211yesmaximal, characteristic, complemented
11#489×C157C211yesmaximal, characteristic, complemented
12C2×Q8×D157C2211yescharacteristic
13C2×D4×D157C2211yescharacteristic
14C2×#14578937C2212yes—
15#19097771C2211yescharacteristic, complemented
16#19097762C2212yescomplemented
17#19097760C2211yescharacteristic
18#19097759C2211yescharacteristic
19#19097758C2211yescharacteristic
20#19097757—21no—
21#19097757—21no—
22#19097751—21no—
23#19097747—21no—
24Q8×Dic157—21no—
25#19097736—21no—
26#19097729C2212yes—
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
37C8×Dic157C2212yescomplemented
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—21no—
46C2×Dic314C2311yescharacteristic
47Q8×D157—24no—
48C2×D628C2311yescharacteristic
49C2×C22⋊D157—21no—
50D4×D157—24no—
51C2×C4×D157—21no—
52C2×C4×D157D411yescharacteristic
53C2×C4×D157D411yescharacteristic
54C2×C4×D157C2311yescharacteristic
55#14578937—24no—
56C2×#11929434C2312yes—
57C4⋊Dic157—22no—
58C4⋊Dic157—21no—
59C4⋊Dic157C2312yes—
60#14578933—22no—
61C22⋊Dic157—21no—
62C22⋊Dic157—21no—
63#14578931—22no—
64#14578931—21no—
65#14578931—21no—
66#14578931—21no—
67#14578931—21no—
68C4×Dic157—22no—
69C4×Dic157D411yescharacteristic
70C4×Dic157D411yescharacteristic
71C4×Dic157C2311yescharacteristic
72C2×Q8×C157—21no—
73C2×Q8×C157C2311yescharacteristic
74C2×D4×C157C2311yescharacteristic
75C2×C8×C157C2312yes—
76C4⋊C4×C157—21no—
77C4⋊C4×C157C2312yes—
78C22⋊C4×C157—21no—
79C22⋊C4×C157—21no—
80C42×C157C2311yescharacteristic
81C22×D157—21no—
82C22×D157—42no—
83C22×D157—42no—
84C22×D157—42no—
85C22×D157C2×D411yescharacteristic
86Dic314—22no—
87Dic314—42no—
88Dic314—42no—
89D628—22no—
90C22⋊D157—42no—
91C2×Dic157—21no—
92C2×Dic157—22no—
93C2×Dic157C2×D411yescharacteristic
94C2×Dic157C2×D411yescharacteristic
95C2×Dic157C2×D411yescharacteristic
96C4×D157—41no—
97C4×D157—41no—
98C4×D157—42no—
99C4×D157#4114yes—
100#11929434—22no—
101C23×C157—21no—
102Q8×C157—22no—
103Q8×C157—42no—
104D4×C157—22no—
105C2×C4×C157—21no—
106C2×C4×C157—22no—
107C2×C4×C157C2×D411yescharacteristic
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2411yescharacteristic
110C8×C157—22no—
111C22×C157—41no—
112C22×C157—41no—
113C22×C157—41no—
114C22×C157C22×D411yescharacteristic
115C2×D157—21no—
116C2×D157—21no—
117C2×D157—41no—
118C2×D157—41no—
119C2×D157—41no—
120C2×D157—84no—
121C2×D157#12412yes—
122C4×C157—21no—
123C4×C157—21no—
124C4×C157—41no—
125C4×C157—42no—
126C4×C157C2×#4112yes—
127Dic157—21no—
128Dic157—21no—
129Dic157—41no—
130Dic157—42no—
131Dic157#12412yes—
132C2×C157—81no—
133C2×C157C2×#12411yescharacteristic
134C2×C157#57511yescharacteristic
135C2×C157#57511yescharacteristic
136D157—22no—
137D157—81no—
138C157#314011yescharacteristic, complemented, Sylow
139#3140—1571nomaximal, Sylow
140#533—1571no—
141C2×#124—1571no—
142#489—1571no—
143#489—1571no—
144#489—1571no—
145#489D15711yescharacteristic, complemented
146#488—1572no—
147#488—1572no—
148#419—1572no—
149#418—1572no—
150C4×M16—1571no—
151C22×Q8—1571no—
152C22×D4—1571no—
153C2×M16—1572no—
154C4⋊Q8—1571no—
155C4⋊Q8C2×D15711yescharacteristic, complemented
156#125—1572no—
157#124—3141no—
158#124—3141no—
159#124—1571no—
160#124—1571no—
161#124—1571no—
162#124C2×D15711yescharacteristic
163C22⋊Q8—3141no—
164C22⋊Q8—3141no—
165C4×Q8—3141no—
166C4×Q8—3141no—
167#117—1572no—
168C2×C22⋊C4—3141no—
169C2×C22⋊C4—3141no—
170C2×C42—1571no—
171#103—1572no—
172#103—1572no—
173#103—1572no—
174#103C2×D15712yescomplemented
175#102—1572no—
176#102—1572no—
177#102—1572no—
178#102C2×D15712yes—
179#97—1571no—
180#97C2×D15711yescharacteristic, complemented
181C4×C8—1572no—
182C24—3141no—
183C2×Q8—21no—
184C2×Q8—3141no—
185C2×Q8—3141no—
186C2×Q8—3144no—
187C2×Q8—1571no—
188C2×Q8C22×D15711yescharacteristic
189C2×D4—3144no—
190C2×D4—3141no—
191C2×D4—1571no—
192C2×D4C22×D15711yescharacteristic
193C22×C4—3141no—
194C22×C4—1571no—
195C22×C4—1571no—
196C22×C4—1571no—
197M16—3144no—
198C2×C8—1572no—
199C2×C8C22×D15712yes—
200C4⋊C4—21no—
201C4⋊C4—3142no—
202C4⋊C4—3142no—
203C4⋊C4—3141no—
204C4⋊C4—1572no—
205C4⋊C4C22×D15712yes—
206C22⋊C4—21no—
207C22⋊C4—21no—
208C22⋊C4—3142no—
209C22⋊C4—3141no—
210C22⋊C4—3141no—
211C22⋊C4—3141no—
212C22⋊C4—3141no—
213C22⋊C4—3141no—
214C22⋊C4—3141no—
215C42—3142no—
216C42—1571no—
217C42—1571no—
218C42—1571no—
219C42C22×D15711yescharacteristic
220C23—21no—
221C23—3141no—
222C23—6282no—
223C23—6282no—
224C23—6282no—
225C23—1571no—
226Q8—22no—
227Q8—42no—
228Q8—3142no—
229Q8—6282no—
230Q8—6282no—
231D4—22no—
232D4—3142no—
233D4—6282no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—3142no—
237C2×C4—3141no—
238C2×C4—6281no—
239C2×C4—6281no—
240C2×C4—6282no—
241C2×C4—1571no—
242C2×C4—1571no—
243C2×C4—1571no—
244C2×C4—1574no—
245C2×C4D4×D15711yescharacteristic
246C2×C4D4×D15711yescharacteristic
247C2×C4C23×D15711yescharacteristic
248C8—22no—
249C8—3142no—
250C22—41no—
251C22—41no—
252C22—41no—
253C22—3141no—
254C22—3141no—
255C22—6281no—
256C22—6281no—
257C22—6281no—
258C22—1572no—
259C22—12564no—
260C22C2×D4×D15711yescharacteristic
261C4—21no—
262C4—21no—
263C4—41no—
264C4—42no—
265C4—3141no—
266C4—3141no—
267C4—6281no—
268C4—6282no—
269C4—1572no—
270C4#41×D15712yes—
271C2—81no—
272C2—3142no—
273C2—12561no—
274C2#124×D15711yescharacteristic
275C2#2375806911yescharacteristic
276C2#2375807211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#489×C157C2
socleC22×C157C22×D4
prime radicalC2×D4×D157C22
hypercenter#489D157
hypocenterC157#3140
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. #23757828C2semidirect
3. #23757635C2semidirect
4. #23757633C2semidirect
5. #23757632C2semidirect
6. #23757629C2semidirect
7. #23757626C2semidirect
8. #23757280C2semidirect
10. #23757225C2semidirect
11. #489×C157C2semidirect
15. #19097771C22semidirect
16. #19097762C22semidirect
30. #19097696C22semidirect
31. #19097695C22semidirect
32. #19097694C22semidirect
36. #19097682C22semidirect
37. C8×Dic157C22semidirect
38. C4⋊Q8×C157C22semidirect
40. #103×C157C22semidirect
42. #97×C157C22semidirect
138. C157#3140semidirect
145. #489D157semidirect
155. C4⋊Q8C2×D157semidirect
174. #103C2×D157semidirect
180. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578282
#124×D1572
#237576352
#237576332
#237576322
#237576292
#237576262
#237572802
#237572762
#237572252
#489×C1572
#3140157