Finite Group Databasebeta

← #34110599

Subgroups — #34110599

Order 20096.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#19097803C212yesmaximal, complemented
2C2×#19097798C212yesmaximal, complemented
3#23757824C211yesmaximal, characteristic, complemented
4#124×D157C211yesmaximal, characteristic, complemented
5#23757665C211yesmaximal, characteristic, complemented
6#23757663C211yesmaximal, characteristic, complemented
7#23757662C211yesmaximal, characteristic, complemented
8#23757658C212yesmaximal, complemented
9#23757654C212yesmaximal, complemented
10#23757225C211yesmaximal, characteristic, complemented
11#497×C157C211yesmaximal, characteristic, complemented
12C2×#14578955C2212yes—
13C2×Q8×D157C2211yescharacteristic
14C2×D4×D157C2211yescharacteristic
15#19097803—28no—
16#19097798—28no—
17C2×#14578944C2212yescomplemented
18C2×#14578943C2212yescomplemented
19C2×#14578941C2212yes—
20C2×#14578941C2212yescomplemented
21C2×D1256C2212yes—
22C2×#14578939C2212yes—
23C2×#14578937C2212yes—
24#19097768C2211yescharacteristic, complemented
25#19097760C2211yescharacteristic
26#19097760C2212yescomplemented
27#19097759C2211yescharacteristic
28#19097758C2211yescharacteristic
29#19097757—22no—
30#19097746—22no—
31#19097733—22no—
32C22⋊C4×D157—22no—
33C42×D157C2211yescharacteristic
34#19097682C2211yescharacteristic, complemented
35C8×Dic157C2212yescomplemented
36C2×Q16×C157C2212yescomplemented
37C2×SD16×C157C2212yes—
38C4⋊Q8×C157C2211yescharacteristic, complemented
39#124×C157C2211yescharacteristic
40#97×C157C2211yescharacteristic, complemented
41C23×D157—21no—
42#14578955—24no—
43#14578955—24no—
44C2×Dic314—21no—
45C2×Dic314C2311yescharacteristic
46Q8×D157—24no—
47C2×D628—22no—
48C2×D628—22no—
49C2×D628C2311yescharacteristic
50C2×D628C2312yes—
51C2×C22⋊D157—21no—
52D4×D157—24no—
53C2×C4×D157—21no—
54C2×C4×D157—22no—
55C2×C4×D157D412yes—
56C2×C4×D157C2311yescharacteristic
57#14578944—24no—
58#14578943—24no—
59#14578941—24no—
60#14578941—24no—
61D1256—24no—
62#14578939—24no—
63#14578937D418yescomplemented
64C2×#11929434C2312yes—
65C22⋊Dic157—22no—
66#14578931—22no—
67#14578931—24no—
68#14578931—22no—
69C4×Dic157D412yes—
70C4×Dic157C2311yescharacteristic
71C2×Q8×C157C2311yescharacteristic
72C2×Q8×C157C2312yes—
73C2×D4×C157C2311yescharacteristic
74Q16×C157—24no—
75SD16×C157—24no—
76C2×C8×C157C2312yes—
77C4⋊C4×C157—22no—
78C22⋊C4×C157—22no—
79C42×C157C2311yescharacteristic
80C22×D157—22no—
81C22×D157—21no—
82C22×D157—44no—
83C22×D157—42no—
84C22×D157C2×D411yescharacteristic
85Dic314—22no—
86Dic314—42no—
87D628—22no—
88D628—22no—
89D628—22no—
90D628—44no—
91D628—44no—
92C22⋊D157—42no—
93C2×Dic157—21no—
94C2×Dic157C2×D412yes—
95C2×Dic157C2×D411yescharacteristic
96C4×D157—42no—
97C4×D157—42no—
98C4×D157—44no—
99C4×D157C2×D414yes—
100#11929434C2×D414yes—
101C23×C157—21no—
102Q8×C157—22no—
103Q8×C157—22no—
104Q8×C157—22no—
105D4×C157—22no—
106C2×C4×C157—22no—
107C2×C4×C157—21no—
108C2×C4×C157C2×D412yes—
109C2×C4×C157C2411yescharacteristic
110C8×C157C2×D414yes—
111C22×C157—41no—
112C22×C157—42no—
113C22×C157C22×D411yescharacteristic
114C2×D157—22no—
115C2×D157—42no—
116C2×D157—41no—
117C2×D157—44no—
118C2×D157—42no—
119C2×D157—84no—
120C2×D157#12712yes—
121C4×C157—22no—
122C4×C157—42no—
123C4×C157—41no—
124C4×C157C22×D412yes—
125Dic157—22no—
126Dic157—41no—
127Dic157#12712yes—
128C2×C157—81no—
129C2×C157C2×#12711yescharacteristic
130C2×C157#57512yes—
131D157—22no—
132D157—82no—
133D157—81no—
134C157#315311yescharacteristic, complemented, Sylow
135#3153—1571nomaximal, Sylow
136C2×#137—1572no—
137C2×AGL(1,ℤ/8)—1572no—
138#532—1571no—
139C2×#124—1571no—
140#497—1571no—
141#497D15711yescharacteristic, complemented
142#496—1571no—
143#496—1571no—
144#495—1572no—
145#495—1572no—
146C4×M16—1571no—
147C2×#41—1572no—
148C22×Q8—1571no—
149C22×D4—1571no—
150#137—3148no—
151AGL(1,ℤ/8)—3148no—
152C2×Q16—1572no—
153C2×Q16C2×D15712yescomplemented
154C2×SD16—1572no—
155C2×SD16—1572no—
156C2×SD16—1572no—
157C2×SD16C2×D15712yes—
158C2×D8—1572no—
159C2×D8—1572no—
160C2×M16—1572no—
161C4⋊Q8C2×D15711yescharacteristic, complemented
162#127—1571no—
163#124—3142no—
164#124—1571no—
165#124—1572no—
166#124—1571no—
167#124—1571no—
168#124C2×D15711yescharacteristic
169#121—3142no—
170C4×D4—3142no—
171C2×C22⋊C4—3142no—
172C2×C42—1571no—
173#97—1571no—
174#97C2×D15711yescharacteristic, complemented
175C4×C8—1572no—
176C24—3141no—
177#41—3144no—
178#41—3144no—
179C2×Q8—3141no—
180C2×Q8—3144no—
181C2×Q8—1571no—
182C2×Q8C22×D15711yescharacteristic
183C2×Q8C22×D15712yes—
184C2×D4—3142no—
185C2×D4—3144no—
186C2×D4—3141no—
187C2×D4—3142no—
188C2×D4—1571no—
189C2×D4—1572no—
190C2×D4C22×D15711yescharacteristic
191C22×C4—3141no—
192C22×C4—3142no—
193C22×C4—1572no—
194C22×C4—1571no—
195Q16—24no—
196Q16—3144no—
197SD16—24no—
198SD16—3144no—
199SD16—3144no—
200SD16—3144no—
201D8—3144no—
202D8—3144no—
203M16—1578no—
204C2×C8—1572no—
205C2×C8C22×D15712yes—
206C4⋊C4—22no—
207C22⋊C4—22no—
208C22⋊C4—3142no—
209C22⋊C4—3144no—
210C22⋊C4—3142no—
211C22⋊C4—3142no—
212C42—1572no—
213C42—1571no—
214C42C22×D15711yescharacteristic
215C23—21no—
216C23—3141no—
217C23—3142no—
218C23—6284no—
219C23—6282no—
220C23—1571no—
221Q8—22no—
222Q8—22no—
223Q8—22no—
224Q8—3142no—
225Q8—6282no—
226D4—22no—
227D4—3142no—
228D4—3142no—
229D4—3142no—
230D4—6284no—
231D4—6284no—
232D4—6282no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—3141no—
236C2×C4—6282no—
237C2×C4—6282no—
238C2×C4—6284no—
239C2×C4—1572no—
240C2×C4—1571no—
241C2×C4—1574no—
242C2×C4D4×D15712yes—
243C2×C4C23×D15711yescharacteristic
244C8—1574no—
245C8D4×D15714yes—
246C22—41no—
247C22—42no—
248C22—3142no—
249C22—6282no—
250C22—6281no—
251C22—6284no—
252C22—6282no—
253C22—1572no—
254C22—12564no—
255C22C2×D4×D15711yescharacteristic
256C4—22no—
257C4—42no—
258C4—41no—
259C4—3142no—
260C4—6281no—
261C4—1572no—
262C4C2×D4×D15712yes—
263C2—81no—
264C2—3142no—
265C2—12561no—
266C2—12562no—
267C2#127×D15711yescharacteristic
268C2#2375807212yes—
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#497×C157C2
socleC22×C157C22×D4
hypercenter#497D157
hypocenterC157#3153
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
1. C2×#19097803C2semidirect
2. C2×#19097798C2semidirect
3. #23757824C2semidirect
4. #124×D157C2semidirect
5. #23757665C2semidirect
6. #23757663C2semidirect
7. #23757662C2semidirect
8. #23757658C2semidirect
9. #23757654C2semidirect
10. #23757225C2semidirect
11. #497×C157C2semidirect
17. C2×#14578944C22semidirect
18. C2×#14578943C22semidirect
20. C2×#14578941C22semidirect
24. #19097768C22semidirect
26. #19097760C22semidirect
34. #19097682C22semidirect
35. C8×Dic157C22semidirect
36. C2×Q16×C157C22semidirect
38. C4⋊Q8×C157C22semidirect
40. #97×C157C22semidirect
63. #14578937D4semidirect
134. C157#3153semidirect
141. #497D157semidirect
153. C2×Q16C2×D157semidirect
161. C4⋊Q8C2×D157semidirect
174. #97C2×D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978032
C2×#190977982
#237578242
#124×D1572
#237576652
#237576632
#237576622
#237576582
#237576542
#237572252
#497×C1572
#3153157