Finite Group Databasebeta

← #34110096

Subgroups — #34110096

Order 20096.

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

normal subgroups only

This group has 418 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
1#23757842C211yesmaximal, characteristic
2#23757754C211yesmaximal, characteristic, complemented
3#23757600C212yesmaximal, complemented
4#23757599C212yesmaximal, complemented
5#23757593C212yesmaximal, complemented
6#23757588C212yesmaximal, complemented
7#23757309C211yesmaximal, characteristic, complemented
8#23757279C211yesmaximal, characteristic, complemented
9C2×#19097691C211yesmaximal, characteristic, complemented
10#23757241C211yesmaximal, characteristic, complemented
11#428×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×D628C2211yescharacteristic
14C22×#11929434C2211yescharacteristic
15#19097768—21no—
16#19097759—21no—
17#19097746C2212yes—
18#19097746C2212yes—
19#19097741—22no—
20#19097736—21no—
21#19097733—21no—
22C4×C22⋊D157—22no—
23#19097730C2211yescharacteristic
24#19097729C2212yescomplemented
25C4⋊C4×D157C2212yescomplemented
26#19097704C2211yescharacteristic, complemented
27#19097704C2211yescharacteristic, complemented
28#19097702C2211yescharacteristic, complemented
29#19097702C2211yescharacteristic, complemented
30#19097693C2212yes—
31#19097693C2212yes—
32#19097691C2212yes—
33#19097691C2212yes—
34#19097683C2212yescomplemented
35#19097683C2212yescomplemented
36C2×M16×C157C2211yescharacteristic
37#117×C157C2211yescharacteristic
38C2×C4⋊C4×C157C2211yescharacteristic
39#107×C157C2212yescomplemented
40#106×C157C2212yescomplemented
41C23×D157—21no—
42#14578954—22no—
43#14578954—22no—
44C2×Dic314D411yescharacteristic
45C2×D628—22no—
46C2×D628—22no—
47C2×D628—21no—
48C2×D628—22no—
49C2×D628—22no—
50C2×D628D411yescharacteristic
51C2×D628C2312yes—
52C2×C22⋊D157D412yes—
53C2×C4×D157—22no—
54C2×C4×D157C2312yes—
55C2×#11929434—22no—
56C2×#11929434C2311yescharacteristic
57C2×#11929434C2311yescharacteristic
58C4⋊Dic157C2311yescharacteristic
59C4⋊Dic157C2311yescharacteristic
60#14578933—22no—
61C22⋊Dic157—21no—
62#14578931—22no—
63#14578931—22no—
64#14578931—22no—
65C4×Dic157—22no—
66C4×Dic157—21no—
67C22×C4×C157—21no—
68C22×C4×C157C2311yescharacteristic
69M16×C157—22no—
70C2×C8×C157C2312yes—
71C4⋊C4×C157—21no—
72C4⋊C4×C157C2311yescharacteristic
73C4⋊C4×C157C2312yes—
74C4⋊C4×C157C2311yescharacteristic
75C22⋊C4×C157—21no—
76C42×C157—21no—
77C22×D157—22no—
78C22×D157—42no—
79C22×D157—81no—
80C22×D157—81no—
81C22×D157C2×D412yes—
82Dic314—22no—
83D628—22no—
84D628—22no—
85D628—22no—
86D628—42no—
87D628—82no—
88D628—82no—
89C22⋊D157—24no—
90C2×Dic157—21no—
91C2×Dic157—21no—
92C2×Dic157C2×D412yes—
93C4×D157—22no—
94C4×D157—22no—
95C4×D157—82no—
96#11929434—21no—
97#11929434—21no—
98C23×C157C2×D411yescharacteristic
99C2×C4×C157—22no—
100C2×C4×C157—21no—
101C2×C4×C157—21no—
102C2×C4×C157—81no—
103C2×C4×C157C2×D411yescharacteristic
104C2×C4×C157#4112yes—
105C2×C4×C157#4112yes—
106C2×C4×C157C2411yescharacteristic
107C8×C157—22no—
108C22×C157—21no—
109C22×C157—21no—
110C22×C157#12312yes—
111C22×C157C22×D411yescharacteristic
112C2×D157—22no—
113C2×D157—22no—
114C2×D157—22no—
115C2×D157—42no—
116C2×D157—42no—
117C2×D157—42no—
118C2×D157—84no—
119C4×C157—42no—
120C4×C157—41no—
121C4×C157—41no—
122C4×C157#12312yes—
123C4×C157C2×#4111yescharacteristic
124C4×C157C2×#4111yescharacteristic
125Dic157—22no—
126Dic157—41no—
127Dic157—41no—
128C2×C157—22no—
129C2×C157C2×#12311yescharacteristic
130C2×C157#57611yescharacteristic
131C2×C157#57711yescharacteristic
132D157—42no—
133D157—82no—
134C157#309611yescharacteristic, complemented, Sylow
135#3096—1571nomaximal, Sylow
136#535—1571no—
137#520—1571no—
138#482—1572no—
139#482—1572no—
140#481—1572no—
141#480—1572no—
142#428D15711yescharacteristic, complemented
143#427—1571no—
144#418—1571no—
145C2×#102—1571no—
146#408—1571no—
147C2×#41—1571no—
148C22×D4—1571no—
149C2×M16C2×D15711yescharacteristic
150C22×C8—1571no—
151#127—3141no—
152#124—3141no—
153#121—1572no—
154#121—1572no—
155#120—3142no—
156C4×Q8—3141no—
157C4×D4—3141no—
158C4×D4—3142no—
159#117—1571no—
160#117—1572no—
161#117C2×D15711yescharacteristic
162C2×C4⋊C4—1572no—
163C2×C4⋊C4C2×D15711yescharacteristic
164#107—1571no—
165#107—1571no—
166#107C2×D15712yescomplemented
167#106—1571no—
168#106—1571no—
169#106C2×D15712yescomplemented
170#102—1572no—
171#102—1572no—
172#102—1572no—
173#102—1572no—
174C22⋊C8—1572no—
175C22⋊C8—1572no—
176C24—3141no—
177#41—3142no—
178#41—3142no—
179C2×Q8—1571no—
180C2×D4—3142no—
181C2×D4—3142no—
182C2×D4—3141no—
183C2×D4—3142no—
184C2×D4—3142no—
185C2×D4—1572no—
186C2×D4—1571no—
187C2×D4—1572no—
188C22×C4—21no—
189C22×C4—3142no—
190C22×C4—1572no—
191C22×C4C22×D15711yescharacteristic
192M16—22no—
193C2×C8—3142no—
194C2×C8—1571no—
195C2×C8—1571no—
196C2×C8C22×D15712yes—
197C4⋊C4—21no—
198C4⋊C4—3142no—
199C4⋊C4—1571no—
200C4⋊C4—1571no—
201C4⋊C4C22×D15712yes—
202C4⋊C4C22×D15711yescharacteristic
203C4⋊C4C22×D15711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—3142no—
206C22⋊C4—3142no—
207C22⋊C4—3142no—
208C22⋊C4—3141no—
209C42—21no—
210C42—3142no—
211C42—3141no—
212C23—3142no—
213C23—6282no—
214C23—1572no—
215C23—12561no—
216C23—12561no—
217C23D4×D15711yescharacteristic
218Q8—3142no—
219D4—3144no—
220D4—3142no—
221D4—3142no—
222D4—3142no—
223D4—6282no—
224D4—12562no—
225D4—12562no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—22no—
229C2×C4—81no—
230C2×C4—3142no—
231C2×C4—3142no—
232C2×C4—3141no—
233C2×C4—3141no—
234C2×C4—1572no—
235C2×C4—12562no—
236C2×C4D4×D15711yescharacteristic
237C2×C4#1457895412yes—
238C2×C4#1457895512yes—
239C2×C4C23×D15711yescharacteristic
240C8—22no—
241C8—3141no—
242C8—3141no—
243C22—21no—
244C22—21no—
245C22—3142no—
246C22—3142no—
247C22—3142no—
248C22—6282no—
249C22—6282no—
250C22—6282no—
251C22—12564no—
252C22#1909775612yes—
253C22C2×D4×D15711yescharacteristic
254C4—41no—
255C4—42no—
256C4—41no—
257C4—3142no—
258C4—6281no—
259C4—6281no—
260C4#1909775612yes—
261C4C2×#1457895411yescharacteristic
262C4C2×#1457895511yescharacteristic
263C2—22no—
264C2—6282no—
265C2—12562no—
266C2C2×#1909775611yescharacteristic
267C2#2375807711yescharacteristic
268C2#2375808211yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#428×C157C2
socleC22×C157C22×D4
prime radical#23757842C2
hypercenter#428D157
hypocenterC157#3096
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #23757754C2semidirect
3. #23757600C2semidirect
4. #23757599C2semidirect
5. #23757593C2semidirect
6. #23757588C2semidirect
7. #23757309C2semidirect
8. #23757279C2semidirect
9. C2×#19097691C2semidirect
10. #23757241C2semidirect
11. #428×C157C2semidirect
24. #19097729C22semidirect
25. C4⋊C4×D157C22semidirect
26. #19097704C22semidirect
27. #19097704C22semidirect
28. #19097702C22semidirect
29. #19097702C22semidirect
34. #19097683C22semidirect
35. #19097683C22semidirect
39. #107×C157C22semidirect
40. #106×C157C22semidirect
134. C157#3096semidirect
142. #428D157semidirect
166. #107C2×D157semidirect
169. #106C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578422
#237577542
#237576002
#237575992
#237575932
#237575882
#237573092
#237572792
C2×#190976912
#237572412
#428×C1572
#3096157