Finite Group Databasebeta

← #34110092

Subgroups — #34110092

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#23757840C211yesmaximal, characteristic
2#23757753C211yesmaximal, characteristic, complemented
3#23757596C212yesmaximal, complemented
4#23757595C212yesmaximal, complemented
5#23757590C212yesmaximal, complemented
6#23757585C212yesmaximal, complemented
7#23757315C211yesmaximal, characteristic, complemented
8#23757309C211yesmaximal, characteristic, complemented
9C2×#19097692C211yesmaximal, characteristic, complemented
10#23757241C211yesmaximal, characteristic, complemented
11#418×C157C211yesmaximal, characteristic, complemented
12C2×#14578954C2211yescharacteristic
13C22×#11929434C2211yescharacteristic
14#19097767—21no—
15#19097758—21no—
16#19097742C2212yes—
17#19097742C2212yes—
18#19097740—22no—
19C4×Dic314—21no—
20C4×D628—21no—
21#19097731—22no—
22#19097730C2211yescharacteristic
23#19097729C2212yescomplemented
24C2×C4⋊Dic157C2211yescharacteristic
25C4⋊C4×D157C2212yescomplemented
26#19097705C2212yescomplemented
27#19097704C2211yescharacteristic, complemented
28#19097704C2211yescharacteristic, complemented
29#19097703C2212yescomplemented
30#19097702C2211yescharacteristic, complemented
31#19097702C2211yescharacteristic, complemented
32#19097692C2212yes—
33#19097692C2212yes—
34#19097683C2212yescomplemented
35#19097683C2212yescomplemented
36C22×D4×C157C2211yescharacteristic
37C2×M16×C157C2211yescharacteristic
38#117×C157C2211yescharacteristic
39#102×C157C2212yes—
40#102×C157C2212yes—
41#14578954—22no—
42#14578954—22no—
43C2×Dic314D411yescharacteristic
44C2×D628D411yescharacteristic
45C2×C22⋊D157—22no—
46C2×C22⋊D157—22no—
47C2×C22⋊D157D412yes—
48C22×Dic157—21no—
49C2×C4×D157—22no—
50C2×C4×D157C2312yes—
51C2×#11929434—22no—
52C2×#11929434C2311yescharacteristic
53C2×#11929434C2311yescharacteristic
54C4⋊Dic157—21no—
55C4⋊Dic157C2312yes—
56C4⋊Dic157C2311yescharacteristic
57C4⋊Dic157C2311yescharacteristic
58#14578933—22no—
59C22⋊Dic157—22no—
60C22⋊Dic157—22no—
61C22⋊Dic157—21no—
62#14578931—22no—
63C4×Dic157—22no—
64C4×Dic157—21no—
65C24×C157—21no—
66C2×D4×C157—21no—
67C2×D4×C157—22no—
68C2×D4×C157—22no—
69C2×D4×C157C2312yes—
70C22×C4×C157C2311yescharacteristic
71M16×C157—22no—
72C2×C8×C157C2312yes—
73C4⋊C4×C157C2311yescharacteristic
74C4⋊C4×C157C2311yescharacteristic
75C22⋊C4×C157—21no—
76C42×C157—21no—
77C22×D157C2×D412yes—
78Dic314—22no—
79D628—22no—
80C22⋊D157—24no—
81C22⋊D157—82no—
82C22⋊D157—82no—
83C2×Dic157—22no—
84C2×Dic157—21no—
85C2×Dic157—21no—
86C2×Dic157—81no—
87C2×Dic157C2×D412yes—
88C4×D157—22no—
89C4×D157—22no—
90C4×D157—82no—
91#11929434—21no—
92#11929434—21no—
93C23×C157—22no—
94C23×C157—42no—
95C23×C157—81no—
96C23×C157—81no—
97C23×C157C2×D411yescharacteristic
98D4×C157—22no—
99D4×C157—22no—
100D4×C157—42no—
101C2×C4×C157—21no—
102C2×C4×C157—21no—
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—42no—
111C22×C157—42no—
112C22×C157—42no—
113C22×C157—84no—
114C22×C157#12312yes—
115C22×C157C22×D411yescharacteristic
116C2×D157—22no—
117C2×D157—22no—
118C2×D157—22no—
119C4×C157—41no—
120C4×C157—41no—
121C4×C157#12312yes—
122C4×C157C2×#4111yescharacteristic
123C4×C157C2×#4111yescharacteristic
124Dic157—22no—
125Dic157—42no—
126Dic157—41no—
127Dic157—41no—
128C2×C157—22no—
129C2×C157—82no—
130C2×C157C2×#12311yescharacteristic
131C2×C157#57611yescharacteristic
132C2×C157#57711yescharacteristic
133D157—42no—
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#428—1571no—
143#427—1571no—
144#418D15711yescharacteristic, complemented
145C2×#102—1571no—
146#408—1571no—
147C2×#41—1571no—
148C22×D4C2×D15711yescharacteristic
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—1571no—
163C2×C4⋊C4—1572no—
164#107—1572no—
165#107—1571no—
166#107—1571no—
167#106—1572no—
168#106—1571no—
169#106—1571no—
170#102—1572no—
171#102—1572no—
172#102C2×D15712yes—
173#102C2×D15712yes—
174C22⋊C8—1572no—
175C22⋊C8—1572no—
176C24—21no—
177#41—3142no—
178#41—3142no—
179C2×Q8—1571no—
180C2×D4—21no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—3142no—
184C2×D4—3142no—
185C2×D4—1572no—
186C2×D4—1571no—
187C2×D4C22×D15712yes—
188C22×C4—3142no—
189C22×C4—3141no—
190C22×C4—1572no—
191C22×C4C22×D15711yescharacteristic
192M16—22no—
193C2×C8—3142no—
194C2×C8—1571no—
195C2×C8—1571no—
196C2×C8C22×D15712yes—
197C4⋊C4—3142no—
198C4⋊C4—3141no—
199C4⋊C4—1572no—
200C4⋊C4—1571no—
201C4⋊C4—1571no—
202C4⋊C4C22×D15711yescharacteristic
203C4⋊C4C22×D15711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—3142no—
206C22⋊C4—3141no—
207C22⋊C4—3142no—
208C22⋊C4—3142no—
209C42—21no—
210C42—3142no—
211C42—3141no—
212C23—22no—
213C23—42no—
214C23—81no—
215C23—81no—
216C23—1572no—
217C23D4×D15711yescharacteristic
218Q8—3142no—
219D4—22no—
220D4—22no—
221D4—42no—
222D4—3144no—
223D4—3142no—
224D4—12562no—
225D4—12562no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—3142no—
229C2×C4—3142no—
230C2×C4—3141no—
231C2×C4—3141no—
232C2×C4—3142no—
233C2×C4—1572no—
234C2×C4—12562no—
235C2×C4—12561no—
236C2×C4D4×D15711yescharacteristic
237C2×C4#1457895312yes—
238C2×C4#1457895412yes—
239C2×C4C23×D15711yescharacteristic
240C8—22no—
241C8—3141no—
242C8—3141no—
243C22—21no—
244C22—21no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—84no—
249C22—3142no—
250C22—3142no—
251C22—3142no—
252C22#1909775312yes—
253C22C2×D4×D15711yescharacteristic
254C4—41no—
255C4—41no—
256C4—3142no—
257C4—6281no—
258C4—6282no—
259C4—6281no—
260C4#1909775312yes—
261C4C2×#1457895311yescharacteristic
262C4C2×#1457895411yescharacteristic
263C2—22no—
264C2—82no—
265C2—6282no—
266C2C2×#1909775311yescharacteristic
267C2#2375807711yescharacteristic
268C2#2375808311yescharacteristic
subgroupG/H
centerC22C2×D4×D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#418×C157C2
socleC22×C157C22×D4
prime radical#23757840C2
hypercenter#418D157
hypocenterC157#3096
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. #23757753C2semidirect
3. #23757596C2semidirect
4. #23757595C2semidirect
5. #23757590C2semidirect
6. #23757585C2semidirect
7. #23757315C2semidirect
8. #23757309C2semidirect
9. C2×#19097692C2semidirect
10. #23757241C2semidirect
11. #418×C157C2semidirect
23. #19097729C22semidirect
25. C4⋊C4×D157C22semidirect
26. #19097705C22semidirect
27. #19097704C22semidirect
28. #19097704C22semidirect
29. #19097703C22semidirect
30. #19097702C22semidirect
31. #19097702C22semidirect
34. #19097683C22semidirect
35. #19097683C22semidirect
134. C157#3096semidirect
144. #418D157semidirect

Maximal subgroups

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

subgroupindex
#237578402
#237577532
#237575962
#237575952
#237575902
#237575852
#237573152
#237573092
C2×#190976922
#237572412
#418×C1572
#3096157