Finite Group Databasebeta

← #34110094

Subgroups — #34110094

Order 20096.

418 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 133 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#23757837C211yesmaximal, characteristic
2#23757597C212yesmaximal, complemented
3#23757597C212yesmaximal, complemented
4#23757591C212yesmaximal, complemented
5#23757586C212yesmaximal, complemented
6#23757313C211yesmaximal, characteristic, complemented
7#23757309C211yesmaximal, characteristic, complemented
8#23757277C211yesmaximal, characteristic, complemented
9C2×#19097691C211yesmaximal, characteristic, complemented
10#23757240C211yesmaximal, characteristic, complemented
11#520×C157C211yesmaximal, characteristic, complemented
12C22×D628C2211yescharacteristic
13C2×#14578938C2211yescharacteristic
14C22×#11929434C2211yescharacteristic
15#19097767—21no—
16#19097760—21no—
17#19097745C2212yes—
18#19097745C2212yes—
19#19097739—22no—
20#19097730C2211yescharacteristic
21#19097704C2212yescomplemented
22#19097704C2211yescharacteristic, complemented
23#19097704C2211yescharacteristic, complemented
24#19097702C2212yescomplemented
25#19097702C2211yescharacteristic, complemented
26#19097702C2211yescharacteristic, complemented
27#19097691C2212yes—
28#19097691C2212yes—
29#19097691C2212yes—
30#19097691C2212yes—
31C22⋊#11929434C2212yescomplemented
32C22⋊#11929434C2212yescomplemented
33C2×#41×C157C2211yescharacteristic
34C4×Q8×C157—21no—
35C4×D4×C157—21no—
36C4×D4×C157—22no—
37#117×C157C2211yescharacteristic
38#117×C157C2212yescomplemented
39C2×C4⋊C4×C157C2211yescharacteristic
40C2×C4⋊C4×C157C2212yescomplemented
41C23×D157—21no—
42C2×D628—21no—
43C2×D628—22no—
44C2×D628—22no—
45C2×D628C2312yes—
46C2×C22⋊D157—22no—
47C2×C22⋊D157—22no—
48#14578938—22no—
49C2×#11929434—22no—
50C2×#11929434C2311yescharacteristic
51C2×#11929434C2312yes—
52C2×#11929434C2311yescharacteristic
53C4⋊Dic157C2311yescharacteristic
54C4⋊Dic157C2311yescharacteristic
55C22⋊Dic157—21no—
56#14578931—22no—
57#14578931—22no—
58C4×Dic157—21no—
59#41×C157—22no—
60#41×C157—22no—
61C2×Q8×C157D411yescharacteristic
62C2×D4×C157D412yes—
63C2×D4×C157D411yescharacteristic
64C22×C4×C157—21no—
65C22×C4×C157—22no—
66C22×C4×C157C2312yes—
67C22×C4×C157C2311yescharacteristic
68C4⋊C4×C157—21no—
69C4⋊C4×C157—22no—
70C4⋊C4×C157C2312yes—
71C4⋊C4×C157C2311yescharacteristic
72C4⋊C4×C157C2311yescharacteristic
73C22⋊C4×C157—21no—
74C22⋊C4×C157—22no—
75C42×C157—21no—
76C42×C157—22no—
77C22×D157—22no—
78C22×D157—42no—
79C22×D157—81no—
80C22×D157—81no—
81D628—22no—
82D628—22no—
83D628—42no—
84C22⋊D157—82no—
85C22⋊D157—82no—
86C2×Dic157—21no—
87C2×Dic157—21no—
88#11929434—21no—
89#11929434—21no—
90#11929434—22no—
91C23×C157C2×D412yes—
92C23×C157C2×D411yescharacteristic
93Q8×C157—22no—
94D4×C157—24no—
95D4×C157—22no—
96C2×C4×C157—22no—
97C2×C4×C157—22no—
98C2×C4×C157—21no—
99C2×C4×C157—21no—
100C2×C4×C157—22no—
101C2×C4×C157—81no—
102C2×C4×C157—82no—
103C2×C4×C157C2×D412yes—
104C2×C4×C157C2×D411yescharacteristic
105C2×C4×C157#4112yes—
106C2×C4×C157#4112yes—
107C2×C4×C157C2411yescharacteristic
108C22×C157—21no—
109C22×C157—21no—
110C22×C157—22no—
111C22×C157—22no—
112C22×C157—22no—
113C22×C157#12312yes—
114C22×C157C22×D411yescharacteristic
115C2×D157—42no—
116C2×D157—42no—
117C2×D157—42no—
118C2×D157—84no—
119C4×C157—22no—
120C4×C157—41no—
121C4×C157—42no—
122C4×C157—41no—
123C4×C157#12312yes—
124C4×C157C2×#4111yescharacteristic
125C4×C157C2×#4111yescharacteristic
126Dic157—41no—
127Dic157—41no—
128C2×C157—22no—
129C2×C157—42no—
130C2×C157C2×#12311yescharacteristic
131C2×C157#57611yescharacteristic
132C2×C157#57711yescharacteristic
133D157—82no—
134C157#309611yescharacteristic, complemented, Sylow
135#3096—1571nomaximal, Sylow
136#535—1571no—
137#520D15711yescharacteristic, complemented
138#482—1572no—
139#482—1572no—
140#481—1572no—
141#480—1572no—
142#428—1571no—
143#427—1571no—
144#418—1571no—
145C2×#102—1571no—
146#408—1571no—
147C2×#41C2×D15711yescharacteristic
148C22×D4—1571no—
149C2×M16—1571no—
150C22×C8—1571no—
151#127—3141no—
152#124—3141no—
153#121—1572no—
154#121—1572no—
155#120—3142no—
156C4×Q8—21no—
157C4×D4—21no—
158C4×D4—22no—
159#117—1571no—
160#117C2×D15711yescharacteristic
161#117C2×D15712yescomplemented
162C2×C4⋊C4C2×D15711yescharacteristic
163C2×C4⋊C4C2×D15712yescomplemented
164#107—1571no—
165#107—1571no—
166#107—1572no—
167#106—1571no—
168#106—1571no—
169#106—1572no—
170#102—1572no—
171#102—1572no—
172#102—1572no—
173#102—1572no—
174C22⋊C8—1572no—
175C22⋊C8—1572no—
176C24—3141no—
177#41—22no—
178#41—22no—
179C2×Q8C22⋊D15711yescharacteristic
180C2×D4—3142no—
181C2×D4—3142no—
182C2×D4—3141no—
183C2×D4—3142no—
184C2×D4—3142no—
185C2×D4—1572no—
186C2×D4C22⋊D15712yes—
187C2×D4C22⋊D15711yescharacteristic
188C22×C4—21no—
189C22×C4—22no—
190C22×C4C22×D15712yes—
191C22×C4C22×D15711yescharacteristic
192M16—3142no—
193C2×C8—3142no—
194C2×C8—1571no—
195C2×C8—1572no—
196C2×C8—1571no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—1571no—
200C4⋊C4—1571no—
201C4⋊C4C22×D15712yes—
202C4⋊C4C22×D15711yescharacteristic
203C4⋊C4C22×D15711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—22no—
206C22⋊C4—3141no—
207C22⋊C4—3142no—
208C22⋊C4—3142no—
209C42—21no—
210C42—22no—
211C42—3141no—
212C23—3142no—
213C23—6282no—
214C23—12561no—
215C23—12561no—
216C23C2×C22⋊D15712yes—
217C23C2×C22⋊D15711yescharacteristic
218Q8—22no—
219D4—24no—
220D4—22no—
221D4—3142no—
222D4—3142no—
223D4—6282no—
224D4—12562no—
225D4—12562no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—81no—
232C2×C4—82no—
233C2×C4—3141no—
234C2×C4—3141no—
235C2×C4C2×C22⋊D15712yes—
236C2×C4C2×C22⋊D15711yescharacteristic
237C2×C4#1457895412yes—
238C2×C4#1457895412yes—
239C2×C4C23×D15711yescharacteristic
240C8—3141no—
241C8—3141no—
242C8—3142no—
243C22—21no—
244C22—21no—
245C22—22no—
246C22—22no—
247C22—22no—
248C22—6282no—
249C22—6282no—
250C22—6282no—
251C22—12564no—
252C22#1909775412yes—
253C22C22×C22⋊D15711yescharacteristic
254C4—22no—
255C4—41no—
256C4—41no—
257C4—42no—
258C4—6281no—
259C4—6281no—
260C4#1909775412yes—
261C4C2×#1457895411yescharacteristic
262C4C2×#1457895411yescharacteristic
263C2—22no—
264C2—42no—
265C2—12562no—
266C2C2×#1909775411yescharacteristic
267C2#2375807611yescharacteristic
268C2#2375808111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D157
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#520×C157C2
socleC22×C157C22×D4
prime radical#23757837C2
hypercenter#520D157
hypocenterC157#3096
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C2×#41 ◁ #520

Complements

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

NH ≅ G/Ntype
2. #23757597C2semidirect
3. #23757597C2semidirect
4. #23757591C2semidirect
5. #23757586C2semidirect
6. #23757313C2semidirect
7. #23757309C2semidirect
8. #23757277C2semidirect
9. C2×#19097691C2semidirect
10. #23757240C2semidirect
11. #520×C157C2semidirect
21. #19097704C22semidirect
22. #19097704C22semidirect
23. #19097704C22semidirect
24. #19097702C22semidirect
25. #19097702C22semidirect
26. #19097702C22semidirect
31. C22⋊#11929434C22semidirect
32. C22⋊#11929434C22semidirect
38. #117×C157C22semidirect
40. C2×C4⋊C4×C157C22semidirect
134. C157#3096semidirect
137. #520D157semidirect
161. #117C2×D157semidirect
163. C2×C4⋊C4C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237578372
#237575972
#237575912
#237575862
#237573132
#237573092
#237572772
C2×#190976912
#237572402
#520×C1572
#3096157