Finite Group Databasebeta

← #34109493

Subgroups — #34109493

Order 20096.

762 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 161 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#23758071C212yesmaximal, complemented
2#23757840C211yesmaximal, characteristic, complemented
3#23757451C214yesmaximal, complemented
4#23757441C214yesmaximal, complemented
5#23757290C212yesmaximal, complemented
6C2×#19097688C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C157C211yesmaximal, characteristic, complemented
8C2×#14578954C2211yescharacteristic, complemented
9#19097802—22no—
10#19097801C2214yescomplemented
11#19097797—22no—
12#19097796C2214yescomplemented
13#19097791—24no—
14C2×#14578938C2212yescomplemented
15#19097767C2212yescomplemented
16#19097758C2212yescomplemented
17#19097742—22no—
18#19097740—22no—
19#19097730C2211yescharacteristic, complemented
20#19097698C2218yescomplemented
21#19097688C2214yescomplemented
22H(5,𝔽2)×C157—24no—
23H(5,𝔽2)×C157—22no—
24H(5,𝔽2)×C157C2214yes—
25C2×#41×C157—22no—
26C2×#41×C157C2212yes—
27C22×D4×C157—22no—
28C22×D4×C157—22no—
29C22×D4×C157C2211yescharacteristic
30#14578954—22no—
31#14578954—22no—
32C2×Dic314C2311yescharacteristic, complemented
33C2×D628C2311yescharacteristic, complemented
34C2×C22⋊D157—21no—
35C2×C22⋊D157—24no—
36C2×C4×D157—21no—
37#14578944—24no—
38#14578942—24no—
39#14578941—24no—
40#14578939—24no—
41#14578938—22no—
42#14578938C2314yescomplemented
43C2×#11929434—22no—
44C4⋊Dic157—21no—
45C22⋊Dic157—21no—
46C22⋊Dic157—24no—
47C4×Dic157C2312yescomplemented
48C24×C157—22no—
49C24×C157D412yes—
50#41×C157—28no—
51#41×C157—28no—
52#41×C157—22no—
53#41×C157—22no—
54#41×C157D414yescomplemented
55#41×C157C2314yes—
56C2×Q8×C157D412yescomplemented
57C2×D4×C157—24no—
58C2×D4×C157—28no—
59C2×D4×C157—28no—
60C2×D4×C157—22no—
61C2×D4×C157—28no—
62C2×D4×C157—28no—
63C2×D4×C157—22no—
64C2×D4×C157—22no—
65C2×D4×C157D412yes—
66C2×D4×C157D412yescomplemented
67C2×D4×C157C2312yes—
68C22×C4×C157—22no—
69C22×C4×C157—22no—
70C22×C4×C157C2311yescharacteristic
71C22×D157—21no—
72Dic314—22no—
73D628—22no—
74C22⋊D157—41no—
75C22⋊D157—41no—
76C22⋊D157—84no—
77C2×Dic157—21no—
78C2×Dic157—22no—
79C4×D157—42no—
80#11929434—24no—
81C23×C157—28no—
82C23×C157—22no—
83C23×C157—22no—
84C23×C157—22no—
85C23×C157—82no—
86C23×C157—84no—
87C23×C157C2×D414yes—
88C23×C157C2×D411yescharacteristic
89Q8×C157—22no—
90Q8×C157C2×D414yes—
91D4×C157—216no—
92D4×C157—24no—
93D4×C157—24no—
94D4×C157—24no—
95D4×C157—22no—
96D4×C157—24no—
97D4×C157C2×D414yes—
98C2×C4×C157—28no—
99C2×C4×C157—22no—
100C2×C4×C157—24no—
101C2×C4×C157—24no—
102C2×C4×C157C2×D411yescharacteristic
103C2×C4×C157C2×D412yes—
104C2×C4×C157C2×D412yes—
105C2×C4×C157C2411yescharacteristic
106C22×C157—24no—
107C22×C157—24no—
108C22×C157—41no—
109C22×C157—44no—
110C22×C157—44no—
111C22×C157—44no—
112C22×C157—82no—
113C22×C157—88no—
114C22×C157#12011yescharacteristic
115C22×C157#12011yescharacteristic
116C22×C157C22×D411yescharacteristic
117C2×D157—42no—
118C2×D157—41no—
119C4×C157—24no—
120C4×C157#12012yes—
121C4×C157C22×D412yes—
122Dic157—42no—
123Dic157—41no—
124C2×C157—21no—
125C2×C157—21no—
126C2×C157—21no—
127C2×C157—44no—
128C2×C157—44no—
129C2×C157C2×#12011yescharacteristic
130D157—81no—
131C157#301911yescharacteristic, complemented, Sylow
132#3019—1571nomaximal, Sylow
133C2×H(5,𝔽2)D15711yescharacteristic, complemented
134#575—1572no—
135#535—1571no—
136#455—1574no—
137#453—1574no—
138#421—1572no—
139C2×#100—1571no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D15714yes—
143C2×#41—22no—
144C2×#41—1571no—
145C2×#41C2×D15712yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D15711yescharacteristic
149#137—3142no—
150#137—1574no—
151AGL(1,ℤ/8)—3142no—
152AGL(1,ℤ/8)—1574no—
153#135—3144no—
154C2×M16—1572no—
155#127—1572no—
156#124—1572no—
157#121—3142no—
158#120—3142no—
159#117—1571no—
160#104—1578no—
161#100—1574no—
162C24—22no—
163C24C22⋊D15712yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—3142no—
169#41—3142no—
170#41C22⋊D15714yescomplemented
171#41C22×D15714yes—
172C2×Q8—1571no—
173C2×Q8C22⋊D15712yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—3141no—
183C2×D4—3144no—
184C2×D4—1571no—
185C2×D4C22⋊D15712yes—
186C2×D4C22⋊D15712yescomplemented
187C2×D4C22×D15712yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—3141no—
191C22×C4C22×D15711yescharacteristic
192Q16—3144no—
193SD16—3144no—
194SD16—3144no—
195D8—3144no—
196M16—3142no—
197M16—1574no—
198C2×C8—3142no—
199C4⋊C4—3141no—
200C22⋊C4—3141no—
201C22⋊C4—3144no—
202C42—1572no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—3141no—
210C23C2×C22⋊D15714yes—
211C23C2×C22⋊D15711yescharacteristic
212Q8—22no—
213Q8—3142no—
214Q8C2×C22⋊D15714yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—3142no—
222D4—6281no—
223D4—6281no—
224D4—12564no—
225D4C2×C22⋊D15714yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—3142no—
231C2×C4—3141no—
232C2×C4—6282no—
233C2×C4C2×C22⋊D15711yescharacteristic
234C2×C4C2×C22⋊D15712yes—
235C2×C4C2×C22⋊D15712yes—
236C2×C4C23×D15711yescharacteristic
237C8—3144no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—6282no—
247C22—6281no—
248C22#1909774011yescharacteristic
249C22#1909774011yescharacteristic
250C22C22×C22⋊D15711yescharacteristic
251C4—24no—
252C4—6282no—
253C4—6281no—
254C4#1909774012yes—
255C4C22×C22⋊D15712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—12561no—
262C2C2×#1909774011yescharacteristic
subgroupG/H
centerC2C2×#19097740
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
FittingC2×H(5,𝔽2)×C157C2
socleC2×C157C2×#120
hypercenterC2×H(5,𝔽2)D157
hypocenterC157#3019
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #23758071C2semidirect
2. #23757840C2semidirect
3. #23757451C2semidirect
4. #23757441C2semidirect
5. #23757290C2semidirect
6. C2×#19097688C2semidirect
7. C2×H(5,𝔽2)×C157C2semidirect
8. C2×#14578954C22semidirect
10. #19097801C22semidirect
12. #19097796C22semidirect
14. C2×#14578938C22semidirect
15. #19097767C22semidirect
16. #19097758C22semidirect
19. #19097730C22semidirect
20. #19097698C22semidirect
21. #19097688C22semidirect
32. C2×Dic314C23semidirect
33. C2×D628C23semidirect
42. #14578938C23semidirect
47. C4×Dic157C23semidirect
54. #41×C157D4semidirect
56. C2×Q8×C157D4semidirect
66. C2×D4×C157D4semidirect
131. C157#3019semidirect
133. C2×H(5,𝔽2)D157semidirect
170. #41C22⋊D157semidirect
173. C2×Q8C22⋊D157semidirect
186. C2×D4C22⋊D157semidirect

Maximal subgroups

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

subgroupindex
#237580712
#237578402
#237574512
#237574412
#237572902
C2×#190976882
C2×H(5,𝔽2)×C1572
#3019157