Finite Group Databasebeta

← #34109507

Subgroups — #34109507

Order 20096.

736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 145 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#23758109C211yesmaximal, characteristic, complemented
2C2×#19097797C212yesmaximal, complemented
3#23757445C214yesmaximal, complemented
4#23757440C214yesmaximal, complemented
5#23757291C212yesmaximal, complemented
6#23757257C211yesmaximal, characteristic, complemented
7#535×C157C211yesmaximal, characteristic, complemented
8#19097819C2212yescomplemented
9#19097818—22no—
10#19097817—22no—
11#19097816—22no—
12#19097816C2212yescomplemented
13C2×#14578954—22no—
14C2×#14578954C2212yes—
15#41×D157—24no—
16#19097797—24no—
17#19097797C2218yescomplemented
18C2×#14578942—22no—
19C2×#14578939—22no—
20C2×#14578938C2212yescomplemented
21#19097699C2218yescomplemented
22#19097690C2212yescomplemented
23#19097688C2212yescomplemented
24C2×#41×C157C2211yescharacteristic
25C22×D4×C157C2211yescharacteristic, complemented
26#127×C157C2212yescomplemented
27#124×C157C2212yescomplemented
28#121×C157—22no—
29#120×C157—22no—
30#117×C157C2211yescharacteristic, complemented
31#14578955—24no—
32#14578954—24no—
33#14578954—24no—
34#14578954—22no—
35#14578954—24no—
36#14578954—22no—
37#14578954D414yescomplemented
38#14578954C2314yes—
39#14578953—28no—
40#14578953—24no—
41C2×Dic314—22no—
42C2×Dic314D412yescomplemented
43Q8×D157—24no—
44C2×D628—22no—
45C2×D628D412yescomplemented
46C2×C22⋊D157—24no—
47C2×C22⋊D157—22no—
48D4×D157—28no—
49D4×D157—24no—
50C2×C4×D157—24no—
51C2×C4×D157—22no—
52#14578942—28no—
53#14578939—28no—
54#14578938—22no—
55#14578938C2314yes—
56C2×#11929434—22no—
57C24×C157—21no—
58#41×C157—22no—
59#41×C157—22no—
60C2×Q8×C157C2311yescharacteristic
61C2×D4×C157—24no—
62C2×D4×C157—22no—
63C2×D4×C157—22no—
64C2×D4×C157—21no—
65C2×D4×C157D412yes—
66C2×D4×C157C2312yes—
67C2×D4×C157C2311yescharacteristic
68C22×C4×C157D412yes—
69C22×C4×C157C2311yescharacteristic
70C4⋊C4×C157—21no—
71C22⋊C4×C157—21no—
72C22⋊C4×C157—24no—
73C42×C157C2312yes—
74C22×D157—24no—
75C22×D157—22no—
76Dic314—24no—
77Dic314—22no—
78Dic314C2×D414yes—
79D628—24no—
80D628—22no—
81D628C2×D414yes—
82C22⋊D157—28no—
83C22⋊D157—24no—
84C22⋊D157—24no—
85C2×Dic157—24no—
86C2×Dic157—22no—
87C4×D157—28no—
88C4×D157—24no—
89C4×D157—24no—
90#11929434—24no—
91C23×C157—22no—
92C23×C157—42no—
93C23×C157—82no—
94C23×C157C2×D412yes—
95C23×C157C2×D411yescharacteristic
96Q8×C157—22no—
97D4×C157—24no—
98D4×C157—22no—
99D4×C157—24no—
100D4×C157—41no—
101D4×C157—41no—
102D4×C157—84no—
103C2×C4×C157—22no—
104C2×C4×C157—22no—
105C2×C4×C157—22no—
106C2×C4×C157C2×D412yes—
107C2×C4×C157C2×D412yes—
108C2×C4×C157C2×D412yes—
109C2×C4×C157C2×D411yescharacteristic
110C2×C4×C157C2411yescharacteristic
111C22×C157—22no—
112C22×C157—41no—
113C22×C157—42no—
114C22×C157—44no—
115C22×C157—42no—
116C22×C157—84no—
117C22×C157#12012yes—
118C22×C157C22×D411yescharacteristic
119C2×D157—24no—
120C2×D157—82no—
121C2×D157—84no—
122C4×C157—22no—
123C4×C157—42no—
124C4×C157#12011yescharacteristic
125C4×C157#12011yescharacteristic
126C4×C157C22×D412yes—
127Dic157—24no—
128C2×C157—21no—
129C2×C157—22no—
130C2×C157—42no—
131C2×C157—82no—
132C2×C157C2×#12011yescharacteristic
133D157—44no—
134C157#302111yescharacteristic, complemented, Sylow
135#3021—1571nomaximal, Sylow
136#585—1571no—
137C2×AGL(1,ℤ/8)—1572no—
138#535D15711yescharacteristic, complemented
139#454—1574no—
140#453—1574no—
141#421—1572no—
142#413—1571no—
1432−1+4—3142no—
1442−1+4—1572no—
145H(5,𝔽2)—3142no—
146H(5,𝔽2)—3142no—
147H(5,𝔽2)—1572no—
148C2×#41—3144no—
149C2×#41—3142no—
150C2×#41—1572no—
151C2×#41C2×D15711yescharacteristic
152C22×D4C2×D15711yescharacteristic, complemented
153AGL(1,ℤ/8)—3144no—
154AGL(1,ℤ/8)—1578no—
155C2×SD16—3142no—
156C2×D8—3142no—
157C2×M16—1572no—
158#127C2×D15712yescomplemented
159#124C2×D15712yescomplemented
160#121—22no—
161#120—22no—
162#117C2×D15711yescharacteristic, complemented
163#104—1578no—
164#101—1572no—
165#100—1572no—
166C24—21no—
167#41—22no—
168#41—22no—
169#41—3144no—
170#41—3144no—
171#41—3148no—
172#41—3144no—
173#41—3144no—
174#41—3142no—
175#41—3142no—
176#41—3144no—
177#41—1574no—
178#41—1574no—
179C2×Q8—3142no—
180C2×Q8—3144no—
181C2×Q8—1572no—
182C2×Q8C22×D15711yescharacteristic
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—21no—
187C2×D4—3142no—
188C2×D4—3148no—
189C2×D4—3144no—
190C2×D4—3144no—
191C2×D4—3142no—
192C2×D4—1572no—
193C2×D4C22⋊D15712yes—
194C2×D4C22×D15712yes—
195C2×D4C22×D15711yescharacteristic
196C22×C4—3144no—
197C22×C4—3142no—
198C22×C4C22⋊D15712yes—
199C22×C4C22×D15711yescharacteristic
200SD16—3148no—
201D8—3148no—
202M16—3142no—
203M16—1574no—
204C2×C8—3142no—
205C4⋊C4—21no—
206C22⋊C4—21no—
207C22⋊C4—24no—
208C42C22×D15712yes—
209C23—22no—
210C23—42no—
211C23—82no—
212C23—3144no—
213C23—3142no—
214C23C2×C22⋊D15712yes—
215C23C2×C22⋊D15711yescharacteristic
216Q8—22no—
217Q8—3144no—
218Q8—3142no—
219Q8—1574no—
220D4—24no—
221D4—22no—
222D4—24no—
223D4—41no—
224D4—41no—
225D4—84no—
226D4—3144no—
227D4—3148no—
228D4—3144no—
229D4—3144no—
230D4—3142no—
231D4—1574no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—3148no—
236C2×C4—3144no—
237C2×C4—3142no—
238C2×C4—3144no—
239C2×C4—3144no—
240C2×C4D4×D15712yes—
241C2×C4D4×D15712yes—
242C2×C4C2×C22⋊D15712yes—
243C2×C4C2×C22⋊D15711yescharacteristic
244C2×C4C23×D15711yescharacteristic
245C8—3144no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—44no—
250C22—42no—
251C22—84no—
252C22—3144no—
253C22—12562no—
254C22—12564no—
255C22#1909773912yes—
256C22C22×C22⋊D15711yescharacteristic
257C4—22no—
258C4—42no—
259C4—3144no—
260C4#1909773911yescharacteristic
261C4#1909773911yescharacteristic
262C4C2×D4×D15712yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—6284no—
268C2C2×#1909773911yescharacteristic
subgroupG/H
centerC2C2×#19097739
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#535×C157C2
socleC2×C157C2×#120
hypercenter#535D157
hypocenterC157#3021
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #23758109C2semidirect
2. C2×#19097797C2semidirect
3. #23757445C2semidirect
4. #23757440C2semidirect
5. #23757291C2semidirect
6. #23757257C2semidirect
7. #535×C157C2semidirect
8. #19097819C22semidirect
12. #19097816C22semidirect
17. #19097797C22semidirect
20. C2×#14578938C22semidirect
21. #19097699C22semidirect
22. #19097690C22semidirect
23. #19097688C22semidirect
25. C22×D4×C157C22semidirect
26. #127×C157C22semidirect
27. #124×C157C22semidirect
30. #117×C157C22semidirect
37. #14578954D4semidirect
42. C2×Dic314D4semidirect
45. C2×D628D4semidirect
134. C157#3021semidirect
138. #535D157semidirect
152. C22×D4C2×D157semidirect
158. #127C2×D157semidirect
159. #124C2×D157semidirect
162. #117C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
C2×#190977972
#237574452
#237574402
#237572912
#237572572
#535×C1572
#3021157