Finite Group Databasebeta

← #34109510

Subgroups — #34109510

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×#19097799C212yesmaximal, complemented
3#23757837C211yesmaximal, characteristic, complemented
4#23757448C214yesmaximal, complemented
5#23757443C214yesmaximal, complemented
6#23757289C212yesmaximal, complemented
7#413×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—
16C22×D628C2211yescharacteristic, complemented
17#19097799—24no—
18#19097799C2218yescomplemented
19C2×#14578943—22no—
20C2×D1256—22no—
21#19097767C2212yescomplemented
22#19097760C2212yescomplemented
23#19097745—22no—
24#19097739—22no—
25#19097730C2211yescharacteristic, complemented
26#19097697C2218yescomplemented
27C2×#41×C157C2211yescharacteristic
28C2×M16×C157C2212yescomplemented
29#101×C157C2212yescomplemented
30#100×C157C2212yescomplemented
31C23×D157—21no—
32#14578955—24no—
33#14578954—24no—
34#14578954—24no—
35#14578954—22no—
36#14578954—24no—
37#14578954—22no—
38#14578954D414yescomplemented
39#14578954C2314yes—
40#14578953—28no—
41#14578953—24no—
42C2×Dic314—22no—
43C2×Dic314D412yescomplemented
44Q8×D157—24no—
45C2×D628—22no—
46C2×D628—22no—
47C2×D628—22no—
48C2×D628—21no—
49C2×D628D412yescomplemented
50C2×D628C2312yes—
51C2×C22⋊D157—24no—
52C2×C22⋊D157—22no—
53C2×C22⋊D157—24no—
54D4×D157—28no—
55D4×D157—24no—
56C2×C4×D157—24no—
57C2×C4×D157—22no—
58#14578943—28no—
59D1256—28no—
60C4⋊Dic157—21no—
61C22⋊Dic157—21no—
62#14578931—24no—
63C4×Dic157C2312yes—
64#41×C157—22no—
65#41×C157—22no—
66C2×Q8×C157C2311yescharacteristic
67C2×D4×C157D412yes—
68C2×D4×C157C2311yescharacteristic
69C22×C4×C157D412yes—
70C22×C4×C157C2311yescharacteristic
71M16×C157—22no—
72M16×C157C2314yes—
73C2×C8×C157—22no—
74C22×D157—24no—
75C22×D157—22no—
76C22×D157—22no—
77C22×D157—42no—
78C22×D157—82no—
79Dic314—24no—
80Dic314—22no—
81Dic314C2×D414yes—
82D628—24no—
83D628—22no—
84D628—24no—
85D628—41no—
86D628—41no—
87D628C2×D414yes—
88C22⋊D157—28no—
89C22⋊D157—24no—
90C22⋊D157—24no—
91C22⋊D157—84no—
92C2×Dic157—24no—
93C2×Dic157—22no—
94C2×Dic157—22no—
95C4×D157—28no—
96C4×D157—24no—
97C4×D157—24no—
98C23×C157C2×D412yes—
99C23×C157C2×D411yescharacteristic
100Q8×C157—22no—
101D4×C157—24no—
102D4×C157—22no—
103C2×C4×C157—22no—
104C2×C4×C157—22no—
105C2×C4×C157C2×D412yes—
106C2×C4×C157C2×D412yes—
107C2×C4×C157C2×D412yes—
108C2×C4×C157C2×D411yescharacteristic
109C2×C4×C157C2411yescharacteristic
110C8×C157—24no—
111C22×C157—22no—
112C22×C157—41no—
113C22×C157—42no—
114C22×C157#12012yes—
115C22×C157C22×D411yescharacteristic
116C2×D157—24no—
117C2×D157—44no—
118C2×D157—42no—
119C2×D157—82no—
120C2×D157—84no—
121C2×D157—84no—
122C4×C157—22no—
123C4×C157#12011yescharacteristic
124C4×C157#12011yescharacteristic
125C4×C157C22×D412yes—
126Dic157—24no—
127Dic157—42no—
128C2×C157—21no—
129C2×C157—22no—
130C2×C157—42no—
131C2×C157C2×#12011yescharacteristic
132D157—44no—
133D157—82no—
134C157#302111yescharacteristic, complemented, Sylow
135#3021—1571nomaximal, Sylow
136#585—1571no—
137C2×AGL(1,ℤ/8)—1572no—
138#535—1571no—
139#454—1574no—
140#453—1574no—
141#421—1572no—
142#413D15711yescharacteristic, complemented
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×D4—1571no—
153AGL(1,ℤ/8)—3144no—
154AGL(1,ℤ/8)—1578no—
155C2×SD16—3142no—
156C2×D8—3142no—
157C2×M16C2×D15712yescomplemented
158#127—1572no—
159#124—1572no—
160#121—3142no—
161#120—3142no—
162#117—1571no—
163#104—1578no—
164#101C2×D15712yescomplemented
165#100C2×D15712yescomplemented
166C24—3141no—
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—3142no—
184C2×D4—3148no—
185C2×D4—3144no—
186C2×D4—3144no—
187C2×D4—3142no—
188C2×D4—3144no—
189C2×D4—3142no—
190C2×D4—3142no—
191C2×D4—3141no—
192C2×D4—1572no—
193C2×D4—1572no—
194C2×D4D62812yes—
195C2×D4C22×D15711yescharacteristic
196C22×C4—3144no—
197C22×C4—3142no—
198C22×C4D62812yes—
199C22×C4C22×D15711yescharacteristic
200SD16—3148no—
201D8—3148no—
202M16—22no—
203M16C22×D15714yes—
204C2×C8—22no—
205C4⋊C4—3141no—
206C22⋊C4—3141no—
207C22⋊C4—3144no—
208C42—1572no—
209C23—3144no—
210C23—3142no—
211C23—3142no—
212C23—6282no—
213C23—12562no—
214C23C2×D62812yes—
215C23C2×D62811yescharacteristic
216Q8—22no—
217Q8—3144no—
218Q8—3142no—
219Q8—1574no—
220D4—24no—
221D4—22no—
222D4—3144no—
223D4—3148no—
224D4—3144no—
225D4—3144no—
226D4—3142no—
227D4—3144no—
228D4—6281no—
229D4—6281no—
230D4—1574no—
231D4—12564no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—3148no—
235C2×C4—3144no—
236C2×C4—3142no—
237C2×C4—3144no—
238C2×C4—3144no—
239C2×C4—3142no—
240C2×C4D4×D15712yes—
241C2×C4D4×D15712yes—
242C2×C4C2×D62812yes—
243C2×C4C2×D62811yescharacteristic
244C2×C4C23×D15711yescharacteristic
245C8—24no—
246C22—22no—
247C22—41no—
248C22—42no—
249C22—3144no—
250C22—6284no—
251C22—6282no—
252C22—12562no—
253C22—12564no—
254C22—12564no—
255C22#1909774112yes—
256C22C22×D62811yescharacteristic
257C4—22no—
258C4—3144no—
259C4—6282no—
260C4#1909774111yescharacteristic
261C4#1909774111yescharacteristic
262C4C2×D4×D15712yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—6284no—
267C2—12562no—
268C2C2×#1909774111yescharacteristic
subgroupG/H
centerC2C2×#19097741
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#413×C157C2
socleC2×C157C2×#120
hypercenter#413D157
hypocenterC157#3021
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #23758109C2semidirect
2. C2×#19097799C2semidirect
3. #23757837C2semidirect
4. #23757448C2semidirect
5. #23757443C2semidirect
6. #23757289C2semidirect
7. #413×C157C2semidirect
8. #19097819C22semidirect
12. #19097816C22semidirect
16. C22×D628C22semidirect
18. #19097799C22semidirect
21. #19097767C22semidirect
22. #19097760C22semidirect
25. #19097730C22semidirect
26. #19097697C22semidirect
28. C2×M16×C157C22semidirect
29. #101×C157C22semidirect
30. #100×C157C22semidirect
38. #14578954D4semidirect
43. C2×Dic314D4semidirect
49. C2×D628D4semidirect
134. C157#3021semidirect
142. #413D157semidirect
157. C2×M16C2×D157semidirect
164. #101C2×D157semidirect
165. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237581092
C2×#190977992
#237578372
#237574482
#237574432
#237572892
#413×C1572
#3021157