Finite Group Databasebeta

← #34111930

Subgroups — #34111930

Order 20096.

460 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 460 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#23758085C212yesmaximal, complemented
2#23758080C212yesmaximal, complemented
3#23757646C211yesmaximal, characteristic, complemented
4#23757537C212yesmaximal, complemented
5#23757536C212yesmaximal, complemented
6#23757457C212yesmaximal, complemented
7#23757442C212yesmaximal, complemented
8#23757391C211yesmaximal, characteristic, complemented
9#443×C157C211yesmaximal, characteristic, complemented
10#19097818C2212yes—
11#19097817C2212yescomplemented
12#19097804—22no—
13#19097804C2212yescomplemented
14#19097803—22no—
15#19097801C2212yescomplemented
16#19097799—22no—
17#19097799C2212yes—
18#19097796C2212yes—
19#19097795—22no—
20#19097793—22no—
21#19097793C2211yescharacteristic
22#19097792—22no—
23C2×#14578943—22no—
24C2×#14578943C2212yescomplemented
25C2×#14578943C2212yescomplemented
26SD16×D157—22no—
27#19097777C2212yescomplemented
28#19097771C2211yescharacteristic, complemented
29#19097768C2211yescharacteristic
30#19097707C2211yescharacteristic, complemented
31#19097699C2212yescomplemented
32#19097689C2212yescomplemented
33#19097687C2212yescomplemented
34#135×C157C2211yescharacteristic
35#131×C157C2212yescomplemented
36#108×C157C2211yescharacteristic, complemented
37#104×C157C2212yescomplemented
38C4×C8×C157C2211yescharacteristic, complemented
39#14578955—22no—
40#14578954—22no—
41#14578954—22no—
42#14578954C2312yes—
43#14578953—22no—
44#14578953—42no—
45C2×Dic314—22no—
46C2×Dic314C2312yes—
47Q8×D157—22no—
48C2×D628—22no—
49C2×D628—41no—
50C2×D628C2312yes—
51D4×D157—22no—
52D4×D157—42no—
53Dic628—22no—
54Dic628D411yescharacteristic, complemented
55#14578944—22no—
56#14578943—22no—
57#14578943—22no—
58#14578943—22no—
59#14578943—22no—
60#14578943—22no—
61#14578943—22no—
62#14578943D412yescomplemented
63#14578942—22no—
64#14578941—22no—
65D1256—22no—
66D1256D411yescharacteristic, complemented
67#14578939—22no—
68#14578938C2312yes—
69#14578937—22no—
70C8×D157—22no—
71C4⋊Dic157—41no—
72#41×C157C2312yes—
73Q16×C157D411yescharacteristic, complemented
74SD16×C157D412yescomplemented
75D8×C157D411yescharacteristic, complemented
76M16×C157—22no—
77M16×C157C2312yes—
78C2×C8×C157—22no—
79C2×C8×C157C2311yescharacteristic
80C2×C8×C157C2311yescharacteristic
81C42×C157C2311yescharacteristic
82C22×D157—22no—
83C22×D157—42no—
84Dic314—22no—
85Dic314—22no—
86Dic314—42no—
87Dic314C2×D412yes—
88D628—22no—
89D628—22no—
90D628—42no—
91D628—82no—
92D628C2×D412yes—
93C22⋊D157—22no—
94C22⋊D157—42no—
95C22⋊D157—42no—
96C2×Dic157—22no—
97C2×Dic157—42no—
98C4×D157—22no—
99C4×D157—42no—
100C4×D157—42no—
101#11929434—22no—
102Q8×C157C2×D412yes—
103D4×C157—22no—
104D4×C157C2×D412yes—
105C2×C4×C157—21no—
106C2×C4×C157—22no—
107C2×C4×C157C2411yescharacteristic
108C8×C157—22no—
109C8×C157C2×D411yescharacteristic
110C8×C157C2×D411yescharacteristic
111C8×C157C2×D411yescharacteristic
112C8×C157C2×D411yescharacteristic
113C22×C157—22no—
114C22×C157H(5,𝔽2)11yescharacteristic
115C2×D157—22no—
116C2×D157—42no—
117C2×D157—82no—
118C2×D157—162no—
119C4×C157—22no—
120C4×C157—22no—
121C4×C157C22×D411yescharacteristic
122C4×C157C22×D411yescharacteristic
123Dic157—22no—
124Dic157—42no—
125C2×C157—21no—
126C2×C157—42no—
127C2×C157D4211yescharacteristic
128D157—42no—
129D157—82no—
130C157#329111yescharacteristic, complemented, Sylow
131#3291—1571nomaximal, Sylow
132#577—1572no—
133#577—1572no—
134#492—1571no—
135#471—1572no—
136#471—1572no—
137#456—1572no—
138#453—1572no—
139#443—1571no—
140#443D15711yescharacteristic, complemented
1412−1+4—1572no—
142H(5,𝔽2)—1572no—
143#137—3142no—
144#137—3142no—
145#137—1572no—
146#137—1572no—
147AGL(1,ℤ/8)—3142no—
148AGL(1,ℤ/8)—3142no—
149AGL(1,ℤ/8)—1572no—
150AGL(1,ℤ/8)—1572no—
151#135—3142no—
152#135—3142no—
153#135—1571no—
154#135C2×D15711yescharacteristic
155C2×SD16—3142no—
156C2×SD16—3142no—
157C2×SD16—1572no—
158C2×SD16—1572no—
159#131—1572no—
160#131C2×D15712yescomplemented
161C4⋊Q8—1571no—
162#127—1571no—
163#108—1571no—
164#108C2×D15711yescharacteristic, complemented
165#104—1572no—
166#104C2×D15712yescomplemented
167#101—1572no—
168#100—1572no—
169C4×C8C2×D15711yescharacteristic, complemented
170#41—3142no—
171#41—3142no—
172#41—3142no—
173#41—3142no—
174#41—6282no—
175#41—1572no—
176#41C22×D15712yes—
177C2×Q8—3142no—
178C2×Q8—3142no—
179C2×Q8—1572no—
180C2×D4—3142no—
181C2×D4—3142no—
182C2×D4—6281no—
183C2×D4—6282no—
184C2×D4—1572no—
185Q16—3142no—
186Q16—3142no—
187Q16—1571no—
188Q16D62811yescharacteristic, complemented
189SD16—3142no—
190SD16—3142no—
191SD16—3142no—
192SD16—3142no—
193SD16—3142no—
194SD16—3142no—
195SD16—3142no—
196SD16—3142no—
197SD16—1572no—
198SD16D62812yescomplemented
199D8—3142no—
200D8—3142no—
201D8—1571no—
202D8D62811yescharacteristic, complemented
203M16—22no—
204M16—3142no—
205M16—1572no—
206M16C22×D15712yes—
207C2×C8—22no—
208C2×C8—3142no—
209C2×C8C22×D15711yescharacteristic
210C2×C8C22×D15711yescharacteristic
211C4⋊C4—6281no—
212C42C22×D15711yescharacteristic
213C23—3142no—
214C23—6282no—
215Q8—3142no—
216Q8—3142no—
217Q8—6282no—
218Q8—1572no—
219Q8C2×D62812yes—
220D4—22no—
221D4—3142no—
222D4—3142no—
223D4—3142no—
224D4—6282no—
225D4—6282no—
226D4—6282no—
227D4—1572no—
228D4—12562no—
229D4C2×D62812yes—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—3142no—
233C2×C4—3142no—
234C2×C4—6282no—
235C2×C4—6282no—
236C2×C4—6282no—
237C2×C4C23×D15711yescharacteristic
238C8—22no—
239C8—3142no—
240C8D4×D15711yescharacteristic
241C8D4×D15711yescharacteristic
242C8C2×D62811yescharacteristic
243C8C2×D62811yescharacteristic
244C22—22no—
245C22—3142no—
246C22—6282no—
247C22—25122no—
248C22—12562no—
249C22#1909781711yescharacteristic
250C4—22no—
251C4—22no—
252C4—3142no—
253C4—6282no—
254C4C2×D4×D15711yescharacteristic
255C4C22×D62811yescharacteristic
256C2—21no—
257C2—42no—
258C2—6282no—
259C2—12562no—
260C2D4×D62811yescharacteristic
subgroupG/H
centerC2D4×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#443×C157C2
socleC2×C157D42
hypercenter#443D157
hypocenterC157#3291
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #23758085C2semidirect
2. #23758080C2semidirect
3. #23757646C2semidirect
4. #23757537C2semidirect
5. #23757536C2semidirect
6. #23757457C2semidirect
7. #23757442C2semidirect
8. #23757391C2semidirect
9. #443×C157C2semidirect
11. #19097817C22semidirect
13. #19097804C22semidirect
15. #19097801C22semidirect
24. C2×#14578943C22semidirect
25. C2×#14578943C22semidirect
27. #19097777C22semidirect
28. #19097771C22semidirect
30. #19097707C22semidirect
31. #19097699C22semidirect
32. #19097689C22semidirect
33. #19097687C22semidirect
35. #131×C157C22semidirect
36. #108×C157C22semidirect
37. #104×C157C22semidirect
38. C4×C8×C157C22semidirect
54. Dic628D4semidirect
62. #14578943D4semidirect
66. D1256D4semidirect
73. Q16×C157D4semidirect
74. SD16×C157D4semidirect
75. D8×C157D4semidirect
130. C157#3291semidirect
140. #443D157semidirect
160. #131C2×D157semidirect
164. #108C2×D157semidirect
166. #104C2×D157semidirect
169. C4×C8C2×D157semidirect
188. Q16D628semidirect
198. SD16D628semidirect
202. D8D628semidirect

Maximal subgroups

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

subgroupindex
#237580852
#237580802
#237576462
#237575372
#237575362
#237574572
#237574422
#237573912
#443×C1572
#3291157