Finite Group Databasebeta

← #34111904

Subgroups — #34111904

Order 20096.

464 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 117 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#23758081C212yesmaximal, complemented
2#23758077C212yesmaximal, complemented
3#23757663C211yesmaximal, characteristic, complemented
4#23757541C212yesmaximal, complemented
5#23757539C212yesmaximal, complemented
6#23757450C211yesmaximal, characteristic, complemented
7#23757442C212yesmaximal, complemented
8#23757392C212yesmaximal, complemented
9#454×C157C211yesmaximal, characteristic, complemented
10#19097817C2212yescomplemented
11#19097816C2211yescharacteristic
12#19097802—22no—
13#19097802C2212yescomplemented
14#19097799C2212yes—
15#19097798—22no—
16#19097796—22no—
17#19097796C2212yescomplemented
18#19097795—22no—
19#19097792—22no—
20#19097791—22no—
21#19097791C2212yes—
22C2×#14578941—22no—
23C2×#14578941C2212yescomplemented
24C2×#14578939C2212yescomplemented
25D8×D157—22no—
26#19097779C2212yescomplemented
27#19097778C2212yescomplemented
28#19097768C2211yescharacteristic, complemented
29#19097706C2212yescomplemented
30#19097699C2212yescomplemented
31#19097688C2211yescharacteristic, complemented
32#19097687C2212yescomplemented
33#19097682C2211yescharacteristic, complemented
342−1+4×C157C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C157C2212yescomplemented
36#124×C157C2211yescharacteristic, complemented
37#104×C157C2212yescomplemented
38#101×C157C2211yescharacteristic, complemented
39#14578955—22no—
40#14578954—22no—
41#14578954C2312yes—
42#14578953—22no—
43C2×D628—22no—
44C2×D628—41no—
45C2×D628C2312yes—
46C2×C22⋊D157—41no—
47D4×D157—22no—
48D4×D157—22no—
49D4×D157—42no—
50#14578944—22no—
51#14578944D412yescomplemented
52#14578943—22no—
53#14578942D412yescomplemented
54#14578941—22no—
55#14578941—22no—
56#14578941—22no—
57#14578941—22no—
58#14578941D412yescomplemented
59D1256—22no—
60#14578939—22no—
61#14578939—22no—
62#14578939—22no—
63#14578939D412yescomplemented
64#14578938—22no—
65#14578938C2312yes—
66#14578937—22no—
67C2×#11929434—22no—
68C2×#11929434C2312yes—
69C8×D157—22no—
70#41×C157—22no—
71#41×C157—41no—
72#41×C157C2312yes—
73C2×Q8×C157—22no—
74C2×Q8×C157C2311yescharacteristic
75C2×D4×C157C2311yescharacteristic
76SD16×C157—22no—
77D8×C157—22no—
78M16×C157C2312yes—
79C22⋊C4×C157—41no—
80C42×C157C2311yescharacteristic
81C22×D157—22no—
82C22×D157—41no—
83C22×D157—42no—
84Dic314C2×D412yes—
85D628—22no—
86D628—22no—
87D628—42no—
88D628—82no—
89D628C2×D412yes—
90C22⋊D157—22no—
91C22⋊D157—42no—
92C22⋊D157—42no—
93C2×Dic157—41no—
94C4×D157—22no—
95C4×D157—42no—
96#11929434—22no—
97#11929434C2×D412yes—
98#11929434C2×D412yes—
99C23×C157—21no—
100Q8×C157—22no—
101Q8×C157—41no—
102Q8×C157C2×D412yes—
103D4×C157—22no—
104D4×C157—22no—
105D4×C157—41no—
106D4×C157C2×D412yes—
107C2×C4×C157—22no—
108C2×C4×C157—21no—
109C2×C4×C157—21no—
110C2×C4×C157—42no—
111C2×C4×C157C2411yescharacteristic
112C8×C157—22no—
113C22×C157—22no—
114C22×C157—41no—
115C22×C157—81no—
116C22×C157H(5,𝔽2)11yescharacteristic
117C2×D157—22no—
118C2×D157—42no—
119C2×D157—82no—
120C2×D157—161no—
121C2×D157—162no—
122C4×C157—22no—
123C4×C157—41no—
124C4×C157—41no—
125C4×C157C22×D412yes—
126Dic157—22no—
127C2×C157—21no—
128C2×C157—42no—
129C2×C157—81no—
130C2×C157D4211yescharacteristic
131D157—42no—
132D157—82no—
133C157#328811yescharacteristic, complemented, Sylow
134#3288—1571nomaximal, Sylow
135#577—1572no—
136#576—1572no—
137#496—1571no—
138#472—1572no—
139#471—1572no—
140#455—1571no—
141#454D15711yescharacteristic, complemented
142#453—1572no—
143#444—1572no—
1442−1+4C2×D15711yescharacteristic, complemented
145H(5,𝔽2)—1572no—
146H(5,𝔽2)—1571no—
147#137—3142no—
148#137—1572no—
149AGL(1,ℤ/8)—3142no—
150AGL(1,ℤ/8)—3142no—
151AGL(1,ℤ/8)—3142no—
152AGL(1,ℤ/8)—1572no—
153AGL(1,ℤ/8)—1572no—
154AGL(1,ℤ/8)C2×D15712yescomplemented
155#135—3142no—
156#135—3142no—
157#135—1572no—
158C2×SD16—3142no—
159C2×SD16—1572no—
160C2×D8—3142no—
161C2×D8—1572no—
162#131—1572no—
163#131—1572no—
164#127—1571no—
165#124C2×D15711yescharacteristic, complemented
166#108—1572no—
167#104—1572no—
168#104C2×D15712yescomplemented
169#101C2×D15711yescharacteristic, complemented
170#100—1571no—
171#100—1572no—
172#97—1571no—
173#41—22no—
174#41—41no—
175#41—3142no—
176#41—3142no—
177#41—3142no—
178#41—1572no—
179#41C22×D15712yes—
180C2×Q8—22no—
181C2×Q8C22×D15711yescharacteristic
182C2×D4—3142no—
183C2×D4—3142no—
184C2×D4—3142no—
185C2×D4—6281no—
186C2×D4—6282no—
187C2×D4—6281no—
188C2×D4—1572no—
189C2×D4C22×D15711yescharacteristic
190Q16—3142no—
191Q16—1572no—
192SD16—22no—
193SD16—3142no—
194SD16—3142no—
195SD16—3142no—
196SD16—3142no—
197SD16—3142no—
198SD16—1572no—
199SD16—1572no—
200D8—22no—
201D8—3142no—
202D8—3142no—
203D8—3142no—
204D8—3142no—
205D8—1572no—
206M16—3142no—
207M16—3142no—
208M16—1572no—
209M16C22×D15712yes—
210C2×C8—3142no—
211C2×C8—3142no—
212C2×C8—1572no—
213C22⋊C4—41no—
214C42C22×D15711yescharacteristic
215C23—21no—
216C23—3142no—
217C23—6282no—
218C23—6281no—
219Q8—22no—
220Q8—41no—
221Q8—1572no—
222Q8D4×D15712yes—
223D4—22no—
224D4—22no—
225D4—41no—
226D4—3142no—
227D4—3142no—
228D4—3142no—
229D4—6282no—
230D4—6282no—
231D4—6282no—
232D4—1572no—
233D4—12562no—
234D4D4×D15712yes—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—3142no—
240C2×C4—6282no—
241C2×C4—6281no—
242C2×C4C23×D15711yescharacteristic
243C8—22no—
244C8—3142no—
245C8—1572no—
246C8—1572no—
247C22—22no—
248C22—41no—
249C22—81no—
250C22—3142no—
251C22—6282no—
252C22—25122no—
253C22—25121no—
254C22—12562no—
255C22#1909781611yescharacteristic
256C4—22no—
257C4—41no—
258C4—41no—
259C4—3142no—
260C4C2×D4×D15712yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—6282no—
265C2—12562no—
266C2#2375790511yescharacteristic
subgroupG/H
centerC2#23757905
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#454×C157C2
socleC2×C157D42
hypercenter#454D157
hypocenterC157#3288
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #454

Complements

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

NH ≅ G/Ntype
1. #23758081C2semidirect
2. #23758077C2semidirect
3. #23757663C2semidirect
4. #23757541C2semidirect
5. #23757539C2semidirect
6. #23757450C2semidirect
7. #23757442C2semidirect
8. #23757392C2semidirect
9. #454×C157C2semidirect
10. #19097817C22semidirect
13. #19097802C22semidirect
17. #19097796C22semidirect
23. C2×#14578941C22semidirect
24. C2×#14578939C22semidirect
26. #19097779C22semidirect
27. #19097778C22semidirect
28. #19097768C22semidirect
29. #19097706C22semidirect
30. #19097699C22semidirect
31. #19097688C22semidirect
32. #19097687C22semidirect
33. #19097682C22semidirect
34. 2−1+4×C157C22semidirect
35. AGL(1,ℤ/8)×C157C22semidirect
36. #124×C157C22semidirect
37. #104×C157C22semidirect
38. #101×C157C22semidirect
51. #14578944D4semidirect
53. #14578942D4semidirect
58. #14578941D4semidirect
63. #14578939D4semidirect
133. C157#3288semidirect
141. #454D157semidirect
144. 2−1+4C2×D157semidirect
154. AGL(1,ℤ/8)C2×D157semidirect
165. #124C2×D157semidirect
168. #104C2×D157semidirect
169. #101C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580812
#237580772
#237576632
#237575412
#237575392
#237574502
#237574422
#237573922
#454×C1572
#3288157