Finite Group Databasebeta

← #34111932

Subgroups — #34111932

Order 20096.

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

normal subgroups only

This group has 476 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#23758077C212yesmaximal, complemented
2#23758076C212yesmaximal, complemented
3#23757649C211yesmaximal, characteristic, complemented
4#23757542C212yesmaximal, complemented
5#23757541C212yesmaximal, complemented
6#23757442C212yesmaximal, complemented
7#23757440C211yesmaximal, characteristic, complemented
8#23757387C212yesmaximal, complemented
9#453×C157C211yesmaximal, characteristic, complemented
10#19097817C2212yescomplemented
11#19097816C2211yescharacteristic, complemented
12#19097799C2212yescomplemented
13#19097798—22no—
14#19097797—22no—
15#19097797C2212yescomplemented
16#19097796—22no—
17#19097796C2212yescomplemented
18#19097795—22no—
19#19097792—22no—
20#19097791—22no—
21#19097791C2212yes—
22C2×#14578939—22no—
23C2×#14578939C2212yescomplemented
24C2×#14578939C2212yescomplemented
25D8×D157—22no—
26#19097779C2212yescomplemented
27#19097778C2212yescomplemented
28#19097768C2211yescharacteristic, complemented
29#19097706C2212yescomplemented
30#19097699C2212yescomplemented
31#19097688C2211yescharacteristic, complemented
32#19097687C2212yescomplemented
33C4×#11929434C2211yescharacteristic, complemented
34H(5,𝔽2)×C157C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C157C2212yescomplemented
36#127×C157C2211yescharacteristic, complemented
37#104×C157C2212yescomplemented
38#100×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#14578944D412yescomplemented
51#14578943—22no—
52#14578942—22no—
53#14578942D412yescomplemented
54#14578941—22no—
55#14578941D412yescomplemented
56D1256—22no—
57#14578939—22no—
58#14578939—22no—
59#14578939—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×C157C2312yes—
72C2×D4×C157—22no—
73C2×D4×C157—41no—
74C2×D4×C157—41no—
75C2×D4×C157C2311yescharacteristic
76C2×D4×C157C2311yescharacteristic
77SD16×C157—22no—
78D8×C157—22no—
79M16×C157C2312yes—
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—
100C23×C157—21no—
101C23×C157—41no—
102Q8×C157C2×D412yes—
103D4×C157—22no—
104D4×C157—22no—
105D4×C157—22no—
106D4×C157—42no—
107D4×C157—82no—
108D4×C157C2×D412yes—
109C2×C4×C157—21no—
110C2×C4×C157—22no—
111C2×C4×C157—41no—
112C2×C4×C157C2411yescharacteristic
113C8×C157—22no—
114C22×C157—22no—
115C22×C157—41no—
116C22×C157—41no—
117C22×C157—81no—
118C22×C157—81no—
119C22×C157—161no—
120C22×C157H(5,𝔽2)11yescharacteristic
121C2×D157—22no—
122C2×D157—42no—
123C2×D157—82no—
124C2×D157—161no—
125C2×D157—162no—
126C4×C157—22no—
127C4×C157—22no—
128C4×C157C22×D412yes—
129Dic157—22no—
130C2×C157—21no—
131C2×C157—42no—
132C2×C157—81no—
133C2×C157—81no—
134C2×C157D4211yescharacteristic
135D157—42no—
136D157—82no—
137C157#329211yescharacteristic, complemented, Sylow
138#3292—1571nomaximal, Sylow
139#576—1572no—
140#576—1572no—
141#493—1571no—
142#472—1572no—
143#472—1572no—
144#453—1571no—
145#453—1572no—
146#453D15711yescharacteristic, complemented
147#443—1572no—
148H(5,𝔽2)—1572no—
149H(5,𝔽2)—1571no—
150H(5,𝔽2)C2×D15711yescharacteristic, complemented
151AGL(1,ℤ/8)—3142no—
152AGL(1,ℤ/8)—3142no—
153AGL(1,ℤ/8)—3142no—
154AGL(1,ℤ/8)—3142no—
155AGL(1,ℤ/8)—1572no—
156AGL(1,ℤ/8)—1572no—
157AGL(1,ℤ/8)—1572no—
158AGL(1,ℤ/8)C2×D15712yescomplemented
159#135—3142no—
160#135—3142no—
161#135—1572no—
162C2×D8—3142no—
163C2×D8—3142no—
164C2×D8—1572no—
165C2×D8—1572no—
166#131—1572no—
167#131—1572no—
168#127—1571no—
169#127C2×D15711yescharacteristic, complemented
170#108—1572no—
171#104—1572no—
172#104C2×D15712yescomplemented
173#100—1571no—
174#100—1572no—
175#100C2×D15711yescharacteristic, complemented
176C4×C8—1571no—
177#41—22no—
178#41—3142no—
179#41—3142no—
180#41—3142no—
181#41—1572no—
182#41C22×D15712yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—3142no—
187C2×D4—3142no—
188C2×D4—3142no—
189C2×D4—6281no—
190C2×D4—6282no—
191C2×D4—6281no—
192C2×D4—1572no—
193C2×D4C22×D15711yescharacteristic
194C2×D4C22×D15711yescharacteristic
195Q16—1572no—
196SD16—22no—
197SD16—3142no—
198SD16—3142no—
199SD16—3142no—
200SD16—1572no—
201SD16—1572no—
202D8—22no—
203D8—3142no—
204D8—3142no—
205D8—3142no—
206D8—3142no—
207D8—3142no—
208D8—3142no—
209D8—3142no—
210D8—1572no—
211M16—3142no—
212M16—3142no—
213M16—1572no—
214M16C22×D15712yes—
215C2×C8—3142no—
216C2×C8—3142no—
217C2×C8—1572no—
218C42C22×D15711yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—3142no—
223C23—6282no—
224C23—6281no—
225Q8—1572no—
226Q8D4×D15712yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—3142no—
233D4—3142no—
234D4—3142no—
235D4—6282no—
236D4—6282no—
237D4—6282no—
238D4—1572no—
239D4—12562no—
240D4D4×D15712yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—3142no—
245C2×C4—6282no—
246C2×C4—6281no—
247C2×C4C23×D15711yescharacteristic
248C8—22no—
249C8—3142no—
250C8—1572no—
251C8—1572no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—3142no—
259C22—6282no—
260C22—25122no—
261C22—25121no—
262C22—12562no—
263C22#1909781611yescharacteristic
264C4—22no—
265C4—22no—
266C4—3142no—
267C4C2×D4×D15712yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—6282no—
273C2—12562no—
274C2#2375790511yescharacteristic
subgroupG/H
centerC2#23757905
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#453×C157C2
socleC2×C157D42
hypercenter#453D157
hypocenterC157#3292
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #23758077C2semidirect
2. #23758076C2semidirect
3. #23757649C2semidirect
4. #23757542C2semidirect
5. #23757541C2semidirect
6. #23757442C2semidirect
7. #23757440C2semidirect
8. #23757387C2semidirect
9. #453×C157C2semidirect
10. #19097817C22semidirect
11. #19097816C22semidirect
12. #19097799C22semidirect
15. #19097797C22semidirect
17. #19097796C22semidirect
23. C2×#14578939C22semidirect
24. C2×#14578939C22semidirect
26. #19097779C22semidirect
27. #19097778C22semidirect
28. #19097768C22semidirect
29. #19097706C22semidirect
30. #19097699C22semidirect
31. #19097688C22semidirect
32. #19097687C22semidirect
33. C4×#11929434C22semidirect
34. H(5,𝔽2)×C157C22semidirect
35. AGL(1,ℤ/8)×C157C22semidirect
36. #127×C157C22semidirect
37. #104×C157C22semidirect
38. #100×C157C22semidirect
50. #14578944D4semidirect
53. #14578942D4semidirect
55. #14578941D4semidirect
63. #14578939D4semidirect
137. C157#3292semidirect
146. #453D157semidirect
150. H(5,𝔽2)C2×D157semidirect
158. AGL(1,ℤ/8)C2×D157semidirect
169. #127C2×D157semidirect
172. #104C2×D157semidirect
175. #100C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580772
#237580762
#237576492
#237575422
#237575412
#237574422
#237574402
#237573872
#453×C1572
#3292157