Finite Group Databasebeta

← #34111925

Subgroups — #34111925

Order 20096.

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

normal subgroups only

This group has 454 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#23758086C211yesmaximal, characteristic, complemented
2#23758082C212yesmaximal, complemented
3#23758078C211yesmaximal, characteristic, complemented
4#23757658C211yesmaximal, characteristic, complemented
5#23757544C211yesmaximal, characteristic, complemented
6#23757539C212yesmaximal, complemented
7#23757449C212yesmaximal, complemented
8#23757448C212yesmaximal, complemented
9#23757389C211yesmaximal, characteristic, complemented
10#23757388C211yesmaximal, characteristic, complemented
11#473×C157C211yesmaximal, characteristic, complemented
12#19097819C2212yescomplemented
13#19097817C2212yes—
14#19097803—22no—
15#19097803—22no—
16#19097802C2212yescomplemented
17#19097799—22no—
18#19097799C2212yes—
19#19097799C2212yescomplemented
20#19097798—22no—
21#19097794—22no—
22#19097794—21no—
23#19097793—21no—
24#19097793C2211yescharacteristic
25#19097791C2211yescharacteristic
26Q16×D157—21no—
27C2×#14578941C2212yescomplemented
28SD16×D157—22no—
29C2×D1256—21no—
30C2×D1256C2211yescharacteristic
31#19097779C2212yescomplemented
32#19097777C2211yescharacteristic, complemented
33#19097760C2212yescomplemented
34#19097706C2211yescharacteristic, complemented
35#19097698C2211yescharacteristic
36#19097697C2211yescharacteristic, complemented
37#19097697C2212yescomplemented
38#19097687C2212yescomplemented
39C8×Dic157C2211yescharacteristic, complemented
40#137×C157C2212yes—
41C2×Q16×C157C2211yescharacteristic, complemented
42#131×C157C2211yescharacteristic
43#108×C157C2211yescharacteristic, complemented
44#101×C157C2212yescomplemented
45#14578955—22no—
46#14578955—22no—
47#14578955—22no—
48#14578954—22no—
49#14578954—42no—
50#14578954C2311yescharacteristic
51#14578954C2312yes—
52Q8×D157—22no—
53Q8×D157—22no—
54C2×D628—22no—
55C2×D628C2312yes—
56D4×D157—22no—
57D4×D157—42no—
58Dic628D411yescharacteristic
59#14578944—22no—
60#14578944D411yescharacteristic
61#14578943—22no—
62#14578943—22no—
63#14578943D412yes—
64#14578942D411yescharacteristic
65#14578941—22no—
66#14578941—22no—
67#14578941—22no—
68#14578941D411yescharacteristic
69D1256—22no—
70D1256—22no—
71D1256—21no—
72D1256—21no—
73D1256D411yescharacteristic
74#14578939D411yescharacteristic
75#14578938C2311yescharacteristic
76#14578937—22no—
77#14578937—21no—
78C2×#11929434C2311yescharacteristic
79C8×D157—22no—
80C8×D157—21no—
81#14578931—42no—
82C4×Dic157C2311yescharacteristic
83#41×C157C2311yescharacteristic
84C2×Q8×C157C2312yes—
85Q16×C157—22no—
86Q16×C157—21no—
87Q16×C157—21no—
88SD16×C157—22no—
89M16×C157—21no—
90M16×C157C2311yescharacteristic
91M16×C157C2312yes—
92C2×C8×C157—21no—
93C2×C8×C157C2311yescharacteristic
94C22×D157—22no—
95C22×D157—42no—
96Dic314—42no—
97Dic314C2×D412yes—
98Dic314C2×D411yescharacteristic
99D628—22no—
100D628—22no—
101D628—42no—
102D628—42no—
103D628C2×D412yes—
104D628C2×D411yescharacteristic
105C22⋊D157—22no—
106C22⋊D157—21no—
107C22⋊D157—42no—
108C2×Dic157—21no—
109C4×D157—22no—
110C4×D157—21no—
111C4×D157—42no—
112C4×D157—42no—
113C4×D157—42no—
114#11929434—21no—
115#11929434C2×D411yescharacteristic
116#11929434C2×D411yescharacteristic
117Q8×C157—22no—
118Q8×C157—22no—
119Q8×C157C2×D411yescharacteristic
120D4×C157—21no—
121D4×C157C2×D411yescharacteristic
122C2×C4×C157—21no—
123C2×C4×C157—22no—
124C2×C4×C157C2411yescharacteristic
125C8×C157—21no—
126C8×C157—22no—
127C8×C157C2×D411yescharacteristic
128C8×C157C2×D411yescharacteristic
129C22×C157—21no—
130C22×C157H(5,𝔽2)11yescharacteristic
131C2×D157—21no—
132C2×D157—22no—
133C2×D157—42no—
134C2×D157—82no—
135C2×D157—162no—
136C4×C157—21no—
137C4×C157—42no—
138C4×C157C22×D411yescharacteristic
139C4×C157C22×D411yescharacteristic
140Dic157—22no—
141Dic157—21no—
142Dic157—41no—
143C2×C157—21no—
144C2×C157—41no—
145C2×C157D4211yescharacteristic
146D157—41no—
147D157—42no—
148D157—82no—
149C157#329011yescharacteristic, complemented, Sylow
150#3290—1571nomaximal, Sylow
151#578—1571no—
152#577—1572no—
153#576—1571no—
154#495—1571no—
155#473D15711yescharacteristic, complemented
156#472—1571no—
157#471—1572no—
158#455—1572no—
159#454—1572no—
160#443—1571no—
161#443—1571no—
1622−1+4—1572no—
163H(5,𝔽2)—1572no—
164#137—3142no—
165#137—3142no—
166#137—1572no—
167#137C2×D15712yes—
168AGL(1,ℤ/8)—3142no—
169AGL(1,ℤ/8)—3142no—
170AGL(1,ℤ/8)—1572no—
171AGL(1,ℤ/8)—1572no—
172#135—3141no—
173#135—3142no—
174#135—3141no—
175#135—1571no—
176#135—1571no—
177C2×Q16—3141no—
178C2×Q16C2×D15711yescharacteristic, complemented
179C2×SD16—3142no—
180C2×SD16—1572no—
181C2×D8—3141no—
182C2×D8—1571no—
183#131—1572no—
184#131—1571no—
185#131C2×D15711yescharacteristic
186#124—1572no—
187#108—1571no—
188#108C2×D15711yescharacteristic, complemented
189#104—1571no—
190#104—1571no—
191#104—1572no—
192#101C2×D15712yescomplemented
193#100—1572no—
194C4×C8—1571no—
195#41—3142no—
196#41—3142no—
197#41—3142no—
198#41—3142no—
199#41—6282no—
200#41—1571no—
201#41—1572no—
202#41C22×D15711yescharacteristic
203C2×Q8—3142no—
204C2×Q8—3142no—
205C2×Q8C22×D15712yes—
206C2×D4—3142no—
207C2×D4—3142no—
208C2×D4—6282no—
209C2×D4—1572no—
210Q16—22no—
211Q16—21no—
212Q16—21no—
213Q16—3142no—
214Q16—1571no—
215Q16—1571no—
216SD16—22no—
217SD16—3142no—
218SD16—3142no—
219SD16—3142no—
220SD16—3142no—
221SD16—3142no—
222SD16—1571no—
223SD16—1571no—
224SD16—1572no—
225D8—3142no—
226D8—3142no—
227D8—3141no—
228D8—3141no—
229D8—1571no—
230D8—1571no—
231M16—21no—
232M16—3142no—
233M16—3141no—
234M16—1571no—
235M16C22×D15712yes—
236M16C22×D15711yescharacteristic
237C2×C8—21no—
238C2×C8—3142no—
239C2×C8—3141no—
240C2×C8—1571no—
241C2×C8C22×D15711yescharacteristic
242C22⋊C4—6282no—
243C42—1571no—
244C23—3142no—
245C23—6282no—
246Q8—22no—
247Q8—22no—
248Q8—6282no—
249Q8—1571no—
250Q8—1572no—
251Q8D4×D15711yescharacteristic
252D4—21no—
253D4—3142no—
254D4—3141no—
255D4—3142no—
256D4—3142no—
257D4—6282no—
258D4—6282no—
259D4—6282no—
260D4—1571no—
261D4—1572no—
262D4D4×D15711yescharacteristic
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—3142no—
266C2×C4—3141no—
267C2×C4—3141no—
268C2×C4—6282no—
269C2×C4—6282no—
270C2×C4—6282no—
271C2×C4C23×D15711yescharacteristic
272C8—21no—
273C8—22no—
274C8—3141no—
275C8—1571no—
276C8—1571no—
277C8D4×D15711yescharacteristic
278C8D4×D15711yescharacteristic
279C22—21no—
280C22—3141no—
281C22—3142no—
282C22—6282no—
283C22—25122no—
284C22—12562no—
285C22#1909781711yescharacteristic
286C4—21no—
287C4—42no—
288C4—3142no—
289C4—3141no—
290C4—6281no—
291C4C2×D4×D15711yescharacteristic
292C4C2×D4×D15711yescharacteristic
293C2—21no—
294C2—41no—
295C2—6282no—
296C2—6281no—
297C2—12562no—
298C2#2375790711yescharacteristic
subgroupG/H
centerC2#23757907
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#473×C157C2
socleC2×C157D42
hypercenter#473D157
hypocenterC157#3290
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ #473

Complements

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

NH ≅ G/Ntype
1. #23758086C2semidirect
2. #23758082C2semidirect
3. #23758078C2semidirect
4. #23757658C2semidirect
5. #23757544C2semidirect
6. #23757539C2semidirect
7. #23757449C2semidirect
8. #23757448C2semidirect
9. #23757389C2semidirect
10. #23757388C2semidirect
11. #473×C157C2semidirect
12. #19097819C22semidirect
16. #19097802C22semidirect
19. #19097799C22semidirect
27. C2×#14578941C22semidirect
31. #19097779C22semidirect
32. #19097777C22semidirect
33. #19097760C22semidirect
34. #19097706C22semidirect
36. #19097697C22semidirect
37. #19097697C22semidirect
38. #19097687C22semidirect
39. C8×Dic157C22semidirect
41. C2×Q16×C157C22semidirect
43. #108×C157C22semidirect
44. #101×C157C22semidirect
149. C157#3290semidirect
155. #473D157semidirect
178. C2×Q16C2×D157semidirect
188. #108C2×D157semidirect
192. #101C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580862
#237580822
#237580782
#237576582
#237575442
#237575392
#237574492
#237574482
#237573892
#237573882
#473×C1572
#3290157