Finite Group Databasebeta

← #34111924

Subgroups — #34111924

Order 20096.

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

normal subgroups only

This group has 454 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#23758087C211yesmaximal, characteristic, complemented
2#23758085C212yesmaximal, complemented
3#23758075C211yesmaximal, characteristic, complemented
4#23757657C211yesmaximal, characteristic, complemented
5#23757549C211yesmaximal, characteristic, complemented
6#23757544C211yesmaximal, characteristic, complemented
7#23757536C212yesmaximal, complemented
8#23757452C212yesmaximal, complemented
9#23757447C212yesmaximal, complemented
10#23757391C211yesmaximal, characteristic, complemented
11#443×C157C211yesmaximal, characteristic, complemented
12#19097818C2212yescomplemented
13#19097817C2212yes—
14#19097804—22no—
15#19097804C2212yes—
16#19097801C2212yescomplemented
17#19097800—22no—
18#19097799—22no—
19#19097799C2212yescomplemented
20#19097798—22no—
21#19097796C2212yes—
22#19097794—21no—
23#19097793—22no—
24#19097793C2211yescharacteristic
25#19097790—21no—
26C2×Dic628C2211yescharacteristic, complemented
27Q16×D157—21no—
28C2×#14578943—22no—
29C2×#14578943C2212yescomplemented
30C2×D1256C2211yescharacteristic
31D8×D157—21no—
32#19097777C2211yescharacteristic
33#19097777C2211yescharacteristic, complemented
34#19097759C2212yescomplemented
35#19097707C2211yescharacteristic, complemented
36#19097699C2211yescharacteristic
37#19097699C2211yescharacteristic, complemented
38#19097689C2212yescomplemented
39#19097687C2212yescomplemented
40#135×C157C2211yescharacteristic
41#131×C157C2212yescomplemented
42#108×C157C2211yescharacteristic, complemented
43#104×C157C2212yescomplemented
44C4×C8×C157C2211yescharacteristic, complemented
45#14578955—22no—
46#14578954—22no—
47#14578954—22no—
48#14578954C2311yescharacteristic
49#14578954C2311yescharacteristic
50#14578953—22no—
51#14578953—42no—
52C2×Dic314—22no—
53C2×Dic314C2312yes—
54Q8×D157—22no—
55C2×D628—22no—
56C2×D628C2312yes—
57D4×D157—22no—
58D4×D157—42no—
59Dic628—22no—
60Dic628—21no—
61Dic628—21no—
62Dic628D411yescharacteristic
63#14578944—22no—
64#14578943—22no—
65#14578943—22no—
66#14578943—22no—
67#14578943—22no—
68#14578943D411yescharacteristic
69#14578943D411yescharacteristic
70#14578942—22no—
71#14578941—22no—
72D1256—22no—
73D1256—21no—
74D1256—21no—
75D1256D411yescharacteristic
76#14578939—22no—
77#14578938C2311yescharacteristic
78#14578938C2311yescharacteristic
79#14578937—21no—
80#14578937—21no—
81C8×D157—21no—
82C8×D157—21no—
83#14578931—42no—
84#41×C157C2312yes—
85Q16×C157D411yescharacteristic
86SD16×C157D412yes—
87D8×C157D411yescharacteristic
88M16×C157—22no—
89M16×C157C2312yes—
90C2×C8×C157—22no—
91C2×C8×C157C2311yescharacteristic
92C2×C8×C157C2311yescharacteristic
93C42×C157C2311yescharacteristic
94C22×D157—22no—
95C22×D157—42no—
96Dic314—22no—
97Dic314—22no—
98Dic314—42no—
99Dic314C2×D411yescharacteristic
100Dic314C2×D411yescharacteristic
101D628—22no—
102D628—22no—
103D628—42no—
104D628C2×D411yescharacteristic
105D628C2×D411yescharacteristic
106C22⋊D157—21no—
107C22⋊D157—21no—
108C22⋊D157—42no—
109C22⋊D157—42no—
110C2×Dic157—22no—
111C2×Dic157—42no—
112C4×D157—21no—
113C4×D157—21no—
114C4×D157—42no—
115C4×D157—42no—
116#11929434—21no—
117#11929434—21no—
118Q8×C157C2×D412yes—
119D4×C157—22no—
120D4×C157C2×D412yes—
121C2×C4×C157—22no—
122C2×C4×C157—21no—
123C2×C4×C157C2411yescharacteristic
124C8×C157—22no—
125C8×C157C2×D411yescharacteristic
126C8×C157C2×D411yescharacteristic
127C8×C157C2×D411yescharacteristic
128C8×C157C2×D411yescharacteristic
129C22×C157—22no—
130C22×C157H(5,𝔽2)11yescharacteristic
131C2×D157—21no—
132C2×D157—21no—
133C2×D157—42no—
134C2×D157—82no—
135C2×D157—162no—
136C4×C157—22no—
137C4×C157—41no—
138C4×C157C22×D411yescharacteristic
139C4×C157C22×D411yescharacteristic
140Dic157—21no—
141Dic157—21no—
142Dic157—42no—
143C2×C157—21no—
144C2×C157—42no—
145C2×C157D4211yescharacteristic
146D157—41no—
147D157—41no—
148D157—82no—
149C157#329011yescharacteristic, complemented, Sylow
150#3290—1571nomaximal, Sylow
151#578—1571no—
152#577—1572no—
153#576—1571no—
154#495—1571no—
155#473—1571no—
156#472—1571no—
157#471—1572no—
158#455—1572no—
159#454—1572no—
160#443—1571no—
161#443D15711yescharacteristic, complemented
1622−1+4—1572no—
163H(5,𝔽2)—1572no—
164#137—3142no—
165#137—3142no—
166#137—1572no—
167#137—1572no—
168AGL(1,ℤ/8)—3142no—
169AGL(1,ℤ/8)—3142no—
170AGL(1,ℤ/8)—1572no—
171AGL(1,ℤ/8)—1572no—
172#135—3142no—
173#135—3141no—
174#135—3141no—
175#135—1571no—
176#135C2×D15711yescharacteristic
177C2×Q16—3141no—
178C2×Q16—1571no—
179C2×SD16—3142no—
180C2×SD16—1572no—
181C2×D8—3141no—
182C2×D8—1571no—
183#131—1571no—
184#131—1571no—
185#131C2×D15712yescomplemented
186#124—1572no—
187#108—1571no—
188#108C2×D15711yescharacteristic, complemented
189#104—1571no—
190#104—1571no—
191#104C2×D15712yescomplemented
192#101—1572no—
193#100—1572no—
194C4×C8C2×D15711yescharacteristic, complemented
195#41—3142no—
196#41—3142no—
197#41—3142no—
198#41—3142no—
199#41—6282no—
200#41—1571no—
201#41—1571no—
202#41C22×D15712yes—
203C2×Q8—3142no—
204C2×Q8—3142no—
205C2×Q8—1572no—
206C2×D4—3142no—
207C2×D4—3142no—
208C2×D4—6282no—
209C2×D4—1572no—
210Q16—3142no—
211Q16—3142no—
212Q16—3141no—
213Q16—3141no—
214Q16—1571no—
215Q16D62811yescharacteristic
216SD16—3142no—
217SD16—3142no—
218SD16—3142no—
219SD16—3142no—
220SD16—3142no—
221SD16—3142no—
222SD16—1571no—
223SD16—1571no—
224SD16D62812yes—
225D8—3142no—
226D8—3142no—
227D8—3141no—
228D8—3141no—
229D8—1571no—
230D8D62811yescharacteristic
231M16—22no—
232M16—3141no—
233M16—3141no—
234M16—1571no—
235M16—1571no—
236M16C22×D15712yes—
237C2×C8—22no—
238C2×C8—3141no—
239C2×C8—3141no—
240C2×C8C22×D15711yescharacteristic
241C2×C8C22×D15711yescharacteristic
242C22⋊C4—6282no—
243C42C22×D15711yescharacteristic
244C23—3142no—
245C23—6282no—
246Q8—3142no—
247Q8—3142no—
248Q8—6282no—
249Q8—1571no—
250Q8—1571no—
251Q8C2×D62812yes—
252D4—22no—
253D4—3141no—
254D4—3141no—
255D4—3142no—
256D4—3142no—
257D4—6282no—
258D4—6282no—
259D4—6282no—
260D4—1571no—
261D4—1571no—
262D4C2×D62812yes—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—3141no—
266C2×C4—3141no—
267C2×C4—3142no—
268C2×C4—6282no—
269C2×C4—6282no—
270C2×C4—6282no—
271C2×C4C23×D15711yescharacteristic
272C8—22no—
273C8—3141no—
274C8—3141no—
275C8D4×D15711yescharacteristic
276C8D4×D15711yescharacteristic
277C8C2×D62811yescharacteristic
278C8C2×D62811yescharacteristic
279C22—22no—
280C22—3141no—
281C22—3141no—
282C22—6282no—
283C22—25122no—
284C22—12562no—
285C22#1909781711yescharacteristic
286C4—22no—
287C4—41no—
288C4—3141no—
289C4—3141no—
290C4—6282no—
291C4C2×D4×D15711yescharacteristic
292C4C22×D62811yescharacteristic
293C2—21no—
294C2—42no—
295C2—6281no—
296C2—6281no—
297C2—12562no—
298C2D4×D62811yescharacteristic
subgroupG/H
centerC2D4×D628
commutator, cosocleC2×C4×C157C24
FrattiniC2×C4C23×D157
Fitting#443×C157C2
socleC2×C157D42
hypercenter#443D157
hypocenterC157#3290
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. #23758087C2semidirect
2. #23758085C2semidirect
3. #23758075C2semidirect
4. #23757657C2semidirect
5. #23757549C2semidirect
6. #23757544C2semidirect
7. #23757536C2semidirect
8. #23757452C2semidirect
9. #23757447C2semidirect
10. #23757391C2semidirect
11. #443×C157C2semidirect
12. #19097818C22semidirect
16. #19097801C22semidirect
19. #19097799C22semidirect
26. C2×Dic628C22semidirect
29. C2×#14578943C22semidirect
33. #19097777C22semidirect
34. #19097759C22semidirect
35. #19097707C22semidirect
37. #19097699C22semidirect
38. #19097689C22semidirect
39. #19097687C22semidirect
41. #131×C157C22semidirect
42. #108×C157C22semidirect
43. #104×C157C22semidirect
44. C4×C8×C157C22semidirect
149. C157#3290semidirect
161. #443D157semidirect
185. #131C2×D157semidirect
188. #108C2×D157semidirect
191. #104C2×D157semidirect
194. C4×C8C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580872
#237580852
#237580752
#237576572
#237575492
#237575442
#237575362
#237574522
#237574472
#237573912
#443×C1572
#3290157