Finite Group Databasebeta

← #34113276

Subgroups — #34113276

Order 20096.

370 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 113 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#23758075C211yesmaximal, characteristic, complemented
2#23758074C211yesmaximal, characteristic, complemented
3#23757711C211yesmaximal, characteristic, complemented
4#23757710C212yesmaximal, complemented
5#23757707C212yesmaximal, complemented
6#23757704C212yesmaximal, complemented
7#23757703C211yesmaximal, characteristic, complemented
8C2×#19097712C211yesmaximal, characteristic, complemented
9D16×D157C212yesmaximal, complemented
10#23757691C211yesmaximal, characteristic, complemented
11#509×C157C211yesmaximal, characteristic, complemented
12#19097817—21no—
13#19097816—21no—
14#19097798—22no—
15#19097797—21no—
16#19097796—21no—
17#19097795—22no—
18#19097794C2212yescomplemented
19#19097793C2211yescharacteristic, complemented
20#19097790C2212yescomplemented
21C2×D1256C2211yescharacteristic, complemented
22D8×D157C2212yescomplemented
23D8×D157C2212yescomplemented
24#19097777C2211yescharacteristic
25#19097717C2211yescharacteristic, complemented
26#19097716C2212yescomplemented
27#19097715C2211yescharacteristic, complemented
28#19097714C2211yescharacteristic, complemented
29D2512C2212yescomplemented
30#19097712C2211yescharacteristic, complemented
31#19097712C2212yescomplemented
32#19097712C2212yescomplemented
33#19097710C2212yescomplemented
34C2×#14578929C2211yescharacteristic, complemented
35C16×D157C2212yescomplemented
36#135×C157C2211yescharacteristic, complemented
37C2×D8×C157C2211yescharacteristic, complemented
38SD32×C157C2212yescomplemented
39D16×C157C2212yescomplemented
40M32×C157C2211yescharacteristic, complemented
41#14578955—22no—
42#14578954—21no—
43#14578954—41no—
44#14578954D411yescharacteristic
45#14578953—22no—
46#14578953—41no—
47C2×D628—21no—
48C2×D628—41no—
49C2×C22⋊D157—42no—
50D4×D157—22no—
51D4×D157—22no—
52D4×D157—41no—
53D4×D157—42no—
54Dic628C2311yescharacteristic
55#14578943—21no—
56#14578942—22no—
57#14578941—22no—
58D1256—21no—
59D1256C2312yes—
60D1256C2311yescharacteristic
61#14578939—22no—
62#14578939—22no—
63#14578938D411yescharacteristic
64#14578937D412yes—
65C8×D157C2312yes—
66#14578929C2311yescharacteristic
67#14578929C2311yescharacteristic
68#41×C157—21no—
69C2×D4×C157—21no—
70Q16×C157C2311yescharacteristic
71SD16×C157—21no—
72D8×C157—21no—
73D8×C157C2311yescharacteristic
74D8×C157C2312yes—
75C2×C8×C157C2311yescharacteristic
76C16×C157C2312yes—
77C22×D157—41no—
78C22×D157—42no—
79C22×D157—42no—
80Dic314—21no—
81Dic314D811yescharacteristic
82D628—21no—
83D628—22no—
84D628—42no—
85D628—41no—
86D628D811yescharacteristic
87C22⋊D157—41no—
88C22⋊D157—42no—
89C22⋊D157—42no—
90C22⋊D157—42no—
91C22⋊D157D812yes—
92C2×Dic157—42no—
93C4×D157—41no—
94C4×D157—42no—
95C4×D157C2×D412yes—
96#11929434C2×D412yes—
97C23×C157—41no—
98Q8×C157—21no—
99D4×C157—21no—
100D4×C157—22no—
101D4×C157—41no—
102D4×C157—41no—
103C2×C4×C157—41no—
104C2×C4×C157C2×D411yescharacteristic
105C8×C157C2×D411yescharacteristic
106C8×C157C2411yescharacteristic
107C22×C157—41no—
108C22×C157—42no—
109C22×C157—161no—
110C22×C157C2×D811yescharacteristic
111C2×D157—41no—
112C2×D157—42no—
113C2×D157—162no—
114C2×D157—162no—
115C2×D157—161no—
116C2×D157C2×D812yes—
117C4×C157—41no—
118C4×C157C2×D811yescharacteristic
119C4×C157C22×D411yescharacteristic
120Dic157—41no—
121Dic157C2×D812yes—
122C2×C157—21no—
123C2×C157—81no—
124C2×C157—82no—
125C2×C157C22×D811yescharacteristic
126D157—22no—
127D157—81no—
128D157—82no—
129C157#341511yescharacteristic, complemented, Sylow
130#3415—1571nomaximal, Sylow
131#576—1571no—
132#576—1571no—
133#509—1572no—
134#509—1572no—
135#509—1571no—
136#509D15711yescharacteristic, complemented
137#508—1571no—
138#508—1572no—
139C2×D16—1571no—
140C2×D16—1572no—
141#504—1571no—
142H(5,𝔽2)—3141no—
143H(5,𝔽2)—3141no—
144AGL(1,ℤ/8)—3142no—
145AGL(1,ℤ/8)—3142no—
146AGL(1,ℤ/8)—3141no—
147AGL(1,ℤ/8)—3141no—
148#135—1572no—
149#135—1572no—
150#135—1571no—
151#135C2×D15711yescharacteristic, complemented
152C2×D8—1572no—
153C2×D8—1572no—
154C2×D8—1571no—
155C2×D8C2×D15711yescharacteristic, complemented
156#131—1571no—
157Q32—1571no—
158SD32—1571no—
159SD32—1571no—
160SD32—1572no—
161SD32C2×D15712yescomplemented
162D16—1572no—
163D16—1571no—
164D16—1572no—
165D16—1572no—
166D16C2×D15712yescomplemented
167M32—1572no—
168M32C2×D15711yescharacteristic, complemented
169C2×C16—1571no—
170C2×C16—1572no—
171#41—21no—
172#41—3142no—
173#41—3142no—
174#41—3141no—
175#41—6281no—
176#41—6281no—
177#41—1571no—
178C2×D4—21no—
179C2×D4—3142no—
180C2×D4—3142no—
181C2×D4—3141no—
182C2×D4—6281no—
183C2×D4—6282no—
184C2×D4—6281no—
185C2×D4—6282no—
186Q16—1571no—
187Q16C22×D15711yescharacteristic
188SD16—21no—
189SD16—3142no—
190SD16—3142no—
191SD16—3141no—
192D8—21no—
193D8—3142no—
194D8—3142no—
195D8—3141no—
196D8—1572no—
197D8—1571no—
198D8C22×D15712yes—
199D8C22×D15711yescharacteristic
200M16—1571no—
201M16—1572no—
202C2×C8—1572no—
203C2×C8C22×D15711yescharacteristic
204C16—1571no—
205C16—1571no—
206C16C22×D15712yes—
207C23—41no—
208C23—6282no—
209C23—6282no—
210C23—6281no—
211Q8—21no—
212Q8—3141no—
213Q8—1571no—
214D4—22no—
215D4—21no—
216D4—41no—
217D4—41no—
218D4—3141no—
219D4—3142no—
220D4—6282no—
221D4—6282no—
222D4—6281no—
223D4—6282no—
224D4—6282no—
225D4—6281no—
226D4—1572no—
227D4—1571no—
228C2×C4—41no—
229C2×C4—6282no—
230C2×C4—6282no—
231C2×C4—6281no—
232C2×C4—1572no—
233C2×C4D4×D15711yescharacteristic
234C8—1572no—
235C8D4×D15711yescharacteristic
236C8C23×D15711yescharacteristic
237C22—41no—
238C22—42no—
239C22—161no—
240C22—6282no—
241C22—6281no—
242C22—1572no—
243C22—25121no—
244C22—25122no—
245C22—25122no—
246C22D8×D15711yescharacteristic
247C4—41no—
248C4—6281no—
249C4—1572no—
250C4D8×D15711yescharacteristic
251C4C2×D4×D15711yescharacteristic
252C2—21no—
253C2—82no—
254C2—81no—
255C2—3142no—
256C2—12562no—
257C2—12561no—
258C2C2×D8×D15711yescharacteristic
subgroupG/H
centerC2C2×D8×D157
commutator, cosocleC8×C157C24
FrattiniC8C23×D157
Fitting#509×C157C2
socleC2×C157C22×D8
hypercenter#509D157
hypocenterC157#3415
LCSG ▷ C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #509

Complements

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

NH ≅ G/Ntype
1. #23758075C2semidirect
2. #23758074C2semidirect
3. #23757711C2semidirect
4. #23757710C2semidirect
5. #23757707C2semidirect
6. #23757704C2semidirect
7. #23757703C2semidirect
8. C2×#19097712C2semidirect
9. D16×D157C2semidirect
10. #23757691C2semidirect
11. #509×C157C2semidirect
18. #19097794C22semidirect
19. #19097793C22semidirect
20. #19097790C22semidirect
21. C2×D1256C22semidirect
22. D8×D157C22semidirect
23. D8×D157C22semidirect
25. #19097717C22semidirect
26. #19097716C22semidirect
27. #19097715C22semidirect
28. #19097714C22semidirect
29. D2512C22semidirect
30. #19097712C22semidirect
31. #19097712C22semidirect
32. #19097712C22semidirect
33. #19097710C22semidirect
34. C2×#14578929C22semidirect
35. C16×D157C22semidirect
36. #135×C157C22semidirect
37. C2×D8×C157C22semidirect
38. SD32×C157C22semidirect
39. D16×C157C22semidirect
40. M32×C157C22semidirect
129. C157#3415semidirect
136. #509D157semidirect
151. #135C2×D157semidirect
155. C2×D8C2×D157semidirect
161. SD32C2×D157semidirect
166. D16C2×D157semidirect
168. M32C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580752
#237580742
#237577112
#237577102
#237577072
#237577042
#237577032
C2×#190977122
D16×D1572
#237576912
#509×C1572
#3415157