Finite Group Databasebeta

← #34113281

Subgroups — #34113281

Order 20096.

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

normal subgroups only

This group has 360 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#23758086C211yesmaximal, characteristic, complemented
2#23758075C211yesmaximal, characteristic, complemented
3#23757715C212yesmaximal, complemented
4#23757711C211yesmaximal, characteristic, complemented
5#23757710C212yesmaximal, complemented
6#23757706C212yesmaximal, complemented
7#23757703C211yesmaximal, characteristic, complemented
8C2×#19097714C211yesmaximal, characteristic, complemented
9SD32×D157C212yesmaximal, complemented
10#23757691C211yesmaximal, characteristic, complemented
11#510×C157C211yesmaximal, characteristic, complemented
12#19097819—21no—
13#19097817—21no—
14#19097803—22no—
15#19097802—21no—
16#19097798—22no—
17#19097796—21no—
18#19097794C2212yes—
19#19097794C2212yescomplemented
20#19097793C2211yescharacteristic, complemented
21Q16×D157C2212yescomplemented
22C2×D1256C2211yescharacteristic
23D8×D157C2212yes—
24#19097777C2211yescharacteristic
25#19097717C2211yescharacteristic, complemented
26#19097716C2212yescomplemented
27#19097715C2211yescharacteristic, complemented
28#19097714C2212yescomplemented
29#19097714C2212yescomplemented
30#19097714C2211yescharacteristic, complemented
31D2512C2212yescomplemented
32#19097712C2211yescharacteristic, complemented
33#19097710C2212yescomplemented
34C2×#14578929C2211yescharacteristic, complemented
35C16×D157C2212yescomplemented
36#135×C157C2211yescharacteristic
37C2×Q16×C157C2211yescharacteristic, complemented
38Q32×C157C2212yescomplemented
39SD32×C157C2212yescomplemented
40M32×C157C2211yescharacteristic, complemented
41#14578955—22no—
42#14578955—22no—
43#14578955—41no—
44#14578954—21no—
45#14578954—41no—
46#14578954—42no—
47#14578954D411yescharacteristic
48Q8×D157—22no—
49Q8×D157—41no—
50C2×D628—21no—
51C2×D628—41no—
52D4×D157—22no—
53D4×D157—42no—
54Dic628C2311yescharacteristic
55#14578944—22no—
56#14578943—21no—
57#14578941—22no—
58#14578941—22no—
59D1256—21no—
60D1256C2311yescharacteristic
61D1256C2312yes—
62#14578939—22no—
63#14578938D411yescharacteristic
64#14578937D412yes—
65C8×D157C2312yes—
66#14578929C2311yescharacteristic
67#14578929C2311yescharacteristic
68#41×C157—21no—
69C2×Q8×C157—21no—
70Q16×C157—21no—
71Q16×C157C2312yes—
72Q16×C157C2311yescharacteristic
73SD16×C157—21no—
74D8×C157C2311yescharacteristic
75C2×C8×C157C2311yescharacteristic
76C16×C157C2312yes—
77C22×D157—41no—
78C22×D157—42no—
79Dic314—21no—
80Dic314—42no—
81Dic314D811yescharacteristic
82D628—22no—
83D628—21no—
84D628—42no—
85D628—42no—
86D628—41no—
87D628D811yescharacteristic
88C22⋊D157—41no—
89C22⋊D157—42no—
90C22⋊D157D812yes—
91C4×D157—42no—
92C4×D157—41no—
93C4×D157—42no—
94C4×D157—42no—
95C4×D157C2×D412yes—
96#11929434C2×D412yes—
97Q8×C157—22no—
98Q8×C157—21no—
99Q8×C157—41no—
100D4×C157—21no—
101D4×C157—41no—
102C2×C4×C157—41no—
103C2×C4×C157—41no—
104C2×C4×C157C2×D411yescharacteristic
105C8×C157C2×D411yescharacteristic
106C8×C157C2411yescharacteristic
107C22×C157—41no—
108C22×C157C2×D811yescharacteristic
109C2×D157—41no—
110C2×D157—42no—
111C2×D157—162no—
112C2×D157—161no—
113C2×D157C2×D812yes—
114C4×C157—41no—
115C4×C157—42no—
116C4×C157C2×D811yescharacteristic
117C4×C157C22×D411yescharacteristic
118Dic157—41no—
119Dic157C2×D812yes—
120C2×C157—21no—
121C2×C157—81no—
122C2×C157C22×D811yescharacteristic
123D157—22no—
124D157—81no—
125D157—82no—
126C157#341611yescharacteristic, complemented, Sylow
127#3416—1571nomaximal, Sylow
128#578—1571no—
129#576—1571no—
130#510—1572no—
131#510D15711yescharacteristic, complemented
132#509—1572no—
133#509—1571no—
134#508—1571no—
135#508—1572no—
136C2×SD32—1571no—
137C2×SD32—1572no—
138#504—1571no—
1392−1+4—3141no—
140H(5,𝔽2)—3141no—
141#137—3142no—
142#137—3141no—
143AGL(1,ℤ/8)—3142no—
144AGL(1,ℤ/8)—3141no—
145#135—1572no—
146#135—1572no—
147#135—1571no—
148#135C2×D15711yescharacteristic
149C2×Q16—1572no—
150C2×Q16C2×D15711yescharacteristic, complemented
151C2×D8—1572no—
152C2×D8—1571no—
153#131—1571no—
154Q32—1571no—
155Q32C2×D15712yescomplemented
156SD32—1572no—
157SD32—1572no—
158SD32—1571no—
159SD32—1571no—
160SD32—1572no—
161SD32C2×D15712yescomplemented
162D16—1571no—
163D16—1572no—
164M32—1572no—
165M32C2×D15711yescharacteristic, complemented
166C2×C16—1571no—
167C2×C16—1572no—
168#41—21no—
169#41—3142no—
170#41—3142no—
171#41—3141no—
172#41—6281no—
173#41—6282no—
174#41—6281no—
175#41—1571no—
176C2×Q8—21no—
177C2×Q8—3142no—
178C2×Q8—6281no—
179C2×D4—3142no—
180C2×D4—3141no—
181C2×D4—6281no—
182C2×D4—6282no—
183Q16—21no—
184Q16—3142no—
185Q16—1571no—
186Q16C22×D15712yes—
187Q16C22×D15711yescharacteristic
188SD16—21no—
189SD16—3142no—
190SD16—3142no—
191SD16—3141no—
192D8—3142no—
193D8—3141no—
194D8—1571no—
195D8—1572no—
196D8C22×D15711yescharacteristic
197M16—1571no—
198M16—1572no—
199C2×C8—1572no—
200C2×C8C22×D15711yescharacteristic
201C16—1571no—
202C16—1571no—
203C16C22×D15712yes—
204C23—6282no—
205C23—6281no—
206Q8—21no—
207Q8—22no—
208Q8—41no—
209Q8—3141no—
210Q8—6282no—
211Q8—1571no—
212D4—21no—
213D4—41no—
214D4—3142no—
215D4—3141no—
216D4—6282no—
217D4—6282no—
218D4—6281no—
219D4—6282no—
220D4—6281no—
221D4—1572no—
222D4—1571no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—6282no—
226C2×C4—6282no—
227C2×C4—6282no—
228C2×C4—6281no—
229C2×C4—1572no—
230C2×C4D4×D15711yescharacteristic
231C8—1572no—
232C8D4×D15711yescharacteristic
233C8C23×D15711yescharacteristic
234C22—41no—
235C22—6281no—
236C22—6282no—
237C22—1572no—
238C22—25121no—
239C22—25122no—
240C22D8×D15711yescharacteristic
241C4—41no—
242C4—42no—
243C4—6281no—
244C4—1572no—
245C4D8×D15711yescharacteristic
246C4C2×D4×D15711yescharacteristic
247C2—21no—
248C2—81no—
249C2—3142no—
250C2—12562no—
251C2—12561no—
252C2C2×D8×D15711yescharacteristic
subgroupG/H
centerC2C2×D8×D157
commutator, cosocleC8×C157C24
FrattiniC8C23×D157
Fitting#510×C157C2
socleC2×C157C22×D8
hypercenter#510D157
hypocenterC157#3416
LCSG ▷ C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #510

Complements

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

NH ≅ G/Ntype
1. #23758086C2semidirect
2. #23758075C2semidirect
3. #23757715C2semidirect
4. #23757711C2semidirect
5. #23757710C2semidirect
6. #23757706C2semidirect
7. #23757703C2semidirect
8. C2×#19097714C2semidirect
9. SD32×D157C2semidirect
10. #23757691C2semidirect
11. #510×C157C2semidirect
19. #19097794C22semidirect
20. #19097793C22semidirect
21. Q16×D157C22semidirect
25. #19097717C22semidirect
26. #19097716C22semidirect
27. #19097715C22semidirect
28. #19097714C22semidirect
29. #19097714C22semidirect
30. #19097714C22semidirect
31. D2512C22semidirect
32. #19097712C22semidirect
33. #19097710C22semidirect
34. C2×#14578929C22semidirect
35. C16×D157C22semidirect
37. C2×Q16×C157C22semidirect
38. Q32×C157C22semidirect
39. SD32×C157C22semidirect
40. M32×C157C22semidirect
126. C157#3416semidirect
131. #510D157semidirect
150. C2×Q16C2×D157semidirect
155. Q32C2×D157semidirect
161. SD32C2×D157semidirect
165. M32C2×D157semidirect

Maximal subgroups

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

subgroupindex
#237580862
#237580752
#237577152
#237577112
#237577102
#237577062
#237577032
C2×#190977142
SD32×D1572
#237576912
#510×C1572
#3416157