Finite Group Databasebeta

← #46848160

Subgroups — #46848160

Order 34688.

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#31774701C212yesmaximal, complemented
2#31774700C212yesmaximal, complemented
3#31774273C211yesmaximal, characteristic, complemented
4#31774166C212yesmaximal, complemented
5#31774165C212yesmaximal, complemented
6#31774066C212yesmaximal, complemented
7#31774064C211yesmaximal, characteristic, complemented
8#31774011C212yesmaximal, complemented
9#453×C271C211yesmaximal, characteristic, complemented
10#22576206C2212yescomplemented
11#22576205C2211yescharacteristic, complemented
12#22576188C2212yescomplemented
13#22576187—22no—
14#22576186—22no—
15#22576186C2212yescomplemented
16#22576185—22no—
17#22576185C2212yescomplemented
18#22576184—22no—
19#22576181—22no—
20#22576180—22no—
21#22576180C2212yes—
22C2×#16864639—22no—
23C2×#16864639C2212yescomplemented
24C2×#16864639C2212yescomplemented
25D8×D271—22no—
26#22576168C2212yescomplemented
27#22576167C2212yescomplemented
28#22576157C2211yescharacteristic, complemented
29#22576095C2212yescomplemented
30#22576088C2212yescomplemented
31#22576077C2211yescharacteristic, complemented
32#22576076C2212yescomplemented
33C4×C271⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C271C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C271C2212yescomplemented
36#127×C271C2211yescharacteristic, complemented
37#104×C271C2212yescomplemented
38#100×C271C2211yescharacteristic, complemented
39#16864655—22no—
40#16864654—22no—
41#16864654C2312yes—
42#16864653—22no—
43C2×D1084—22no—
44C2×D1084—41no—
45C2×D1084C2312yes—
46C2×C22⋊D271—41no—
47D4×D271—22no—
48D4×D271—22no—
49D4×D271—42no—
50#16864644D412yescomplemented
51#16864643—22no—
52#16864642—22no—
53#16864642D412yescomplemented
54#16864641—22no—
55#16864641D412yescomplemented
56D2168—22no—
57#16864639—22no—
58#16864639—22no—
59#16864639—22no—
60#16864639—22no—
61#16864639—22no—
62#16864639—22no—
63#16864639D412yescomplemented
64#16864638—22no—
65#16864638C2312yes—
66#16864637—22no—
67C2×C271⋊C8—22no—
68C2×C271⋊C8C2312yes—
69C8×D271—22no—
70#41×C271—22no—
71#41×C271C2312yes—
72C2×D4×C271—22no—
73C2×D4×C271—41no—
74C2×D4×C271—41no—
75C2×D4×C271C2311yescharacteristic
76C2×D4×C271C2311yescharacteristic
77SD16×C271—22no—
78D8×C271—22no—
79M16×C271C2312yes—
80C42×C271C2311yescharacteristic
81C22×D271—22no—
82C22×D271—41no—
83C22×D271—42no—
84Dic542C2×D412yes—
85D1084—22no—
86D1084—22no—
87D1084—42no—
88D1084—82no—
89D1084C2×D412yes—
90C22⋊D271—22no—
91C22⋊D271—42no—
92C22⋊D271—42no—
93C2×Dic271—41no—
94C4×D271—22no—
95C4×D271—42no—
96C271⋊C8—22no—
97C271⋊C8C2×D412yes—
98C271⋊C8C2×D412yes—
99C23×C271—21no—
100C23×C271—21no—
101C23×C271—41no—
102Q8×C271C2×D412yes—
103D4×C271—22no—
104D4×C271—22no—
105D4×C271—22no—
106D4×C271—42no—
107D4×C271—82no—
108D4×C271C2×D412yes—
109C2×C4×C271—21no—
110C2×C4×C271—22no—
111C2×C4×C271—41no—
112C2×C4×C271C2411yescharacteristic
113C8×C271—22no—
114C22×C271—22no—
115C22×C271—41no—
116C22×C271—41no—
117C22×C271—81no—
118C22×C271—81no—
119C22×C271—161no—
120C22×C271H(5,𝔽2)11yescharacteristic
121C2×D271—22no—
122C2×D271—42no—
123C2×D271—82no—
124C2×D271—161no—
125C2×D271—162no—
126C4×C271—22no—
127C4×C271—22no—
128C4×C271C22×D412yes—
129Dic271—22no—
130C2×C271—21no—
131C2×C271—42no—
132C2×C271—81no—
133C2×C271—81no—
134C2×C271D4211yescharacteristic
135D271—42no—
136D271—82no—
137C271#329211yescharacteristic, complemented, Sylow
138#3292—2711nomaximal, Sylow
139#576—2712no—
140#576—2712no—
141#493—2711no—
142#472—2712no—
143#472—2712no—
144#453—2711no—
145#453—2712no—
146#453D27111yescharacteristic, complemented
147#443—2712no—
148H(5,𝔽2)—2712no—
149H(5,𝔽2)—2711no—
150H(5,𝔽2)C2×D27111yescharacteristic, complemented
151AGL(1,ℤ/8)—2712no—
152AGL(1,ℤ/8)—2712no—
153AGL(1,ℤ/8)—2712no—
154AGL(1,ℤ/8)—5422no—
155AGL(1,ℤ/8)—5422no—
156AGL(1,ℤ/8)—5422no—
157AGL(1,ℤ/8)—5422no—
158AGL(1,ℤ/8)C2×D27112yescomplemented
159#135—2712no—
160#135—5422no—
161#135—5422no—
162C2×D8—2712no—
163C2×D8—2712no—
164C2×D8—5422no—
165C2×D8—5422no—
166#131—2712no—
167#131—2712no—
168#127—2711no—
169#127C2×D27111yescharacteristic, complemented
170#108—2712no—
171#104—2712no—
172#104C2×D27112yescomplemented
173#100—2711no—
174#100—2712no—
175#100C2×D27111yescharacteristic, complemented
176C4×C8—2711no—
177#41—22no—
178#41—2712no—
179#41—5422no—
180#41—5422no—
181#41—5422no—
182#41C22×D27112yes—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—2712no—
187C2×D4—5422no—
188C2×D4—5422no—
189C2×D4—5422no—
190C2×D4—10841no—
191C2×D4—10842no—
192C2×D4—10841no—
193C2×D4C22×D27111yescharacteristic
194C2×D4C22×D27111yescharacteristic
195Q16—2712no—
196SD16—22no—
197SD16—2712no—
198SD16—2712no—
199SD16—5422no—
200SD16—5422no—
201SD16—5422no—
202D8—22no—
203D8—2712no—
204D8—5422no—
205D8—5422no—
206D8—5422no—
207D8—5422no—
208D8—5422no—
209D8—5422no—
210D8—5422no—
211M16—2712no—
212M16—5422no—
213M16—5422no—
214M16C22×D27112yes—
215C2×C8—2712no—
216C2×C8—5422no—
217C2×C8—5422no—
218C42C22×D27111yescharacteristic
219C23—21no—
220C23—21no—
221C23—41no—
222C23—5422no—
223C23—10842no—
224C23—10841no—
225Q8—2712no—
226Q8D4×D27112yes—
227D4—22no—
228D4—22no—
229D4—22no—
230D4—42no—
231D4—82no—
232D4—2712no—
233D4—5422no—
234D4—5422no—
235D4—5422no—
236D4—10842no—
237D4—10842no—
238D4—10842no—
239D4—21682no—
240D4D4×D27112yes—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—41no—
244C2×C4—5422no—
245C2×C4—10842no—
246C2×C4—10841no—
247C2×C4C23×D27111yescharacteristic
248C8—22no—
249C8—2712no—
250C8—2712no—
251C8—5422no—
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—161no—
258C22—5422no—
259C22—10842no—
260C22—21682no—
261C22—43362no—
262C22—43361no—
263C22#2257620511yescharacteristic
264C4—22no—
265C4—22no—
266C4—5422no—
267C4C2×D4×D27112yes—
268C2—21no—
269C2—42no—
270C2—81no—
271C2—81no—
272C2—10842no—
273C2—21682no—
274C2#3177452911yescharacteristic
subgroupG/H
centerC2#31774529
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#453×C271C2
socleC2×C271D42
hypercenter#453D271
hypocenterC271#3292
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #31774701C2semidirect
2. #31774700C2semidirect
3. #31774273C2semidirect
4. #31774166C2semidirect
5. #31774165C2semidirect
6. #31774066C2semidirect
7. #31774064C2semidirect
8. #31774011C2semidirect
9. #453×C271C2semidirect
10. #22576206C22semidirect
11. #22576205C22semidirect
12. #22576188C22semidirect
15. #22576186C22semidirect
17. #22576185C22semidirect
23. C2×#16864639C22semidirect
24. C2×#16864639C22semidirect
26. #22576168C22semidirect
27. #22576167C22semidirect
28. #22576157C22semidirect
29. #22576095C22semidirect
30. #22576088C22semidirect
31. #22576077C22semidirect
32. #22576076C22semidirect
33. C4×C271⋊C8C22semidirect
34. H(5,𝔽2)×C271C22semidirect
35. AGL(1,ℤ/8)×C271C22semidirect
36. #127×C271C22semidirect
37. #104×C271C22semidirect
38. #100×C271C22semidirect
50. #16864644D4semidirect
53. #16864642D4semidirect
55. #16864641D4semidirect
63. #16864639D4semidirect
137. C271#3292semidirect
146. #453D271semidirect
150. H(5,𝔽2)C2×D271semidirect
158. AGL(1,ℤ/8)C2×D271semidirect
169. #127C2×D271semidirect
172. #104C2×D271semidirect
175. #100C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747012
#317747002
#317742732
#317741662
#317741652
#317740662
#317740642
#317740112
#453×C2712
#3292271