Finite Group Databasebeta

← #46848159

Subgroups — #46848159

Order 34688.

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

normal subgroups only

This group has 460 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#31774708C212yesmaximal, complemented
2#31774707C212yesmaximal, complemented
3#31774269C211yesmaximal, characteristic, complemented
4#31774159C212yesmaximal, complemented
5#31774158C212yesmaximal, complemented
6#31774081C212yesmaximal, complemented
7#31774064C211yesmaximal, characteristic, complemented
8#31774011C212yesmaximal, complemented
9#453×C271C211yesmaximal, characteristic, complemented
10#22576207C2212yes—
11#22576205C2211yescharacteristic, complemented
12#22576193C2212yescomplemented
13#22576190—22no—
14#22576190C2212yescomplemented
15#22576189—22no—
16#22576186—22no—
17#22576186C2212yes—
18#22576184—22no—
19#22576180—22no—
20#22576180C2212yes—
21#22576179—22no—
22C2×#16864642—22no—
23C2×#16864642C2212yescomplemented
24C2×#16864642C2212yescomplemented
25SD16×D271—22no—
26#22576168C2212yescomplemented
27#22576167C2212yescomplemented
28#22576160C2211yescharacteristic, complemented
29#22576095C2212yescomplemented
30#22576088C2212yescomplemented
31#22576078C2212yescomplemented
32#22576077C2211yescharacteristic, complemented
33C4×C271⋊C8C2211yescharacteristic, complemented
34H(5,𝔽2)×C271C2211yescharacteristic, complemented
35AGL(1,ℤ/8)×C271C2212yes—
36#127×C271C2211yescharacteristic
37#104×C271C2212yescomplemented
38#100×C271C2211yescharacteristic, complemented
39#16864654—22no—
40#16864654C2312yes—
41#16864653—22no—
42#16864653—22no—
43#16864653—42no—
44C2×Dic542—22no—
45C2×Dic542C2312yes—
46Q8×D271—22no—
47C2×C22⋊D271—41no—
48D4×D271—22no—
49Dic1084—22no—
50#16864644—22no—
51#16864644D412yescomplemented
52#16864643—22no—
53#16864642—22no—
54#16864642—22no—
55#16864642—22no—
56#16864642—22no—
57#16864642—22no—
58#16864642—22no—
59#16864642D412yescomplemented
60#16864641D412yescomplemented
61#16864639—22no—
62#16864639D412yescomplemented
63#16864638—22no—
64#16864638C2312yes—
65#16864637—22no—
66C2×C271⋊C8—22no—
67C2×C271⋊C8C2312yes—
68C8×D271—22no—
69C4⋊Dic271—41no—
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—41no—
82Dic542—22no—
83Dic542—22no—
84Dic542—42no—
85Dic542C2×D412yes—
86D1084C2×D412yes—
87C22⋊D271—22no—
88C22⋊D271—42no—
89C22⋊D271—42no—
90C2×Dic271—22no—
91C2×Dic271—41no—
92C2×Dic271—42no—
93C4×D271—22no—
94C4×D271—42no—
95C271⋊C8—22no—
96C271⋊C8C2×D412yes—
97C271⋊C8C2×D412yes—
98C23×C271—21no—
99C23×C271—21no—
100C23×C271—41no—
101Q8×C271C2×D412yes—
102D4×C271—22no—
103D4×C271—22no—
104D4×C271—22no—
105D4×C271—42no—
106D4×C271—82no—
107D4×C271C2×D412yes—
108C2×C4×C271—21no—
109C2×C4×C271—22no—
110C2×C4×C271—41no—
111C2×C4×C271C2411yescharacteristic
112C8×C271—22no—
113C22×C271—22no—
114C22×C271—41no—
115C22×C271—41no—
116C22×C271—81no—
117C22×C271—81no—
118C22×C271—161no—
119C22×C271H(5,𝔽2)11yescharacteristic
120C2×D271—22no—
121C2×D271—161no—
122C4×C271—22no—
123C4×C271—22no—
124C4×C271C22×D412yes—
125Dic271—22no—
126Dic271—42no—
127C2×C271—21no—
128C2×C271—42no—
129C2×C271—81no—
130C2×C271—81no—
131C2×C271D4211yescharacteristic
132D271—42no—
133C271#329111yescharacteristic, complemented, Sylow
134#3291—2711nomaximal, Sylow
135#577—2712no—
136#577—2712no—
137#492—2711no—
138#471—2712no—
139#471—2712no—
140#456—2712no—
141#453—2711no—
142#453D27111yescharacteristic, complemented
143#443—2712no—
1442−1+4—2712no—
145H(5,𝔽2)—2711no—
146H(5,𝔽2)C2×D27111yescharacteristic, complemented
147#137—2712no—
148#137—2712no—
149#137—5422no—
150#137—5422no—
151AGL(1,ℤ/8)—2712no—
152AGL(1,ℤ/8)—5422no—
153AGL(1,ℤ/8)—5422no—
154AGL(1,ℤ/8)C2×D27112yes—
155#135—2712no—
156#135—5422no—
157#135—5422no—
158C2×SD16—2712no—
159C2×SD16—2712no—
160C2×SD16—5422no—
161C2×SD16—5422no—
162#131—2712no—
163#131—2712no—
164C4⋊Q8—2711no—
165#127C2×D27111yescharacteristic
166#108—2712no—
167#104—2712no—
168#104C2×D27112yescomplemented
169#101—2712no—
170#100—2711no—
171#100C2×D27111yescharacteristic, complemented
172C4×C8—2711no—
173#41—22no—
174#41—2712no—
175#41—5422no—
176#41—5422no—
177#41—5422no—
178#41—10842no—
179#41C22×D27112yes—
180C2×Q8—2712no—
181C2×Q8—5422no—
182C2×Q8—5422no—
183C2×D4—22no—
184C2×D4—41no—
185C2×D4—41no—
186C2×D4—5422no—
187C2×D4—10841no—
188C2×D4C22×D27111yescharacteristic
189C2×D4C22×D27111yescharacteristic
190Q16—2712no—
191Q16—5422no—
192Q16—5422no—
193SD16—22no—
194SD16—2712no—
195SD16—2712no—
196SD16—5422no—
197SD16—5422no—
198SD16—5422no—
199SD16—5422no—
200SD16—5422no—
201SD16—5422no—
202SD16—5422no—
203D8—22no—
204D8—2712no—
205D8—5422no—
206M16—2712no—
207M16—5422no—
208M16—5422no—
209M16C22×D27112yes—
210C2×C8—2712no—
211C2×C8—5422no—
212C2×C8—5422no—
213C4⋊C4—10841no—
214C42C22×D27111yescharacteristic
215C23—21no—
216C23—21no—
217C23—41no—
218C23—10841no—
219Q8—2712no—
220Q8—5422no—
221Q8—5422no—
222Q8—10842no—
223Q8D4×D27112yes—
224D4—22no—
225D4—22no—
226D4—22no—
227D4—42no—
228D4—82no—
229D4—2712no—
230D4—5422no—
231D4—10842no—
232D4—10842no—
233D4D4×D27112yes—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—41no—
237C2×C4—5422no—
238C2×C4—5422no—
239C2×C4—10842no—
240C2×C4—10841no—
241C2×C4—10842no—
242C2×C4C23×D27111yescharacteristic
243C8—22no—
244C8—2712no—
245C8—2712no—
246C8—5422no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—81no—
251C22—81no—
252C22—161no—
253C22—5422no—
254C22—43361no—
255C22#2257620511yescharacteristic
256C4—22no—
257C4—22no—
258C4—5422no—
259C4—10842no—
260C4C2×D4×D27112yes—
261C2—21no—
262C2—42no—
263C2—81no—
264C2—81no—
265C2—10842no—
266C2#3177452911yescharacteristic
subgroupG/H
centerC2#31774529
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#453×C271C2
socleC2×C271D42
hypercenter#453D271
hypocenterC271#3291
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. #31774708C2semidirect
2. #31774707C2semidirect
3. #31774269C2semidirect
4. #31774159C2semidirect
5. #31774158C2semidirect
6. #31774081C2semidirect
7. #31774064C2semidirect
8. #31774011C2semidirect
9. #453×C271C2semidirect
11. #22576205C22semidirect
12. #22576193C22semidirect
14. #22576190C22semidirect
23. C2×#16864642C22semidirect
24. C2×#16864642C22semidirect
26. #22576168C22semidirect
27. #22576167C22semidirect
28. #22576160C22semidirect
29. #22576095C22semidirect
30. #22576088C22semidirect
31. #22576078C22semidirect
32. #22576077C22semidirect
33. C4×C271⋊C8C22semidirect
34. H(5,𝔽2)×C271C22semidirect
37. #104×C271C22semidirect
38. #100×C271C22semidirect
51. #16864644D4semidirect
59. #16864642D4semidirect
60. #16864641D4semidirect
62. #16864639D4semidirect
133. C271#3291semidirect
142. #453D271semidirect
146. H(5,𝔽2)C2×D271semidirect
168. #104C2×D271semidirect
171. #100C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747082
#317747072
#317742692
#317741592
#317741582
#317740812
#317740642
#317740112
#453×C2712
#3291271