Finite Group Databasebeta

← #46845767

Subgroups — #46845767

Order 34688.

772 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 145 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#31774733C211yesmaximal, characteristic, complemented
2#31774466C212yesmaximal, complemented
3#31774461C211yesmaximal, characteristic, complemented
4#31774085C218yesmaximal, complemented
5#31773870C212yesmaximal, complemented
6#410×C271C211yesmaximal, characteristic, complemented
7#22576208—21no—
8#22576207—22no—
9#22576206—22no—
10#22576205—21no—
11#22576205C2214yes—
12C2×#16864654—22no—
13C2×#16864654C2212yes—
14#41×D271—24no—
15C22×D1084C2211yescharacteristic, complemented
16#22576157—22no—
17#22576156—21no—
18#22576149—21no—
19#22576148—22no—
20#22576135—24no—
21#22576134—22no—
22#22576130C2218yescomplemented
23#22576128C2214yescomplemented
24#22576119C2211yescharacteristic, complemented
25#22576074C2218yescomplemented
26C2×#41×C271C2211yescharacteristic
27#117×C271C2212yescomplemented
28#99×C271C2214yescomplemented
29C23×D271C2312yes—
30#16864655—24no—
31#16864654—24no—
32#16864654—24no—
33#16864654—22no—
34#16864654—22no—
35#16864654—24no—
36#16864654—22no—
37#16864654—22no—
38#16864653—24no—
39#16864653—28no—
40C2×Dic542—22no—
41C2×Dic542D412yescomplemented
42Q8×D271—24no—
43C2×D1084—22no—
44C2×D1084—28no—
45C2×D1084—21no—
46C2×D1084—22no—
47C2×D1084—22no—
48C2×D1084—21no—
49C2×D1084D412yescomplemented
50C2×C22⋊D271—24no—
51C2×C22⋊D271—24no—
52C2×C22⋊D271C2314yes—
53D4×D271—24no—
54D4×D271—28no—
55C2×C4×D271—24no—
56C2×C4×D271D414yescomplemented
57C4⋊Dic271—21no—
58C22⋊Dic271C2312yes—
59(C2×D271)⋊C4—28no—
60(C2×D271)⋊C4—24no—
61C4×Dic271—21no—
62#41×C271—22no—
63#41×C271—22no—
64C2×Q8×C271D411yescharacteristic, complemented
65C2×D4×C271D411yescharacteristic, complemented
66C2×D4×C271C2312yes—
67C22×C4×C271D412yescomplemented
68C22×C4×C271C2311yescharacteristic
69C4⋊C4×C271—22no—
70C22⋊C4×C271C2314yes—
71C42×C271—22no—
72C22×D271—24no—
73C22×D271—22no—
74C22×D271—24no—
75C22×D271—82no—
76C22×D271C2×D414yes—
77Dic542—24no—
78Dic542—22no—
79Dic542—22no—
80D1084—24no—
81D1084—22no—
82D1084—22no—
83D1084—41no—
84D1084—41no—
85D1084—42no—
86D1084—88no—
87C22⋊D271—28no—
88C22⋊D271—24no—
89C22⋊D271—24no—
90C22⋊D271—84no—
91C2×Dic271—24no—
92C2×Dic271—22no—
93C2×Dic271C2×D414yes—
94C4×D271—28no—
95C4×D271—24no—
96C4×D271—24no—
97C23×C271C2×D412yes—
98C23×C271C2411yescharacteristic
99Q8×C271—22no—
100D4×C271—24no—
101D4×C271—22no—
102C2×C4×C271—22no—
103C2×C4×C271—22no—
104C2×C4×C271—24no—
105C2×C4×C271C2×D412yes—
106C2×C4×C271C2×D412yes—
107C2×C4×C271C2×D412yes—
108C2×C4×C271C2×D411yescharacteristic
109C2×C4×C271C2×D411yescharacteristic
110C22×C271—22no—
111C22×C271—41no—
112C22×C271—42no—
113C22×C271C22×D412yes—
114C22×C271C22×D411yescharacteristic
115C2×D271—24no—
116C2×D271—44no—
117C2×D271—44no—
118C2×D271—48no—
119C2×D271—42no—
120C2×D271—84no—
121C4×C271—22no—
122C4×C271—44no—
123C4×C271#12011yescharacteristic
124C4×C271#12011yescharacteristic
125C4×C271#12012yes—
126Dic271—24no—
127Dic271—42no—
128C2×C271—21no—
129C2×C271—22no—
130C2×C271—42no—
131C2×C271C2×#12011yescharacteristic
132D271—44no—
133D271—82no—
134C271#302711yescharacteristic, complemented, Sylow
135#3027—2711nomaximal, Sylow
136#585—2711no—
137#535—2711no—
138#535—2712no—
139B(4,𝔽2)—2718no—
140#410—2712no—
141#410D27111yescharacteristic, complemented
1422−1+4—5422no—
1432−1+4—5421no—
144H(5,𝔽2)—2714no—
145H(5,𝔽2)—5422no—
146H(5,𝔽2)—5421no—
147C2×#41—2712no—
148C2×#41—5424no—
149C2×#41—5422no—
150C2×#41C2×D27111yescharacteristic
151C22×D4—2711no—
152#127—5422no—
153#127—5421no—
154#124—5422no—
155#124—5421no—
156#121—5424no—
157#121—5422no—
158#120—2714no—
159#120—2718no—
160#117—2711no—
161#117C2×D27112yescomplemented
162#99—2718no—
163#99C2×D27114yescomplemented
164C24—2712no—
165#41—22no—
166#41—22no—
167#41—5424no—
168#41—5424no—
169#41—5428no—
170#41—5424no—
171#41—5424no—
172#41—5422no—
173#41—5422no—
174#41—5422no—
175#41—5422no—
176#41—5424no—
177C2×Q8—2712no—
178C2×Q8—5422no—
179C2×Q8—5424no—
180C2×Q8D108411yescharacteristic, complemented
181C2×D4—2714no—
182C2×D4—2712no—
183C2×D4—5422no—
184C2×D4—5428no—
185C2×D4—5428no—
186C2×D4—5424no—
187C2×D4—5424no—
188C2×D4—5424no—
189C2×D4—5422no—
190C2×D4—5422no—
191C2×D4—5421no—
192C2×D4—5421no—
193C2×D4D108411yescharacteristic, complemented
194C2×D4C22×D27112yes—
195C22×C4—2714no—
196C22×C4—5424no—
197C22×C4D108412yescomplemented
198C22×C4C22×D27111yescharacteristic
199C4⋊C4—22no—
200C4⋊C4—5421no—
201C22⋊C4—2712no—
202C22⋊C4—5428no—
203C22⋊C4—5424no—
204C22⋊C4C22×D27114yes—
205C42—22no—
206C42—5421no—
207C23—2714no—
208C23—5424no—
209C23—5424no—
210C23—5422no—
211C23—21682no—
212C23C2×D108412yes—
213C23C23×D27111yescharacteristic
214Q8—22no—
215Q8—5424no—
216Q8—5422no—
217Q8—5422no—
218D4—24no—
219D4—22no—
220D4—5424no—
221D4—5428no—
222D4—5424no—
223D4—5424no—
224D4—5422no—
225D4—5422no—
226D4—10841no—
227D4—10841no—
228D4—10842no—
229D4—21688no—
230D4—21684no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—24no—
234C2×C4—2714no—
235C2×C4—5428no—
236C2×C4—5424no—
237C2×C4—5424no—
238C2×C4—5424no—
239C2×C4—5422no—
240C2×C4D4×D27112yes—
241C2×C4D4×D27112yes—
242C2×C4C2×D108412yes—
243C2×C4C2×D108411yescharacteristic
244C2×C4C2×D108411yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—5424no—
249C22—10844no—
250C22—10848no—
251C22—10844no—
252C22—10842no—
253C22—21684no—
254C22C2×D4×D27112yes—
255C22C22×D108411yescharacteristic
256C4—22no—
257C4—44no—
258C4—5424no—
259C4—10842no—
260C4#2257613011yescharacteristic
261C4#2257613011yescharacteristic
262C4#2257613012yes—
263C2—22no—
264C2—21no—
265C2—42no—
266C2—10844no—
267C2—21682no—
268C2C2×#2257613011yescharacteristic
subgroupG/H
centerC2C2×#22576130
commutator, cosocleC23×C271C24
FrattiniC23C23×D271
Fitting#410×C271C2
socleC2×C271C2×#120
hypercenter#410D271
hypocenterC271#3027
LCSG ▷ C23×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #31774733C2semidirect
2. #31774466C2semidirect
3. #31774461C2semidirect
4. #31774085C2semidirect
5. #31773870C2semidirect
6. #410×C271C2semidirect
15. C22×D1084C22semidirect
22. #22576130C22semidirect
23. #22576128C22semidirect
24. #22576119C22semidirect
25. #22576074C22semidirect
27. #117×C271C22semidirect
28. #99×C271C22semidirect
41. C2×Dic542D4semidirect
49. C2×D1084D4semidirect
56. C2×C4×D271D4semidirect
64. C2×Q8×C271D4semidirect
65. C2×D4×C271D4semidirect
67. C22×C4×C271D4semidirect
134. C271#3027semidirect
141. #410D271semidirect
161. #117C2×D271semidirect
163. #99C2×D271semidirect
180. C2×Q8D1084semidirect
193. C2×D4D1084semidirect
197. C22×C4D1084semidirect

Maximal subgroups

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

subgroupindex
#317747332
#317744662
#317744612
#317740852
#317738702
#410×C2712
#3027271