Finite Group Databasebeta

← #46845742

Subgroups — #46845742

Order 34688.

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

normal subgroups only

This group has 704 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
2C2×#22576191C212yesmaximal, complemented
3#31774079C214yesmaximal, complemented
4#31774074C214yesmaximal, complemented
5#31773915C212yesmaximal, complemented
6#31773881C211yesmaximal, characteristic, complemented
7#536×C271C211yesmaximal, characteristic, complemented
8#22576208C2212yes—
9#22576207—22no—
10#22576206—22no—
11#22576205—22no—
12#22576205C2212yes—
13C2×#16864654—22no—
14C2×#16864654C2212yes—
15#41×D271—24no—
16#22576191—24no—
17#22576191C2218yescomplemented
18C2×#16864644—22no—
19C2×#16864641—22no—
20C2×#16864638C2212yescomplemented
21#22576088C2218yescomplemented
22#22576079C2212yescomplemented
23#22576077C2212yescomplemented
24C2×#41×C271C2211yescharacteristic
25C22×Q8×C271C2211yescharacteristic, complemented
26C4⋊Q8×C271C2212yescomplemented
27#124×C271C2212yescomplemented
28#123×C271—22no—
29C22⋊Q8×C271—22no—
30#117×C271C2211yescharacteristic, complemented
31#16864655—24no—
32#16864654—24no—
33#16864654—24no—
34#16864654—22no—
35#16864654—24no—
36#16864654—22no—
37#16864654D414yes—
38#16864654C2314yes—
39#16864653—28no—
40#16864653—24no—
41C2×Dic542—22no—
42C2×Dic542D412yes—
43Q8×D271—24no—
44C2×D1084—22no—
45C2×D1084D412yes—
46C2×C22⋊D271—24no—
47C2×C22⋊D271—22no—
48D4×D271—28no—
49D4×D271—24no—
50C2×C4×D271—24no—
51C2×C4×D271—22no—
52#16864644—28no—
53#16864641—28no—
54#16864638—22no—
55#16864638C2314yes—
56C2×C271⋊C8—22no—
57#41×C271—22no—
58#41×C271—22no—
59C2×Q8×C271—21no—
60C2×Q8×C271—22no—
61C2×Q8×C271—22no—
62C2×Q8×C271C2312yes—
63C2×Q8×C271C2311yescharacteristic
64C2×D4×C271D412yes—
65C2×D4×C271C2311yescharacteristic
66C22×C4×C271—21no—
67C22×C4×C271D412yes—
68C22×C4×C271C2311yescharacteristic
69C4⋊C4×C271—21no—
70C4⋊C4×C271—24no—
71C22⋊C4×C271—21no—
72C22⋊C4×C271—24no—
73C42×C271C2312yes—
74C22×D271—24no—
75C22×D271—22no—
76Dic542—24no—
77Dic542—22no—
78Dic542C2×D414yes—
79D1084—24no—
80D1084—22no—
81D1084C2×D414yes—
82C22⋊D271—28no—
83C22⋊D271—24no—
84C22⋊D271—24no—
85C2×Dic271—24no—
86C2×Dic271—22no—
87C4×D271—28no—
88C4×D271—24no—
89C4×D271—24no—
90C271⋊C8—24no—
91C23×C271C2×D412yes—
92C23×C271C2×D411yescharacteristic
93Q8×C271—22no—
94Q8×C271—24no—
95Q8×C271—41no—
96Q8×C271—41no—
97D4×C271—24no—
98D4×C271—22no—
99C2×C4×C271—22no—
100C2×C4×C271—22no—
101C2×C4×C271—22no—
102C2×C4×C271—22no—
103C2×C4×C271—42no—
104C2×C4×C271C2×D412yes—
105C2×C4×C271C2×D412yes—
106C2×C4×C271C2×D412yes—
107C2×C4×C271C2×D411yescharacteristic
108C2×C4×C271C2411yescharacteristic
109C22×C271—22no—
110C22×C271—41no—
111C22×C271—42no—
112C22×C271#12012yes—
113C22×C271C22×D411yescharacteristic
114C2×D271—24no—
115C2×D271—82no—
116C2×D271—84no—
117C4×C271—22no—
118C4×C271—42no—
119C4×C271—42no—
120C4×C271#12011yescharacteristic
121C4×C271#12011yescharacteristic
122C4×C271C22×D412yes—
123Dic271—24no—
124C2×C271—21no—
125C2×C271—22no—
126C2×C271—42no—
127C2×C271C2×#12011yescharacteristic
128D271—44no—
129C271#302211yescharacteristic, complemented, Sylow
130#3022—2711nomaximal, Sylow
131#585—2711no—
132C2×#137—2712no—
133#536D27111yescharacteristic, complemented
134#456—2714no—
135#455—2714no—
136#421—2712no—
137#413—2711no—
1382−1+4—2712no—
1392−1+4—5422no—
140H(5,𝔽2)—2712no—
141H(5,𝔽2)—5422no—
142H(5,𝔽2)—5422no—
143C2×#41—2712no—
144C2×#41—5424no—
145C2×#41—5422no—
146C2×#41C2×D27111yescharacteristic
147C22×Q8C2×D27111yescharacteristic, complemented
148#137—2718no—
149#137—5424no—
150C2×Q16—5422no—
151C2×SD16—5422no—
152C2×M16—2712no—
153C4⋊Q8C2×D27112yescomplemented
154#124C2×D27112yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D27111yescharacteristic, complemented
158#104—2718no—
159#101—2712no—
160#100—2712no—
161#41—22no—
162#41—22no—
163#41—2714no—
164#41—2714no—
165#41—5424no—
166#41—5424no—
167#41—5424no—
168#41—5428no—
169#41—5424no—
170#41—5422no—
171#41—5422no—
172#41—5424no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—2712no—
177C2×Q8—5422no—
178C2×Q8—5424no—
179C2×Q8C22×D27112yes—
180C2×Q8C22×D27111yescharacteristic
181C2×D4—2712no—
182C2×D4—5422no—
183C2×D4—5424no—
184C2×D4—5428no—
185C2×D4—5422no—
186C2×D4—5424no—
187C2×D4C22⋊D27112yes—
188C2×D4C22×D27111yescharacteristic
189C22×C4—21no—
190C22×C4—5424no—
191C22×C4—5422no—
192C22×C4C22⋊D27112yes—
193C22×C4C22×D27111yescharacteristic
194Q16—5428no—
195SD16—5428no—
196M16—2714no—
197M16—5422no—
198C2×C8—5422no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D27112yes—
204C23—5424no—
205C23—5422no—
206C23C2×C22⋊D27112yes—
207C23C2×C22⋊D27111yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—2714no—
213Q8—5424no—
214Q8—5422no—
215D4—24no—
216D4—22no—
217D4—2714no—
218D4—5424no—
219D4—5428no—
220D4—5424no—
221D4—5424no—
222D4—5422no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—5428no—
229C2×C4—5424no—
230C2×C4—5422no—
231C2×C4—5424no—
232C2×C4—5424no—
233C2×C4D4×D27112yes—
234C2×C4D4×D27112yes—
235C2×C4C2×C22⋊D27112yes—
236C2×C4C2×C22⋊D27111yescharacteristic
237C2×C4C23×D27111yescharacteristic
238C8—5424no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—5424no—
243C22—21682no—
244C22—21684no—
245C22#2257612812yes—
246C22C22×C22⋊D27111yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—5424no—
251C4#2257612811yescharacteristic
252C4#2257612811yescharacteristic
253C4C2×D4×D27112yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—10844no—
258C2C2×#2257612811yescharacteristic
subgroupG/H
centerC2C2×#22576128
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#536×C271C2
socleC2×C271C2×#120
prime radical#31774733C2
hypercenter#536D271
hypocenterC271#3022
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#22576191C2semidirect
3. #31774079C2semidirect
4. #31774074C2semidirect
5. #31773915C2semidirect
6. #31773881C2semidirect
7. #536×C271C2semidirect
17. #22576191C22semidirect
20. C2×#16864638C22semidirect
21. #22576088C22semidirect
22. #22576079C22semidirect
23. #22576077C22semidirect
25. C22×Q8×C271C22semidirect
26. C4⋊Q8×C271C22semidirect
27. #124×C271C22semidirect
30. #117×C271C22semidirect
129. C271#3022semidirect
133. #536D271semidirect
147. C22×Q8C2×D271semidirect
153. C4⋊Q8C2×D271semidirect
154. #124C2×D271semidirect
157. #117C2×D271semidirect

Maximal subgroups

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

subgroupindex
#317747332
C2×#225761912
#317740792
#317740742
#317739152
#317738812
#536×C2712
#3022271