Finite Group Databasebeta

← #46820263

Subgroups — #46820263

Order 34432.

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#31771441C211yesmaximal, characteristic
2C2×#22575484C212yesmaximal, complemented
3#31770787C214yesmaximal, complemented
4#31770782C214yesmaximal, complemented
5#31770623C212yesmaximal, complemented
6#31770589C211yesmaximal, characteristic, complemented
7#536×C269C211yesmaximal, characteristic, complemented
8#22575501C2212yes—
9#22575500—22no—
10#22575499—22no—
11#22575498—22no—
12#22575498C2212yes—
13C2×#16864402—22no—
14C2×#16864402C2212yes—
15#41×D269—24no—
16#22575484—24no—
17#22575484C2218yescomplemented
18C2×#16864392—22no—
19C2×#16864389—22no—
20C2×#16864386C2212yescomplemented
21#22575381C2218yescomplemented
22#22575372C2212yescomplemented
23#22575370C2212yescomplemented
24C2×#41×C269C2211yescharacteristic
25C22×Q8×C269C2211yescharacteristic, complemented
26C4⋊Q8×C269C2212yescomplemented
27#124×C269C2212yescomplemented
28#123×C269—22no—
29C22⋊Q8×C269—22no—
30#117×C269C2211yescharacteristic, complemented
31#16864403—24no—
32#16864402—24no—
33#16864402—24no—
34#16864402—22no—
35#16864402—24no—
36#16864402—22no—
37#16864402D414yes—
38#16864402C2314yes—
39#16864401—28no—
40#16864401—24no—
41C2×Dic538—22no—
42C2×Dic538D412yes—
43Q8×D269—24no—
44C2×D1076—22no—
45C2×D1076D412yes—
46C2×C22⋊D269—24no—
47C2×C22⋊D269—22no—
48D4×D269—28no—
49D4×D269—24no—
50C2×C4×D269—24no—
51C2×C4×D269—22no—
52#16864392—28no—
53#16864389—28no—
54#16864386—22no—
55#16864386C2314yes—
56C2×#14524130—22no—
57#41×C269—22no—
58#41×C269—22no—
59C2×Q8×C269—21no—
60C2×Q8×C269—22no—
61C2×Q8×C269—22no—
62C2×Q8×C269C2312yes—
63C2×Q8×C269C2311yescharacteristic
64C2×D4×C269D412yes—
65C2×D4×C269C2311yescharacteristic
66C22×C4×C269—21no—
67C22×C4×C269D412yes—
68C22×C4×C269C2311yescharacteristic
69C4⋊C4×C269—21no—
70C4⋊C4×C269—24no—
71C22⋊C4×C269—21no—
72C22⋊C4×C269—24no—
73C42×C269C2312yes—
74C22×D269—24no—
75C22×D269—22no—
76Dic538—24no—
77Dic538—22no—
78Dic538C2×D414yes—
79D1076—24no—
80D1076—22no—
81D1076C2×D414yes—
82C22⋊D269—28no—
83C22⋊D269—24no—
84C22⋊D269—24no—
85C2×Dic269—24no—
86C2×Dic269—22no—
87C4×D269—28no—
88C4×D269—24no—
89C4×D269—24no—
90#14524130—24no—
91C23×C269C2×D412yes—
92C23×C269C2×D411yescharacteristic
93Q8×C269—22no—
94Q8×C269—24no—
95Q8×C269—41no—
96Q8×C269—41no—
97D4×C269—24no—
98D4×C269—22no—
99C2×C4×C269—22no—
100C2×C4×C269—22no—
101C2×C4×C269—22no—
102C2×C4×C269—22no—
103C2×C4×C269—42no—
104C2×C4×C269C2×D412yes—
105C2×C4×C269C2×D412yes—
106C2×C4×C269C2×D412yes—
107C2×C4×C269C2×D411yescharacteristic
108C2×C4×C269C2411yescharacteristic
109C22×C269—22no—
110C22×C269—41no—
111C22×C269—42no—
112C22×C269#12012yes—
113C22×C269C22×D411yescharacteristic
114C2×D269—24no—
115C2×D269—82no—
116C2×D269—84no—
117C4×C269—22no—
118C4×C269—42no—
119C4×C269—42no—
120C4×C269#12011yescharacteristic
121C4×C269#12011yescharacteristic
122C4×C269C22×D412yes—
123Dic269—24no—
124C2×C269—21no—
125C2×C269—22no—
126C2×C269—42no—
127C2×C269C2×#12011yescharacteristic
128D269—44no—
129C269#302211yescharacteristic, complemented, Sylow
130#3022—2691nomaximal, Sylow
131#585—2691no—
132C2×#137—2692no—
133#536D26911yescharacteristic, complemented
134#456—2694no—
135#455—2694no—
136#421—2692no—
137#413—2691no—
1382−1+4—2692no—
1392−1+4—5382no—
140H(5,𝔽2)—2692no—
141H(5,𝔽2)—5382no—
142H(5,𝔽2)—5382no—
143C2×#41—2692no—
144C2×#41—5384no—
145C2×#41—5382no—
146C2×#41C2×D26911yescharacteristic
147C22×Q8C2×D26911yescharacteristic, complemented
148#137—2698no—
149#137—5384no—
150C2×Q16—5382no—
151C2×SD16—5382no—
152C2×M16—2692no—
153C4⋊Q8C2×D26912yescomplemented
154#124C2×D26912yescomplemented
155#123—22no—
156C22⋊Q8—22no—
157#117C2×D26911yescharacteristic, complemented
158#104—2698no—
159#101—2692no—
160#100—2692no—
161#41—22no—
162#41—22no—
163#41—2694no—
164#41—2694no—
165#41—5384no—
166#41—5384no—
167#41—5384no—
168#41—5388no—
169#41—5384no—
170#41—5382no—
171#41—5382no—
172#41—5384no—
173C2×Q8—21no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—2692no—
177C2×Q8—5382no—
178C2×Q8—5384no—
179C2×Q8C22×D26912yes—
180C2×Q8C22×D26911yescharacteristic
181C2×D4—2692no—
182C2×D4—5382no—
183C2×D4—5384no—
184C2×D4—5388no—
185C2×D4—5382no—
186C2×D4—5384no—
187C2×D4C22⋊D26912yes—
188C2×D4C22×D26911yescharacteristic
189C22×C4—21no—
190C22×C4—5384no—
191C22×C4—5382no—
192C22×C4C22⋊D26912yes—
193C22×C4C22×D26911yescharacteristic
194Q16—5388no—
195SD16—5388no—
196M16—2694no—
197M16—5382no—
198C2×C8—5382no—
199C4⋊C4—21no—
200C4⋊C4—24no—
201C22⋊C4—21no—
202C22⋊C4—24no—
203C42C22×D26912yes—
204C23—5384no—
205C23—5382no—
206C23C2×C22⋊D26912yes—
207C23C2×C22⋊D26911yescharacteristic
208Q8—22no—
209Q8—24no—
210Q8—41no—
211Q8—41no—
212Q8—2694no—
213Q8—5384no—
214Q8—5382no—
215D4—24no—
216D4—22no—
217D4—2694no—
218D4—5384no—
219D4—5388no—
220D4—5384no—
221D4—5384no—
222D4—5382no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—42no—
228C2×C4—5388no—
229C2×C4—5384no—
230C2×C4—5382no—
231C2×C4—5384no—
232C2×C4—5384no—
233C2×C4D4×D26912yes—
234C2×C4D4×D26912yes—
235C2×C4C2×C22⋊D26912yes—
236C2×C4C2×C22⋊D26911yescharacteristic
237C2×C4C23×D26911yescharacteristic
238C8—5384no—
239C22—22no—
240C22—41no—
241C22—42no—
242C22—5384no—
243C22—21522no—
244C22—21524no—
245C22#2257542112yes—
246C22C22×C22⋊D26911yescharacteristic
247C4—22no—
248C4—42no—
249C4—42no—
250C4—5384no—
251C4#2257542111yescharacteristic
252C4#2257542111yescharacteristic
253C4C2×D4×D26912yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—10764no—
258C2C2×#2257542111yescharacteristic
subgroupG/H
centerC2C2×#22575421
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#536×C269C2
socleC2×C269C2×#120
prime radical#31771441C2
hypercenter#536D269
hypocenterC269#3022
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#22575484C2semidirect
3. #31770787C2semidirect
4. #31770782C2semidirect
5. #31770623C2semidirect
6. #31770589C2semidirect
7. #536×C269C2semidirect
17. #22575484C22semidirect
20. C2×#16864386C22semidirect
21. #22575381C22semidirect
22. #22575372C22semidirect
23. #22575370C22semidirect
25. C22×Q8×C269C22semidirect
26. C4⋊Q8×C269C22semidirect
27. #124×C269C22semidirect
30. #117×C269C22semidirect
129. C269#3022semidirect
133. #536D269semidirect
147. C22×Q8C2×D269semidirect
153. C4⋊Q8C2×D269semidirect
154. #124C2×D269semidirect
157. #117C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317714412
C2×#225754842
#317707872
#317707822
#317706232
#317705892
#536×C2692
#3022269