Finite Group Databasebeta

← #46820286

Subgroups — #46820286

Order 34432.

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#31771441C211yesmaximal, characteristic, complemented
2#31771172C212yesmaximal, complemented
3#31770791C218yesmaximal, complemented
4#31770579C211yesmaximal, characteristic, complemented
5#31770578C212yesmaximal, complemented
6#535×C269C211yesmaximal, characteristic, complemented
7#22575501—21no—
8#22575500—22no—
9#22575499—22no—
10#22575498—21no—
11#22575498C2214yes—
12C2×#16864402—22no—
13C2×#16864402C2212yes—
14#41×D269—24no—
15#22575449—22no—
16#22575440—22no—
17#22575424—24no—
18#22575422C2218yescomplemented
19#22575412C2212yescomplemented
20#22575368C2214yescomplemented
21#22575367C2218yescomplemented
22C2×#41×C269C2211yescharacteristic
23C22×D4×C269C2211yescharacteristic, complemented
24#127×C269—21no—
25#124×C269—21no—
26#121×C269—22no—
27#120×C269C2214yescomplemented
28#117×C269C2211yescharacteristic, complemented
29#16864403—24no—
30#16864402—24no—
31#16864402—24no—
32#16864402—22no—
33#16864402—22no—
34#16864402—24no—
35#16864402—22no—
36#16864402—22no—
37#16864401—24no—
38#16864401—28no—
39C2×Dic538—22no—
40C2×Dic538D412yescomplemented
41Q8×D269—24no—
42C2×D1076—22no—
43C2×D1076D412yescomplemented
44C2×C22⋊D269—28no—
45C2×C22⋊D269—24no—
46C2×C22⋊D269C2314yes—
47D4×D269—24no—
48D4×D269—28no—
49C2×C4×D269—24no—
50C2×C4×D269D414yescomplemented
51C4⋊Dic269—22no—
52C22⋊Dic269—28no—
53C22⋊Dic269C2314yes—
54C4×Dic269—22no—
55C24×C269C2312yes—
56#41×C269—22no—
57#41×C269—22no—
58C2×Q8×C269D411yescharacteristic, complemented
59C2×D4×C269—24no—
60C2×D4×C269—22no—
61C2×D4×C269—22no—
62C2×D4×C269—21no—
63C2×D4×C269—21no—
64C2×D4×C269D411yescharacteristic, complemented
65C2×D4×C269C2312yes—
66C22×C4×C269D412yescomplemented
67C22×C4×C269C2311yescharacteristic
68C4⋊C4×C269—21no—
69C22⋊C4×C269—24no—
70C22⋊C4×C269C2312yes—
71C42×C269—21no—
72C22×D269—24no—
73C22×D269C2×D414yes—
74Dic538—24no—
75Dic538—22no—
76Dic538—22no—
77D1076—24no—
78D1076—22no—
79D1076—22no—
80C22⋊D269—28no—
81C22⋊D269—24no—
82C22⋊D269—24no—
83C22⋊D269—88no—
84C2×Dic269—24no—
85C2×Dic269—24no—
86C2×Dic269C2×D414yes—
87C4×D269—28no—
88C4×D269—24no—
89C4×D269—24no—
90C23×C269—22no—
91C23×C269—24no—
92C23×C269—82no—
93C23×C269C2×D412yes—
94C23×C269C2411yescharacteristic
95Q8×C269—22no—
96D4×C269—24no—
97D4×C269—22no—
98D4×C269—41no—
99D4×C269—41no—
100D4×C269—42no—
101D4×C269—84no—
102C2×C4×C269—22no—
103C2×C4×C269—22no—
104C2×C4×C269—22no—
105C2×C4×C269C2×D412yes—
106C2×C4×C269C2×D412yes—
107C2×C4×C269C2×D412yes—
108C2×C4×C269C2×D411yescharacteristic
109C2×C4×C269C2×D411yescharacteristic
110C22×C269—22no—
111C22×C269—41no—
112C22×C269—42no—
113C22×C269—44no—
114C22×C269—48no—
115C22×C269—42no—
116C22×C269C22×D412yes—
117C22×C269C22×D411yescharacteristic
118C2×D269—24no—
119C2×D269—44no—
120C2×D269—84no—
121C4×C269—22no—
122C4×C269—42no—
123C4×C269#12011yescharacteristic
124C4×C269#12011yescharacteristic
125C4×C269#12012yes—
126Dic269—24no—
127Dic269—44no—
128C2×C269—22no—
129C2×C269—21no—
130C2×C269—42no—
131C2×C269—82no—
132C2×C269C2×#12011yescharacteristic
133D269—44no—
134C269#302711yescharacteristic, complemented, Sylow
135#3027—2691nomaximal, Sylow
136#585—2691no—
137#535—2692no—
138#535D26911yescharacteristic, complemented
139B(4,𝔽2)—2698no—
140#410—2691no—
141#410—2692no—
1422−1+4—5382no—
1432−1+4—5381no—
144H(5,𝔽2)—2694no—
145H(5,𝔽2)—5382no—
146H(5,𝔽2)—5381no—
147C2×#41—2692no—
148C2×#41—5384no—
149C2×#41—5382no—
150C2×#41C2×D26911yescharacteristic
151C22×D4C2×D26911yescharacteristic, complemented
152#127—21no—
153#127—5382no—
154#124—21no—
155#124—5382no—
156#121—22no—
157#121—5384no—
158#120—2698no—
159#120C2×D26914yescomplemented
160#117—2692no—
161#117C2×D26911yescharacteristic, complemented
162#99—2694no—
163#99—2698no—
164C24C22×D26912yes—
165#41—22no—
166#41—22no—
167#41—5384no—
168#41—5384no—
169#41—5388no—
170#41—5384no—
171#41—5384no—
172#41—5382no—
173#41—5382no—
174#41—5382no—
175#41—5382no—
176#41—5384no—
177C2×Q8—2692no—
178C2×Q8—5382no—
179C2×Q8—5384no—
180C2×Q8C22⋊D26911yescharacteristic, complemented
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—2694no—
187C2×D4—2692no—
188C2×D4—5382no—
189C2×D4—5388no—
190C2×D4—5384no—
191C2×D4—5384no—
192C2×D4—5388no—
193C2×D4C22⋊D26911yescharacteristic, complemented
194C2×D4C22×D26912yes—
195C22×C4—2694no—
196C22×C4—5384no—
197C22×C4C22⋊D26912yescomplemented
198C22×C4C22×D26911yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—5382no—
201C22⋊C4—24no—
202C22⋊C4—2694no—
203C22⋊C4—5388no—
204C22⋊C4C22×D26912yes—
205C42—21no—
206C42—5382no—
207C23—24no—
208C23—22no—
209C23—82no—
210C23—2694no—
211C23—5384no—
212C23C2×C22⋊D26912yes—
213C23C23×D26911yescharacteristic
214Q8—22no—
215Q8—5384no—
216Q8—5382no—
217Q8—5382no—
218D4—24no—
219D4—22no—
220D4—41no—
221D4—41no—
222D4—42no—
223D4—84no—
224D4—5384no—
225D4—5388no—
226D4—5384no—
227D4—5384no—
228D4—5382no—
229D4—5382no—
230D4—21528no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—2694no—
235C2×C4—5388no—
236C2×C4—5384no—
237C2×C4—5384no—
238C2×C4—5384no—
239C2×C4—5384no—
240C2×C4D4×D26912yes—
241C2×C4D4×D26912yes—
242C2×C4C2×C22⋊D26912yes—
243C2×C4C2×C22⋊D26911yescharacteristic
244C2×C4C2×C22⋊D26911yescharacteristic
245C22—22no—
246C22—41no—
247C22—42no—
248C22—44no—
249C22—48no—
250C22—42no—
251C22—5384no—
252C22—10764no—
253C22—21524no—
254C22C2×D4×D26912yes—
255C22C22×C22⋊D26911yescharacteristic
256C4—22no—
257C4—42no—
258C4—5384no—
259C4—10764no—
260C4#2257542111yescharacteristic
261C4#2257542111yescharacteristic
262C4#2257542112yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—82no—
267C2—10764no—
268C2C2×#2257542111yescharacteristic
subgroupG/H
centerC2C2×#22575421
commutator, cosocleC23×C269C24
FrattiniC23C23×D269
Fitting#535×C269C2
socleC2×C269C2×#120
hypercenter#535D269
hypocenterC269#3027
LCSG ▷ C23×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. #31771441C2semidirect
2. #31771172C2semidirect
3. #31770791C2semidirect
4. #31770579C2semidirect
5. #31770578C2semidirect
6. #535×C269C2semidirect
18. #22575422C22semidirect
19. #22575412C22semidirect
20. #22575368C22semidirect
21. #22575367C22semidirect
23. C22×D4×C269C22semidirect
27. #120×C269C22semidirect
28. #117×C269C22semidirect
40. C2×Dic538D4semidirect
43. C2×D1076D4semidirect
50. C2×C4×D269D4semidirect
58. C2×Q8×C269D4semidirect
64. C2×D4×C269D4semidirect
66. C22×C4×C269D4semidirect
134. C269#3027semidirect
138. #535D269semidirect
151. C22×D4C2×D269semidirect
159. #120C2×D269semidirect
161. #117C2×D269semidirect
180. C2×Q8C22⋊D269semidirect
193. C2×D4C22⋊D269semidirect
197. C22×C4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
#317714412
#317711722
#317707912
#317705792
#317705782
#535×C2692
#3027269