Finite Group Databasebeta

← #46820276

Subgroups — #46820276

Order 34432.

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

normal subgroups only

This group has 766 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#31771116C212yesmaximal, complemented
3#31770794C214yesmaximal, complemented
4#31770791C214yesmaximal, complemented
5#31770579C211yesmaximal, characteristic, complemented
6#31770578C212yesmaximal, complemented
7#525×C269C211yesmaximal, characteristic, complemented
8#22575501—21no—
9#22575500—22no—
10#22575499—22no—
11#22575498—21no—
12#22575498C2214yes—
13C2×#16864402—22no—
14C2×#16864402C2212yes—
15#41×D269—24no—
16#22575436C2214yescomplemented
17#22575432—22no—
18#22575427—22no—
19#22575422C2214yescomplemented
20C4×C22⋊D269—24no—
21#22575412C2212yescomplemented
22#22575368C2214yescomplemented
23#22575367C2218yescomplemented
24C2×#41×C269C2211yescharacteristic
25C23×C4×C269C2211yescharacteristic, complemented
26#123×C269C2212yescomplemented
27C22⋊Q8×C269—21no—
28#121×C269—21no—
29#120×C269C2212yescomplemented
30C4×D4×C269—22no—
31#117×C269C2211yescharacteristic, complemented
32#16864403—24no—
33#16864402—24no—
34#16864402—24no—
35#16864402—22no—
36#16864402—22no—
37#16864402—24no—
38#16864402—22no—
39#16864402—22no—
40#16864401—24no—
41#16864401—28no—
42C2×Dic538—22no—
43C2×Dic538D412yescomplemented
44Q8×D269—24no—
45C2×D1076—22no—
46C2×D1076D412yescomplemented
47C2×C22⋊D269—24no—
48C2×C22⋊D269—24no—
49C2×C22⋊D269C2314yes—
50D4×D269—24no—
51D4×D269—28no—
52C2×C4×D269—24no—
53C2×C4×D269D414yescomplemented
54C4⋊Dic269—22no—
55#16864381—24no—
56C22⋊Dic269—24no—
57C22⋊Dic269C2314yes—
58#16864379—24no—
59C4×Dic269—22no—
60C24×C269C2311yescharacteristic
61#41×C269—22no—
62#41×C269—22no—
63C2×Q8×C269D411yescharacteristic, complemented
64C2×D4×C269—22no—
65C2×D4×C269D411yescharacteristic, complemented
66C2×D4×C269C2312yes—
67C22×C4×C269—22no—
68C22×C4×C269—22no—
69C22×C4×C269—21no—
70C22×C4×C269—21no—
71C22×C4×C269D412yescomplemented
72C22×C4×C269C2311yescharacteristic
73C22×C4×C269C2311yescharacteristic
74C4⋊C4×C269—21no—
75C4⋊C4×C269—22no—
76C22⋊C4×C269—22no—
77C22⋊C4×C269—22no—
78C22⋊C4×C269C2312yes—
79C42×C269—21no—
80C22×D269—24no—
81C22×D269C2×D414yes—
82Dic538—24no—
83Dic538—22no—
84Dic538—22no—
85D1076—24no—
86D1076—22no—
87D1076—22no—
88C22⋊D269—28no—
89C22⋊D269—24no—
90C22⋊D269—24no—
91C22⋊D269—48no—
92C2×Dic269—24no—
93C2×Dic269—24no—
94C2×Dic269C2×D414yes—
95C4×D269—28no—
96C4×D269—24no—
97C4×D269—24no—
98C23×C269—21no—
99C23×C269—22no—
100C23×C269—42no—
101C23×C269C2×D412yes—
102C23×C269C2411yescharacteristic
103Q8×C269—22no—
104D4×C269—24no—
105D4×C269—22no—
106D4×C269—44no—
107C2×C4×C269—22no—
108C2×C4×C269—22no—
109C2×C4×C269—22no—
110C2×C4×C269—21no—
111C2×C4×C269—22no—
112C2×C4×C269—41no—
113C2×C4×C269—42no—
114C2×C4×C269—41no—
115C2×C4×C269C2×D412yes—
116C2×C4×C269C2×D412yes—
117C2×C4×C269C2×D412yes—
118C2×C4×C269C2×D411yescharacteristic
119C2×C4×C269C2×D411yescharacteristic
120C22×C269—22no—
121C22×C269—24no—
122C22×C269—22no—
123C22×C269—41no—
124C22×C269—42no—
125C22×C269—42no—
126C22×C269—41no—
127C22×C269—41no—
128C22×C269C22×D412yes—
129C22×C269C22×D411yescharacteristic
130C2×D269—24no—
131C2×D269—44no—
132C2×D269—84no—
133C4×C269—22no—
134C4×C269—42no—
135C4×C269—41no—
136C4×C269#12011yescharacteristic
137C4×C269#12012yes—
138C4×C269#12011yescharacteristic
139Dic269—24no—
140Dic269—44no—
141C2×C269—22no—
142C2×C269—21no—
143C2×C269—42no—
144C2×C269—42no—
145C2×C269C2×#12011yescharacteristic
146D269—44no—
147C269#302511yescharacteristic, complemented, Sylow
148#3025—2691nomaximal, Sylow
149#585—2691no—
150#525—2692no—
151#525D26911yescharacteristic, complemented
152#458—2694no—
153B(4,𝔽2)—2694no—
154#410—2691no—
155#410—2692no—
1562−1+4—5382no—
1572−1+4—5381no—
158H(5,𝔽2)—2694no—
159H(5,𝔽2)—5382no—
160H(5,𝔽2)—5381no—
161C2×#41—2692no—
162C2×#41—5384no—
163C2×#41—5382no—
164C2×#41C2×D26911yescharacteristic
165C23×C4C2×D26911yescharacteristic, complemented
166#123—2694no—
167#123C2×D26912yescomplemented
168C22⋊Q8—21no—
169C22⋊Q8—5382no—
170#121—21no—
171#121—5382no—
172#120—2694no—
173#120C2×D26912yescomplemented
174C4×D4—22no—
175C4×D4—5384no—
176#117—2692no—
177#117C2×D26911yescharacteristic, complemented
178#99—2694no—
179#99—2698no—
180C24C22×D26911yescharacteristic
181#41—22no—
182#41—22no—
183#41—5384no—
184#41—5384no—
185#41—5388no—
186#41—5384no—
187#41—5384no—
188#41—5382no—
189#41—5382no—
190#41—5382no—
191#41—5382no—
192#41—5384no—
193C2×Q8—2692no—
194C2×Q8—5382no—
195C2×Q8—5384no—
196C2×Q8C22⋊D26911yescharacteristic, complemented
197C2×D4—22no—
198C2×D4—2694no—
199C2×D4—2692no—
200C2×D4—5382no—
201C2×D4—5388no—
202C2×D4—5384no—
203C2×D4—5384no—
204C2×D4—5384no—
205C2×D4C22⋊D26911yescharacteristic, complemented
206C2×D4C22×D26912yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—2694no—
212C22×C4—5384no—
213C22×C4C22⋊D26912yescomplemented
214C22×C4C22×D26911yescharacteristic
215C22×C4C22×D26911yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—22no—
218C4⋊C4—5384no—
219C4⋊C4—5382no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—2694no—
223C22⋊C4—5384no—
224C22⋊C4—5384no—
225C22⋊C4C22×D26912yes—
226C42—21no—
227C42—5382no—
228C23—22no—
229C23—21no—
230C23—42no—
231C23—2694no—
232C23—5384no—
233C23C2×C22⋊D26912yes—
234C23C23×D26911yescharacteristic
235Q8—22no—
236Q8—5384no—
237Q8—5382no—
238Q8—5382no—
239D4—24no—
240D4—22no—
241D4—44no—
242D4—5384no—
243D4—5388no—
244D4—5384no—
245D4—5384no—
246D4—5382no—
247D4—5382no—
248D4—10768no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—2694no—
258C2×C4—5388no—
259C2×C4—5384no—
260C2×C4—5384no—
261C2×C4—5384no—
262C2×C4—5384no—
263C2×C4D4×D26912yes—
264C2×C4D4×D26912yes—
265C2×C4C2×C22⋊D26912yes—
266C2×C4C2×C22⋊D26911yescharacteristic
267C2×C4C2×C22⋊D26911yescharacteristic
268C22—22no—
269C22—24no—
270C22—22no—
271C22—41no—
272C22—42no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—5384no—
277C22—10764no—
278C22—21524no—
279C22C2×D4×D26912yes—
280C22C22×C22⋊D26911yescharacteristic
281C4—22no—
282C4—42no—
283C4—41no—
284C4—5384no—
285C4—10764no—
286C4#2257542111yescharacteristic
287C4#2257542112yes—
288C4#2257542111yescharacteristic
289C2—21no—
290C2—22no—
291C2—42no—
292C2—42no—
293C2—10764no—
294C2C2×#2257542111yescharacteristic
subgroupG/H
centerC4#22575421
commutator, cosocleC23×C269C24
FrattiniC23C23×D269
Fitting#525×C269C2
socleC2×C269C2×#120
hypercenter#525D269
hypocenterC269#3025
LCSG ▷ C23×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #31771441C2semidirect
2. #31771116C2semidirect
3. #31770794C2semidirect
4. #31770791C2semidirect
5. #31770579C2semidirect
6. #31770578C2semidirect
7. #525×C269C2semidirect
16. #22575436C22semidirect
19. #22575422C22semidirect
21. #22575412C22semidirect
22. #22575368C22semidirect
23. #22575367C22semidirect
25. C23×C4×C269C22semidirect
26. #123×C269C22semidirect
29. #120×C269C22semidirect
31. #117×C269C22semidirect
43. C2×Dic538D4semidirect
46. C2×D1076D4semidirect
53. C2×C4×D269D4semidirect
63. C2×Q8×C269D4semidirect
65. C2×D4×C269D4semidirect
71. C22×C4×C269D4semidirect
147. C269#3025semidirect
151. #525D269semidirect
165. C23×C4C2×D269semidirect
167. #123C2×D269semidirect
173. #120C2×D269semidirect
177. #117C2×D269semidirect
196. C2×Q8C22⋊D269semidirect
205. C2×D4C22⋊D269semidirect
213. C22×C4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
#317714412
#317711162
#317707942
#317707912
#317705792
#317705782
#525×C2692
#3025269