Finite Group Databasebeta

← #46820279

Subgroups — #46820279

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#31771117C212yesmaximal, complemented
3#31771112C211yesmaximal, characteristic, complemented
4#31770796C214yesmaximal, complemented
5#31770793C214yesmaximal, complemented
6#31770578C212yesmaximal, complemented
7#410×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—
16C22×C4×D269C2211yescharacteristic, complemented
17#22575437C2214yescomplemented
18#22575434C2212yescomplemented
19#22575433—22no—
20#22575429—21no—
21#22575428—22no—
22#22575424—21no—
23#22575423C2214yescomplemented
24#22575421C2212yescomplemented
25C4×D1076—24no—
26C4×C22⋊D269—22no—
27#22575412C2211yescharacteristic, complemented
28#22575367C2218yescomplemented
29C2×#41×C269C2211yescharacteristic
30#117×C269C2212yescomplemented
31#99×C269C2214yescomplemented
32C23×D269C2311yescharacteristic
33#16864403—24no—
34#16864402—24no—
35#16864402—24no—
36#16864402—22no—
37#16864402—22no—
38#16864402—24no—
39#16864402—22no—
40#16864402—22no—
41#16864401—24no—
42#16864401—28no—
43C2×Dic538—22no—
44C2×Dic538D412yescomplemented
45Q8×D269—24no—
46C2×D1076—22no—
47C2×D1076—24no—
48C2×D1076D412yescomplemented
49C2×C22⋊D269—24no—
50C2×C22⋊D269—22no—
51C2×C22⋊D269C2314yes—
52D4×D269—24no—
53D4×D269—28no—
54C22×Dic269C2311yescharacteristic
55C2×C4×D269—24no—
56C2×C4×D269—21no—
57C2×C4×D269—22no—
58C2×C4×D269—21no—
59C2×C4×D269—22no—
60C2×C4×D269D414yescomplemented
61C4⋊Dic269—24no—
62C4⋊Dic269—21no—
63#16864381—22no—
64C22⋊Dic269—22no—
65C22⋊Dic269C2312yes—
66#16864379—24no—
67#16864379—24no—
68#16864379—22no—
69C4×Dic269—21no—
70#41×C269—22no—
71#41×C269—22no—
72C2×Q8×C269D411yescharacteristic, complemented
73C2×D4×C269D411yescharacteristic, complemented
74C2×D4×C269C2312yes—
75C22×C4×C269D412yescomplemented
76C22×C4×C269C2311yescharacteristic
77C4⋊C4×C269—22no—
78C22⋊C4×C269C2314yes—
79C42×C269—22no—
80C22×D269—21no—
81C22×D269—22no—
82C22×D269—24no—
83C22×D269—42no—
84C22×D269C2×D414yes—
85Dic538—24no—
86Dic538—22no—
87Dic538—22no—
88D1076—24no—
89D1076—22no—
90D1076—22no—
91D1076—48no—
92C22⋊D269—28no—
93C22⋊D269—24no—
94C22⋊D269—24no—
95C22⋊D269—44no—
96C2×Dic269—24no—
97C2×Dic269—22no—
98C2×Dic269—22no—
99C2×Dic269—21no—
100C2×Dic269C2×D414yes—
101C4×D269—28no—
102C4×D269—24no—
103C4×D269—24no—
104C4×D269—41no—
105C4×D269—42no—
106C4×D269—41no—
107C23×C269C2×D412yes—
108C23×C269C2411yescharacteristic
109Q8×C269—22no—
110D4×C269—24no—
111D4×C269—22no—
112C2×C4×C269—22no—
113C2×C4×C269—22no—
114C2×C4×C269—24no—
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—41no—
122C22×C269—42no—
123C22×C269C22×D412yes—
124C22×C269C22×D411yescharacteristic
125C2×D269—22no—
126C2×D269—24no—
127C2×D269—24no—
128C2×D269—41no—
129C2×D269—42no—
130C2×D269—41no—
131C2×D269—44no—
132C2×D269—84no—
133C4×C269—22no—
134C4×C269—44no—
135C4×C269#12011yescharacteristic
136C4×C269#12012yes—
137C4×C269#12011yescharacteristic
138Dic269—24no—
139Dic269—42no—
140Dic269—41no—
141C2×C269—21no—
142C2×C269—22no—
143C2×C269—42no—
144C2×C269C2×#12011yescharacteristic
145D269—42no—
146D269—44no—
147C269#302511yescharacteristic, complemented, Sylow
148#3025—2691nomaximal, Sylow
149#585—2691no—
150#525—2691no—
151#525—2692no—
152#458—2694no—
153B(4,𝔽2)—2694no—
154#410—2692no—
155#410D26911yescharacteristic, complemented
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×C4—2691no—
166#123—2692no—
167#123—2694no—
168C22⋊Q8—5382no—
169C22⋊Q8—5381no—
170#121—5382no—
171#121—5381no—
172#120—2692no—
173#120—2694no—
174C4×D4—5384no—
175C4×D4—5382no—
176#117—2691no—
177#117C2×D26912yescomplemented
178#99—2698no—
179#99C2×D26914yescomplemented
180C24—2691no—
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×Q8D107611yescharacteristic, complemented
197C2×D4—2694no—
198C2×D4—2692no—
199C2×D4—5382no—
200C2×D4—5384no—
201C2×D4—5388no—
202C2×D4—5384no—
203C2×D4—5384no—
204C2×D4—5382no—
205C2×D4D107611yescharacteristic, complemented
206C2×D4C22×D26912yes—
207C22×C4—2694no—
208C22×C4—2691no—
209C22×C4—5384no—
210C22×C4—5382no—
211C22×C4—5382no—
212C22×C4—5381no—
213C22×C4—5381no—
214C22×C4D107612yescomplemented
215C22×C4C22×D26911yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—5384no—
218C4⋊C4—5381no—
219C4⋊C4—5382no—
220C22⋊C4—2692no—
221C22⋊C4—5384no—
222C22⋊C4—5384no—
223C22⋊C4—5382no—
224C22⋊C4—5382no—
225C22⋊C4C22×D26914yes—
226C42—22no—
227C42—5381no—
228C23—2694no—
229C23—5384no—
230C23—5382no—
231C23—5381no—
232C23—10762no—
233C23C2×D107612yes—
234C23C23×D26911yescharacteristic
235Q8—22no—
236Q8—5384no—
237Q8—5382no—
238Q8—5382no—
239D4—24no—
240D4—22no—
241D4—5384no—
242D4—5388no—
243D4—5384no—
244D4—5384no—
245D4—5382no—
246D4—5382no—
247D4—10768no—
248D4—10764no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—2694no—
253C2×C4—5388no—
254C2×C4—5384no—
255C2×C4—5384no—
256C2×C4—5384no—
257C2×C4—5382no—
258C2×C4—5382no—
259C2×C4—5381no—
260C2×C4—10761no—
261C2×C4—10762no—
262C2×C4—10761no—
263C2×C4D4×D26912yes—
264C2×C4D4×D26912yes—
265C2×C4C2×D107612yes—
266C2×C4C2×D107611yescharacteristic
267C2×C4C2×D107611yescharacteristic
268C22—22no—
269C22—41no—
270C22—42no—
271C22—5384no—
272C22—5382no—
273C22—5384no—
274C22—10764no—
275C22—10762no—
276C22—10761no—
277C22—10761no—
278C22—21524no—
279C22C2×D4×D26912yes—
280C22C22×D107611yescharacteristic
281C4—22no—
282C4—44no—
283C4—5384no—
284C4—10762no—
285C4—10761no—
286C4#2257542311yescharacteristic
287C4#2257542312yes—
288C4#2257542311yescharacteristic
289C2—22no—
290C2—21no—
291C2—42no—
292C2—10762no—
293C2—10764no—
294C2C2×#2257542311yescharacteristic
subgroupG/H
centerC4#22575423
commutator, cosocleC23×C269C24
FrattiniC23C23×D269
Fitting#410×C269C2
socleC2×C269C2×#120
hypercenter#410D269
hypocenterC269#3025
LCSG ▷ C23×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C4 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
1. #31771441C2semidirect
2. #31771117C2semidirect
3. #31771112C2semidirect
4. #31770796C2semidirect
5. #31770793C2semidirect
6. #31770578C2semidirect
7. #410×C269C2semidirect
16. C22×C4×D269C22semidirect
17. #22575437C22semidirect
18. #22575434C22semidirect
23. #22575423C22semidirect
24. #22575421C22semidirect
27. #22575412C22semidirect
28. #22575367C22semidirect
30. #117×C269C22semidirect
31. #99×C269C22semidirect
44. C2×Dic538D4semidirect
48. C2×D1076D4semidirect
60. C2×C4×D269D4semidirect
72. C2×Q8×C269D4semidirect
73. C2×D4×C269D4semidirect
75. C22×C4×C269D4semidirect
147. C269#3025semidirect
155. #410D269semidirect
177. #117C2×D269semidirect
179. #99C2×D269semidirect
196. C2×Q8D1076semidirect
205. C2×D4D1076semidirect
214. C22×C4D1076semidirect

Maximal subgroups

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

subgroupindex
#317714412
#317711172
#317711122
#317707962
#317707932
#317705782
#410×C2692
#3025269