Finite Group Databasebeta

← #46821026

Subgroups — #46821026

Order 34432.

450 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 117 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#31771141C211yesmaximal, characteristic, complemented
2C2×#22575415C211yesmaximal, characteristic, complemented
3#31770918C211yesmaximal, characteristic, complemented
4#31770904C212yesmaximal, complemented
5#31770823C212yesmaximal, complemented
6#31770755C211yesmaximal, characteristic, complemented
7#31770741C211yesmaximal, characteristic
8#31770716C212yesmaximal, complemented
9#31770708C212yesmaximal, complemented
10#31770659C211yesmaximal, characteristic, complemented
11#483×C269C211yesmaximal, characteristic, complemented
12C22×D1076C2211yescharacteristic
13C22×C4×D269—21no—
14C2×#16864389C2212yes—
15C2×D2152C2212yes—
16#22575448—21no—
17#22575444C2211yescharacteristic, complemented
18#22575442C2211yescharacteristic, complemented
19#22575437—21no—
20#22575427C2211yescharacteristic
21Q8×Dic269C2211yescharacteristic, complemented
22#22575415—21no—
23#22575415—21no—
24#22575415C2212yes—
25#22575415C2212yes—
26#22575415C2211yescharacteristic, complemented
27#22575413—21no—
28#22575411—21no—
29C2×#16864379—21no—
30C2×C4×Dic269C2211yescharacteristic
31#22575386C2212yescomplemented
32#22575382C2212yescomplemented
33#22575375C2212yes—
34#22575373C2212yes—
35#22575373C2212yescomplemented
36C22⋊#14524130C2211yescharacteristic
37#22575363C2212yescomplemented
38C22⋊Q8×C269C2211yescharacteristic
39C2×C4⋊C4×C269C2211yescharacteristic
40#106×C269C2212yescomplemented
41#103×C269C2212yescomplemented
42C22⋊C8×C269C2211yescharacteristic
43C23×D269—21no—
44C2×D1076—24no—
45C2×D1076—21no—
46C2×D1076C2311yescharacteristic
47C2×D1076C2312yes—
48C2×C22⋊D269—21no—
49C22×Dic269D411yescharacteristic
50C22×Dic269D411yescharacteristic
51C2×C4×D269—21no—
52C2×C4×D269—22no—
53C2×C4×D269—22no—
54C2×C4×D269—44no—
55#16864389—24no—
56D2152—24no—
57C2×#14524130C2312yes—
58C4⋊Dic269—21no—
59C4⋊Dic269—21no—
60C4⋊Dic269C2311yescharacteristic
61#16864381—21no—
62#16864379—22no—
63#16864379—22no—
64#16864379—21no—
65#16864379—21no—
66#16864379—21no—
67#16864379—21no—
68C4×Dic269—21no—
69C4×Dic269D411yescharacteristic
70C4×Dic269D411yescharacteristic
71C4×Dic269C2311yescharacteristic
72C4×Dic269C2311yescharacteristic
73C2×Q8×C269C2311yescharacteristic
74C22×C4×C269—21no—
75C22×C4×C269C2311yescharacteristic
76C2×C8×C269C2312yes—
77C4⋊C4×C269—21no—
78C4⋊C4×C269—21no—
79C4⋊C4×C269C2311yescharacteristic
80C4⋊C4×C269C2312yes—
81C22⋊C4×C269—21no—
82C22×D269—24no—
83C22×D269—22no—
84C22×D269—21no—
85C22×D269—42no—
86C22×D269—42no—
87D1076—24no—
88D1076—22no—
89D1076—22no—
90D1076#4114yes—
91C22⋊D269—42no—
92C2×Dic269—22no—
93C2×Dic269—21no—
94C2×Dic269—41no—
95C2×Dic269C2×D411yescharacteristic
96C2×Dic269C2×D411yescharacteristic
97C2×Dic269C2×D411yescharacteristic
98C2×Dic269C2×D411yescharacteristic
99C4×D269—42no—
100C4×D269—44no—
101C4×D269—44no—
102#14524130—22no—
103C23×C269C2×D411yescharacteristic
104Q8×C269—22no—
105C2×C4×C269—22no—
106C2×C4×C269—21no—
107C2×C4×C269—21no—
108C2×C4×C269—22no—
109C2×C4×C269—42no—
110C2×C4×C269C2×D411yescharacteristic
111C2×C4×C269C2411yescharacteristic
112C8×C269—22no—
113C22×C269—21no—
114C22×C269—21no—
115C22×C269AGL(1,ℤ/8)12yes—
116C22×C269C22×D411yescharacteristic
117C2×D269—24no—
118C2×D269—48no—
119C2×D269—42no—
120C2×D269—42no—
121C2×D269—41no—
122C2×D269—41no—
123C2×D269—41no—
124C4×C269—21no—
125C4×C269—41no—
126C4×C269—42no—
127C4×C269—41no—
128C4×C269C2×#4112yes—
129Dic269—21no—
130Dic269—21no—
131Dic269—21no—
132Dic269—21no—
133Dic269—41no—
134C2×C269—22no—
135C2×C269#52511yescharacteristic
136C2×C269C2×AGL(1,ℤ/8)11yescharacteristic
137C2×C269#57511yescharacteristic
138D269—44no—
139D269—81no—
140C269#311411yescharacteristic, complemented, Sylow
141#3114—2691nomaximal, Sylow
142#529—2691no—
143C2×C4×D4—2691no—
144#483D26911yescharacteristic, complemented
145#480—2691no—
146#478—2692no—
147#463—2692no—
148#450—2691no—
149#447—2691no—
150#442—2692no—
151#440—2692no—
152#432—2691no—
153C22×D4—2691no—
154C23×C4—5381no—
155C2×SD16—2692no—
156C2×D8—2692no—
157#126—5381no—
158#125—2691no—
159#124—2691no—
160#123—5381no—
161C22⋊Q8C2×D26911yescharacteristic
162#121—2691no—
163C4×Q8—2691no—
164C4×D4—2692no—
165C4×D4—2692no—
166C4×D4—2691no—
167C4×D4—5381no—
168C4×D4—5381no—
169C4×D4—5381no—
170#117—5381no—
171C2×C4⋊C4C2×D26911yescharacteristic
172C2×C22⋊C4—5381no—
173C2×C42—2691no—
174#107—2692no—
175#106C2×D26912yescomplemented
176C4⋊C8—2692no—
177#103C2×D26912yescomplemented
178#102—2692no—
179#102—2692no—
180#102—2692no—
181C22⋊C8—2691no—
182C22⋊C8C2×D26911yescharacteristic
183#97—2692no—
184C24—5381no—
185C2×Q8C22×D26911yescharacteristic
186C2×D4—2692no—
187C2×D4—2691no—
188C2×D4—5384no—
189C2×D4—5381no—
190C2×D4—5381no—
191C22×C4—21no—
192C22×C4—2691no—
193C22×C4—2691no—
194C22×C4—5381no—
195C22×C4—5382no—
196C22×C4—5382no—
197C22×C4—10764no—
198C22×C4C22×D26911yescharacteristic
199SD16—5384no—
200D8—5384no—
201C2×C8—2692no—
202C2×C8C22×D26912yes—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—2691no—
206C4⋊C4—5381no—
207C4⋊C4—5381no—
208C4⋊C4—5381no—
209C4⋊C4C22×D26911yescharacteristic
210C4⋊C4C22×D26912yes—
211C22⋊C4—21no—
212C22⋊C4—5382no—
213C22⋊C4—5382no—
214C22⋊C4—5381no—
215C22⋊C4—5381no—
216C22⋊C4—5381no—
217C22⋊C4—5381no—
218C42—2691no—
219C42—2691no—
220C42—2691no—
221C42—2691no—
222C42—5381no—
223C23—5384no—
224C23—5382no—
225C23—5381no—
226C23—10762no—
227C23—10762no—
228C23D4×D26911yescharacteristic
229Q8—22no—
230D4—2694no—
231D4—5384no—
232D4—5382no—
233D4—5382no—
234D4—10762no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—42no—
240C2×C4—2691no—
241C2×C4—2691no—
242C2×C4—2691no—
243C2×C4—2691no—
244C2×C4—5382no—
245C2×C4—5381no—
246C2×C4—10764no—
247C2×C4—10764no—
248C2×C4—10762no—
249C2×C4—10761no—
250C2×C4D4×D26911yescharacteristic
251C2×C4C23×D26911yescharacteristic
252C8—22no—
253C8—5382no—
254C22—21no—
255C22—21no—
256C22—5384no—
257C22—10768no—
258C22—10762no—
259C22—10762no—
260C22—10761no—
261C22—10761no—
262C22—10761no—
263C22#2257548012yes—
264C22C2×D4×D26911yescharacteristic
265C4—21no—
266C4—42no—
267C4—41no—
268C4—41no—
269C4—5381no—
270C4—5381no—
271C4—5381no—
272C4—5381no—
273C4—10761no—
274C4#41×D26912yes—
275C2—22no—
276C2—10764no—
277C2—21521no—
278C2#3177111311yescharacteristic
279C2C2×#2257548011yescharacteristic
280C2#3177140411yescharacteristic
subgroupG/H
centerC22C2×D4×D269
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#483×C269C2
socleC22×C269C22×D4
prime radical#31770741C2
hypercenter#483D269
hypocenterC269#3114
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #483

Complements

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

NH ≅ G/Ntype
1. #31771141C2semidirect
2. C2×#22575415C2semidirect
3. #31770918C2semidirect
4. #31770904C2semidirect
5. #31770823C2semidirect
6. #31770755C2semidirect
8. #31770716C2semidirect
9. #31770708C2semidirect
10. #31770659C2semidirect
11. #483×C269C2semidirect
17. #22575444C22semidirect
18. #22575442C22semidirect
21. Q8×Dic269C22semidirect
26. #22575415C22semidirect
31. #22575386C22semidirect
32. #22575382C22semidirect
35. #22575373C22semidirect
37. #22575363C22semidirect
40. #106×C269C22semidirect
41. #103×C269C22semidirect
140. C269#3114semidirect
144. #483D269semidirect
175. #106C2×D269semidirect
177. #103C2×D269semidirect

Maximal subgroups

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

subgroupindex
#317711412
C2×#225754152
#317709182
#317709042
#317708232
#317707552
#317707412
#317707162
#317707082
#317706592
#483×C2692
#3114269