Finite Group Databasebeta

← #46820257

Subgroups — #46820257

Order 34432.

736 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#22575478C212yesmaximal, complemented
2#31771169C211yesmaximal, characteristic, complemented
3#31770778C214yesmaximal, complemented
4#31770773C214yesmaximal, complemented
5#31770622C212yesmaximal, complemented
6#31770589C211yesmaximal, characteristic, complemented
7#585×C269C211yesmaximal, characteristic, complemented
8C22×D1076C2211yescharacteristic, complemented
9#22575478—24no—
10#22575478C2218yescomplemented
11C2×#16864389—22no—
12C2×#16864387—22no—
13C2×#16864386C2212yescomplemented
14#22575449C2212yescomplemented
15#22575442C2212yescomplemented
16#22575427—22no—
17#22575421—22no—
18#22575412C2211yescharacteristic, complemented
19#22575380C2218yescomplemented
20#22575372C2212yescomplemented
21#22575370C2212yescomplemented
222−1+4×C269—22no—
232−1+4×C269C2212yescomplemented
24H(5,𝔽2)×C269—22no—
25H(5,𝔽2)×C269—22no—
26H(5,𝔽2)×C269C2212yescomplemented
27C2×#41×C269—24no—
28C2×#41×C269—22no—
29C2×#41×C269C2212yes—
30C2×#41×C269C2211yescharacteristic
31C23×D269—21no—
32C2×D1076—22no—
33C2×D1076—22no—
34C2×D1076—21no—
35C2×D1076C2312yes—
36C2×C22⋊D269—24no—
37#16864389—28no—
38#16864387—28no—
39#16864386—22no—
40#16864386C2314yes—
41C2×#14524130—22no—
42C4⋊Dic269—21no—
43C22⋊Dic269—21no—
44#16864379—24no—
45C4×Dic269C2312yes—
46#41×C269—24no—
47#41×C269—28no—
48#41×C269—24no—
49#41×C269—24no—
50#41×C269—24no—
51#41×C269—22no—
52#41×C269—22no—
53#41×C269—24no—
54#41×C269—22no—
55#41×C269—22no—
56#41×C269D414yescomplemented
57#41×C269C2314yes—
58C2×Q8×C269—22no—
59C2×Q8×C269—24no—
60C2×Q8×C269D412yescomplemented
61C2×Q8×C269C2311yescharacteristic
62C2×D4×C269—28no—
63C2×D4×C269—22no—
64C2×D4×C269—24no—
65C2×D4×C269—22no—
66C2×D4×C269—24no—
67C2×D4×C269D412yes—
68C2×D4×C269D412yescomplemented
69C2×D4×C269C2311yescharacteristic
70C22×C4×C269—24no—
71C22×C4×C269—22no—
72C22×C4×C269D412yes—
73C22×C4×C269C2311yescharacteristic
74C22×D269—22no—
75C22×D269—42no—
76C22×D269—82no—
77D1076—24no—
78D1076—41no—
79D1076—41no—
80C22⋊D269—84no—
81C2×Dic269—22no—
82#14524130—24no—
83C23×C269—24no—
84C23×C269—22no—
85C23×C269C2×D412yes—
86C23×C269C2×D411yescharacteristic
87Q8×C269—24no—
88Q8×C269—22no—
89Q8×C269—22no—
90Q8×C269C2×D414yes—
91D4×C269—28no—
92D4×C269—24no—
93D4×C269—24no—
94D4×C269—24no—
95D4×C269—24no—
96D4×C269—22no—
97D4×C269—22no—
98D4×C269C2×D414yes—
99C2×C4×C269—24no—
100C2×C4×C269—28no—
101C2×C4×C269—22no—
102C2×C4×C269—24no—
103C2×C4×C269—22no—
104C2×C4×C269—22no—
105C2×C4×C269—24no—
106C2×C4×C269C2×D412yes—
107C2×C4×C269C2×D411yescharacteristic
108C2×C4×C269C2×D412yes—
109C2×C4×C269C2×D412yes—
110C2×C4×C269C2411yescharacteristic
111C22×C269—22no—
112C22×C269—24no—
113C22×C269—41no—
114C22×C269—42no—
115C22×C269—82no—
116C22×C269—84no—
117C22×C269#12012yes—
118C22×C269C22×D411yescharacteristic
119C2×D269—44no—
120C2×D269—42no—
121C2×D269—84no—
122C4×C269—24no—
123C4×C269—22no—
124C4×C269#12011yescharacteristic
125C4×C269#12011yescharacteristic
126C4×C269C22×D412yes—
127Dic269—42no—
128C2×C269—21no—
129C2×C269—22no—
130C2×C269—42no—
131C2×C269—44no—
132C2×C269C2×#12011yescharacteristic
133D269—82no—
134C269#302111yescharacteristic, complemented, Sylow
135#3021—2691nomaximal, Sylow
136#585D26911yescharacteristic, complemented
137C2×AGL(1,ℤ/8)—2692no—
138#535—2691no—
139#454—2694no—
140#453—2694no—
141#421—2692no—
142#413—2691no—
1432−1+4—22no—
1442−1+4C2×D26912yescomplemented
145H(5,𝔽2)—22no—
146H(5,𝔽2)—22no—
147H(5,𝔽2)C2×D26912yescomplemented
148C2×#41—24no—
149C2×#41—22no—
150C2×#41C2×D26912yes—
151C2×#41C2×D26911yescharacteristic
152C22×D4—2691no—
153AGL(1,ℤ/8)—2698no—
154AGL(1,ℤ/8)—5384no—
155C2×SD16—5382no—
156C2×D8—5382no—
157C2×M16—2692no—
158#127—2692no—
159#124—2692no—
160#121—5382no—
161#120—5382no—
162#117—2691no—
163#104—2698no—
164#101—2692no—
165#100—2692no—
166C24—5381no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41C22⋊D26914yescomplemented
178#41C22×D26914yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8C22⋊D26912yescomplemented
182C2×Q8C22×D26911yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—2692no—
189C2×D4—5384no—
190C2×D4—5382no—
191C2×D4—5382no—
192C2×D4—5381no—
193C2×D4C22⋊D26912yes—
194C2×D4C22⋊D26912yescomplemented
195C2×D4C22×D26911yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4C22⋊D26912yes—
199C22×C4C22×D26911yescharacteristic
200SD16—5388no—
201D8—5388no—
202M16—2694no—
203M16—5382no—
204C2×C8—5382no—
205C4⋊C4—5381no—
206C22⋊C4—5381no—
207C22⋊C4—5384no—
208C42—2692no—
209C23—24no—
210C23—22no—
211C23—5382no—
212C23—10762no—
213C23—21522no—
214C23C2×C22⋊D26912yes—
215C23C2×C22⋊D26911yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×C22⋊D26914yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—5384no—
228D4—10761no—
229D4—10761no—
230D4—21524no—
231D4C2×C22⋊D26914yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—5382no—
240C2×C4C2×C22⋊D26912yes—
241C2×C4C2×C22⋊D26911yescharacteristic
242C2×C4C2×C22⋊D26912yes—
243C2×C4C2×C22⋊D26912yes—
244C2×C4C23×D26911yescharacteristic
245C8—5384no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—84no—
252C22—10764no—
253C22—10762no—
254C22—21524no—
255C22#2257542212yes—
256C22C22×C22⋊D26911yescharacteristic
257C4—24no—
258C4—22no—
259C4—10762no—
260C4#2257542211yescharacteristic
261C4#2257542211yescharacteristic
262C4C22×C22⋊D26912yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—21522no—
268C2C2×#2257542211yescharacteristic
subgroupG/H
centerC2C2×#22575422
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#585×C269C2
socleC2×C269C2×#120
hypercenter#585D269
hypocenterC269#3021
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#22575478C2semidirect
2. #31771169C2semidirect
3. #31770778C2semidirect
4. #31770773C2semidirect
5. #31770622C2semidirect
6. #31770589C2semidirect
7. #585×C269C2semidirect
8. C22×D1076C22semidirect
10. #22575478C22semidirect
13. C2×#16864386C22semidirect
14. #22575449C22semidirect
15. #22575442C22semidirect
18. #22575412C22semidirect
19. #22575380C22semidirect
20. #22575372C22semidirect
21. #22575370C22semidirect
23. 2−1+4×C269C22semidirect
26. H(5,𝔽2)×C269C22semidirect
56. #41×C269D4semidirect
60. C2×Q8×C269D4semidirect
68. C2×D4×C269D4semidirect
134. C269#3021semidirect
136. #585D269semidirect
144. 2−1+4C2×D269semidirect
147. H(5,𝔽2)C2×D269semidirect
177. #41C22⋊D269semidirect
181. C2×Q8C22⋊D269semidirect
194. C2×D4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
C2×#225754782
#317711692
#317707782
#317707732
#317706222
#317705892
#585×C2692
#3021269