Finite Group Databasebeta

← #46820242

Subgroups — #46820242

Order 34432.

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

normal subgroups only

This group has 762 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
1#31771403C212yesmaximal, complemented
2#31771172C211yesmaximal, characteristic, complemented
3#31770783C214yesmaximal, complemented
4#31770773C214yesmaximal, complemented
5#31770622C212yesmaximal, complemented
6C2×#22575370C211yesmaximal, characteristic, complemented
7C2×H(5,𝔽2)×C269C211yesmaximal, characteristic, complemented
8C2×#16864402C2211yescharacteristic, complemented
9#22575484—22no—
10#22575483C2214yescomplemented
11#22575479—22no—
12#22575478C2214yescomplemented
13#22575473—24no—
14C2×#16864386C2212yescomplemented
15#22575449C2212yescomplemented
16#22575440C2212yescomplemented
17#22575424—22no—
18#22575422—22no—
19#22575412C2211yescharacteristic, complemented
20#22575380C2218yescomplemented
21#22575370C2214yescomplemented
22H(5,𝔽2)×C269—24no—
23H(5,𝔽2)×C269—22no—
24H(5,𝔽2)×C269C2214yes—
25C2×#41×C269—22no—
26C2×#41×C269C2212yes—
27C22×D4×C269—22no—
28C22×D4×C269—22no—
29C22×D4×C269C2211yescharacteristic
30#16864402—22no—
31#16864402—22no—
32C2×Dic538C2311yescharacteristic, complemented
33C2×D1076C2311yescharacteristic, complemented
34C2×C22⋊D269—21no—
35C2×C22⋊D269—24no—
36C2×C4×D269—21no—
37#16864392—24no—
38#16864390—24no—
39#16864389—24no—
40#16864387—24no—
41#16864386—22no—
42#16864386C2314yescomplemented
43C2×#14524130—22no—
44C4⋊Dic269—21no—
45C22⋊Dic269—21no—
46C22⋊Dic269—24no—
47C4×Dic269C2312yescomplemented
48C24×C269—22no—
49C24×C269D412yes—
50#41×C269—28no—
51#41×C269—28no—
52#41×C269—22no—
53#41×C269—22no—
54#41×C269D414yescomplemented
55#41×C269C2314yes—
56C2×Q8×C269D412yescomplemented
57C2×D4×C269—24no—
58C2×D4×C269—28no—
59C2×D4×C269—28no—
60C2×D4×C269—22no—
61C2×D4×C269—28no—
62C2×D4×C269—28no—
63C2×D4×C269—22no—
64C2×D4×C269—22no—
65C2×D4×C269D412yes—
66C2×D4×C269D412yescomplemented
67C2×D4×C269C2312yes—
68C22×C4×C269—22no—
69C22×C4×C269—22no—
70C22×C4×C269C2311yescharacteristic
71C22×D269—21no—
72Dic538—22no—
73D1076—22no—
74C22⋊D269—41no—
75C22⋊D269—41no—
76C22⋊D269—84no—
77C2×Dic269—22no—
78C2×Dic269—21no—
79C4×D269—42no—
80#14524130—24no—
81C23×C269—28no—
82C23×C269—22no—
83C23×C269—22no—
84C23×C269—22no—
85C23×C269—82no—
86C23×C269—84no—
87C23×C269C2×D414yes—
88C23×C269C2×D411yescharacteristic
89Q8×C269—22no—
90Q8×C269C2×D414yes—
91D4×C269—216no—
92D4×C269—24no—
93D4×C269—24no—
94D4×C269—24no—
95D4×C269—22no—
96D4×C269—24no—
97D4×C269C2×D414yes—
98C2×C4×C269—28no—
99C2×C4×C269—22no—
100C2×C4×C269—24no—
101C2×C4×C269—24no—
102C2×C4×C269C2×D411yescharacteristic
103C2×C4×C269C2×D412yes—
104C2×C4×C269C2×D412yes—
105C2×C4×C269C2411yescharacteristic
106C22×C269—24no—
107C22×C269—24no—
108C22×C269—41no—
109C22×C269—44no—
110C22×C269—44no—
111C22×C269—44no—
112C22×C269—82no—
113C22×C269—88no—
114C22×C269#12011yescharacteristic
115C22×C269#12011yescharacteristic
116C22×C269C22×D411yescharacteristic
117C2×D269—42no—
118C2×D269—41no—
119C4×C269—24no—
120C4×C269#12012yes—
121C4×C269C22×D412yes—
122Dic269—42no—
123Dic269—41no—
124C2×C269—21no—
125C2×C269—21no—
126C2×C269—21no—
127C2×C269—44no—
128C2×C269—44no—
129C2×C269C2×#12011yescharacteristic
130D269—81no—
131C269#301911yescharacteristic, complemented, Sylow
132#3019—2691nomaximal, Sylow
133C2×H(5,𝔽2)D26911yescharacteristic, complemented
134#575—2692no—
135#535—2691no—
136#455—2694no—
137#453—2694no—
138#421—2692no—
139C2×#100—2691no—
140H(5,𝔽2)—24no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D26914yes—
143C2×#41—22no—
144C2×#41—2691no—
145C2×#41C2×D26912yes—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C2×D26911yescharacteristic
149#137—2694no—
150#137—5382no—
151AGL(1,ℤ/8)—2694no—
152AGL(1,ℤ/8)—5382no—
153#135—5384no—
154C2×M16—2692no—
155#127—2692no—
156#124—2692no—
157#121—5382no—
158#120—5382no—
159#117—2691no—
160#104—2698no—
161#100—2694no—
162C24—22no—
163C24C22⋊D26912yes—
164#41—28no—
165#41—28no—
166#41—22no—
167#41—22no—
168#41—5382no—
169#41—5382no—
170#41C22⋊D26914yescomplemented
171#41C22×D26914yes—
172C2×Q8—2691no—
173C2×Q8C22⋊D26912yescomplemented
174C2×D4—24no—
175C2×D4—28no—
176C2×D4—28no—
177C2×D4—22no—
178C2×D4—28no—
179C2×D4—28no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—2691no—
183C2×D4—5381no—
184C2×D4—5384no—
185C2×D4C22⋊D26912yes—
186C2×D4C22⋊D26912yescomplemented
187C2×D4C22×D26912yes—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4—5381no—
191C22×C4C22×D26911yescharacteristic
192Q16—5384no—
193SD16—5384no—
194SD16—5384no—
195D8—5384no—
196M16—2694no—
197M16—5382no—
198C2×C8—5382no—
199C4⋊C4—5381no—
200C22⋊C4—5381no—
201C22⋊C4—5384no—
202C42—2692no—
203C23—28no—
204C23—22no—
205C23—22no—
206C23—22no—
207C23—82no—
208C23—84no—
209C23—5381no—
210C23C2×C22⋊D26914yes—
211C23C2×C22⋊D26911yescharacteristic
212Q8—22no—
213Q8—5382no—
214Q8C2×C22⋊D26914yes—
215D4—216no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—22no—
220D4—24no—
221D4—5382no—
222D4—10761no—
223D4—10761no—
224D4—21524no—
225D4C2×C22⋊D26914yes—
226C2×C4—28no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—5382no—
231C2×C4—5381no—
232C2×C4—10762no—
233C2×C4C2×C22⋊D26911yescharacteristic
234C2×C4C2×C22⋊D26912yes—
235C2×C4C2×C22⋊D26912yes—
236C2×C4C23×D26911yescharacteristic
237C8—5384no—
238C22—24no—
239C22—24no—
240C22—41no—
241C22—44no—
242C22—44no—
243C22—44no—
244C22—82no—
245C22—88no—
246C22—10762no—
247C22—10761no—
248C22#2257542211yescharacteristic
249C22#2257542211yescharacteristic
250C22C22×C22⋊D26911yescharacteristic
251C4—24no—
252C4—10762no—
253C4—10761no—
254C4#2257542212yes—
255C4C22×C22⋊D26912yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—44no—
261C2—21521no—
262C2C2×#2257542211yescharacteristic
subgroupG/H
centerC2C2×#22575422
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
FittingC2×H(5,𝔽2)×C269C2
socleC2×C269C2×#120
hypercenterC2×H(5,𝔽2)D269
hypocenterC269#3019
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #31771403C2semidirect
2. #31771172C2semidirect
3. #31770783C2semidirect
4. #31770773C2semidirect
5. #31770622C2semidirect
6. C2×#22575370C2semidirect
7. C2×H(5,𝔽2)×C269C2semidirect
8. C2×#16864402C22semidirect
10. #22575483C22semidirect
12. #22575478C22semidirect
14. C2×#16864386C22semidirect
15. #22575449C22semidirect
16. #22575440C22semidirect
19. #22575412C22semidirect
20. #22575380C22semidirect
21. #22575370C22semidirect
32. C2×Dic538C23semidirect
33. C2×D1076C23semidirect
42. #16864386C23semidirect
47. C4×Dic269C23semidirect
54. #41×C269D4semidirect
56. C2×Q8×C269D4semidirect
66. C2×D4×C269D4semidirect
131. C269#3019semidirect
133. C2×H(5,𝔽2)D269semidirect
170. #41C22⋊D269semidirect
173. C2×Q8C22⋊D269semidirect
186. C2×D4C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
#317714032
#317711722
#317707832
#317707732
#317706222
C2×#225753702
C2×H(5,𝔽2)×C2692
#3019269