Finite Group Databasebeta

← #46822685

Subgroups — #46822685

Order 34432.

476 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 125 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#31771410C212yesmaximal, complemented
2#31771407C212yesmaximal, complemented
3#31770982C211yesmaximal, characteristic, complemented
4#31770876C212yesmaximal, complemented
5#31770875C212yesmaximal, complemented
6#31770774C214yesmaximal, complemented
7#31770723C211yesmaximal, characteristic, complemented
8#443×C269C211yesmaximal, characteristic, complemented
9#22575499C2214yescomplemented
10#22575481—24no—
11#22575481C2214yescomplemented
12#22575480—24no—
13#22575478C2214yescomplemented
14#22575476—22no—
15#22575475—22no—
16#22575475C2211yescharacteristic
17C2×D2152—22no—
18C2×D2152C2212yescomplemented
19C2×D2152C2212yescomplemented
20D8×D269—22no—
21#22575459C2212yescomplemented
22#22575450C2212yescomplemented
23#22575389C2211yescharacteristic, complemented
24#22575381C2212yescomplemented
25#22575369C2214yescomplemented
26#135×C269C2211yescharacteristic
27#131×C269C2212yescomplemented
28#108×C269C2211yescharacteristic, complemented
29#104×C269C2212yescomplemented
30C4×C8×C269C2211yescharacteristic, complemented
31#16864403—24no—
32#16864402—24no—
33#16864402C2312yes—
34C2×D1076—24no—
35C2×D1076—42no—
36C2×D1076C2314yes—
37D4×D269—24no—
38D4×D269—44no—
39Dic1076D411yescharacteristic, complemented
40#16864391—24no—
41#16864391D412yescomplemented
42#16864389—24no—
43D2152—24no—
44D2152—22no—
45D2152—22no—
46D2152—22no—
47D2152—22no—
48D2152D411yescharacteristic, complemented
49#16864387—24no—
50#16864386C2312yes—
51#16864385—22no—
52C8×D269—22no—
53#41×C269C2312yes—
54Q16×C269D411yescharacteristic, complemented
55SD16×C269D412yescomplemented
56D8×C269D411yescharacteristic, complemented
57M16×C269—22no—
58M16×C269C2312yes—
59C2×C8×C269—22no—
60C2×C8×C269C2311yescharacteristic
61C2×C8×C269C2311yescharacteristic
62C42×C269C2311yescharacteristic
63C22×D269—24no—
64C22×D269—44no—
65Dic538C2×D412yes—
66D1076—24no—
67D1076—24no—
68D1076—44no—
69D1076—84no—
70D1076C2×D412yes—
71C22⋊D269—22no—
72C22⋊D269—44no—
73C4×D269—22no—
74C4×D269—44no—
75#14524130—22no—
76Q8×C269C2×D412yes—
77D4×C269—22no—
78D4×C269C2×D412yes—
79C2×C4×C269—21no—
80C2×C4×C269—22no—
81C2×C4×C269C2411yescharacteristic
82C8×C269—22no—
83C8×C269C2×D411yescharacteristic
84C8×C269C2×D411yescharacteristic
85C8×C269C2×D411yescharacteristic
86C8×C269C2×D411yescharacteristic
87C22×C269—22no—
88C22×C269H(5,𝔽2)11yescharacteristic
89C2×D269—22no—
90C2×D269—44no—
91C2×D269—84no—
92C2×D269—164no—
93C4×C269—22no—
94C4×C269—22no—
95C4×C269C22×D411yescharacteristic
96C4×C269C22×D411yescharacteristic
97Dic269—22no—
98C2×C269—21no—
99C2×C269—42no—
100C2×C269D4211yescharacteristic
101D269—42no—
102D269—84no—
103C269#329211yescharacteristic, complemented, Sylow
104#3292—2691nomaximal, Sylow
105#576—2692no—
106#576—2692no—
107#493—2691no—
108#472—2692no—
109#472—2692no—
110#453—2694no—
111#443—2691no—
112#443D26911yescharacteristic, complemented
113H(5,𝔽2)—2694no—
114AGL(1,ℤ/8)—2694no—
115AGL(1,ℤ/8)—2694no—
116AGL(1,ℤ/8)—5384no—
117AGL(1,ℤ/8)—5384no—
118#135—2691no—
119#135—5382no—
120#135—5382no—
121#135C2×D26911yescharacteristic
122C2×D8—2692no—
123C2×D8—2692no—
124C2×D8—5382no—
125C2×D8—5382no—
126#131—2692no—
127#131C2×D26912yescomplemented
128#127—2692no—
129#108—2691no—
130#108C2×D26911yescharacteristic, complemented
131#104—2692no—
132#104C2×D26912yescomplemented
133#100—2694no—
134C4×C8C2×D26911yescharacteristic, complemented
135#41—2692no—
136#41—5384no—
137#41—5384no—
138#41C22×D26912yes—
139C2×D4—2694no—
140C2×D4—5384no—
141C2×D4—5384no—
142C2×D4—10762no—
143C2×D4—10764no—
144Q16—2691no—
145Q16D107611yescharacteristic, complemented
146SD16—2692no—
147SD16—5384no—
148SD16—5384no—
149SD16D107612yescomplemented
150D8—2691no—
151D8—5384no—
152D8—5384no—
153D8—5382no—
154D8—5382no—
155D8—5382no—
156D8—5382no—
157D8D107611yescharacteristic, complemented
158M16—22no—
159M16—2692no—
160M16—5382no—
161M16C22×D26912yes—
162C2×C8—22no—
163C2×C8—5382no—
164C2×C8C22×D26911yescharacteristic
165C2×C8C22×D26911yescharacteristic
166C42C22×D26911yescharacteristic
167C23—5384no—
168C23—10764no—
169Q8—2692no—
170Q8C2×D107612yes—
171D4—22no—
172D4—2692no—
173D4—5382no—
174D4—5384no—
175D4—5384no—
176D4—10764no—
177D4—10764no—
178D4—21524no—
179D4C2×D107612yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—5382no—
183C2×C4—10764no—
184C2×C4C23×D26911yescharacteristic
185C8—22no—
186C8—5382no—
187C8D4×D26911yescharacteristic
188C8D4×D26911yescharacteristic
189C8C2×D107611yescharacteristic
190C8C2×D107611yescharacteristic
191C22—22no—
192C22—5382no—
193C22—10764no—
194C22—21524no—
195C22—43044no—
196C22#2257549911yescharacteristic
197C4—22no—
198C4—22no—
199C4—5382no—
200C4C2×D4×D26911yescharacteristic
201C4C22×D107611yescharacteristic
202C2—21no—
203C2—42no—
204C2—10762no—
205C2—21524no—
206C2D4×D107611yescharacteristic
subgroupG/H
centerC2D4×D1076
commutator, cosocleC2×C4×C269C24
FrattiniC2×C4C23×D269
Fitting#443×C269C2
socleC2×C269D42
hypercenter#443D269
hypocenterC269#3292
LCSG ▷ C2×C4×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #31771410C2semidirect
2. #31771407C2semidirect
3. #31770982C2semidirect
4. #31770876C2semidirect
5. #31770875C2semidirect
6. #31770774C2semidirect
7. #31770723C2semidirect
8. #443×C269C2semidirect
9. #22575499C22semidirect
11. #22575481C22semidirect
13. #22575478C22semidirect
18. C2×D2152C22semidirect
19. C2×D2152C22semidirect
21. #22575459C22semidirect
22. #22575450C22semidirect
23. #22575389C22semidirect
24. #22575381C22semidirect
25. #22575369C22semidirect
27. #131×C269C22semidirect
28. #108×C269C22semidirect
29. #104×C269C22semidirect
30. C4×C8×C269C22semidirect
39. Dic1076D4semidirect
41. #16864391D4semidirect
48. D2152D4semidirect
54. Q16×C269D4semidirect
55. SD16×C269D4semidirect
56. D8×C269D4semidirect
103. C269#3292semidirect
112. #443D269semidirect
127. #131C2×D269semidirect
130. #108C2×D269semidirect
132. #104C2×D269semidirect
134. C4×C8C2×D269semidirect
145. Q16D1076semidirect
149. SD16D1076semidirect
157. D8D1076semidirect

Maximal subgroups

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

subgroupindex
#317714102
#317714072
#317709822
#317708762
#317708752
#317707742
#317707232
#443×C2692
#3292269