Finite Group Databasebeta

← #46848164

Subgroups — #46848164

Order 34688.

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#31774702C212yesmaximal, complemented
2#31774699C212yesmaximal, complemented
3#31774274C211yesmaximal, characteristic, complemented
4#31774168C212yesmaximal, complemented
5#31774167C212yesmaximal, complemented
6#31774066C214yesmaximal, complemented
7#31774015C211yesmaximal, characteristic, complemented
8#443×C271C211yesmaximal, characteristic, complemented
9#22576206C2214yescomplemented
10#22576188—24no—
11#22576188C2214yescomplemented
12#22576187—24no—
13#22576185C2214yescomplemented
14#22576183—22no—
15#22576182—22no—
16#22576182C2211yescharacteristic
17C2×D2168—22no—
18C2×D2168C2212yescomplemented
19C2×D2168C2212yescomplemented
20D8×D271—22no—
21#22576166C2212yescomplemented
22#22576157C2212yescomplemented
23#22576096C2211yescharacteristic, complemented
24#22576088C2212yescomplemented
25#22576076C2214yescomplemented
26#135×C271C2211yescharacteristic
27#131×C271C2212yescomplemented
28#108×C271C2211yescharacteristic, complemented
29#104×C271C2212yescomplemented
30C4×C8×C271C2211yescharacteristic, complemented
31#16864655—24no—
32#16864654—24no—
33#16864654C2312yes—
34C2×D1084—24no—
35C2×D1084—42no—
36C2×D1084C2314yes—
37D4×D271—24no—
38D4×D271—44no—
39Dic1084D411yescharacteristic, complemented
40#16864643—24no—
41#16864643D412yescomplemented
42#16864641—24no—
43D2168—24no—
44D2168—22no—
45D2168—22no—
46D2168—22no—
47D2168—22no—
48D2168D411yescharacteristic, complemented
49#16864639—24no—
50#16864638C2312yes—
51#16864637—22no—
52C8×D271—22no—
53#41×C271C2312yes—
54Q16×C271D411yescharacteristic, complemented
55SD16×C271D412yescomplemented
56D8×C271D411yescharacteristic, complemented
57M16×C271—22no—
58M16×C271C2312yes—
59C2×C8×C271—22no—
60C2×C8×C271C2311yescharacteristic
61C2×C8×C271C2311yescharacteristic
62C42×C271C2311yescharacteristic
63C22×D271—24no—
64C22×D271—44no—
65Dic542C2×D412yes—
66D1084—24no—
67D1084—24no—
68D1084—44no—
69D1084—84no—
70D1084C2×D412yes—
71C22⋊D271—22no—
72C22⋊D271—44no—
73C4×D271—22no—
74C4×D271—44no—
75C271⋊C8—22no—
76Q8×C271C2×D412yes—
77D4×C271—22no—
78D4×C271C2×D412yes—
79C2×C4×C271—21no—
80C2×C4×C271—22no—
81C2×C4×C271C2411yescharacteristic
82C8×C271—22no—
83C8×C271C2×D411yescharacteristic
84C8×C271C2×D411yescharacteristic
85C8×C271C2×D411yescharacteristic
86C8×C271C2×D411yescharacteristic
87C22×C271—22no—
88C22×C271H(5,𝔽2)11yescharacteristic
89C2×D271—22no—
90C2×D271—44no—
91C2×D271—84no—
92C2×D271—164no—
93C4×C271—22no—
94C4×C271—22no—
95C4×C271C22×D411yescharacteristic
96C4×C271C22×D411yescharacteristic
97Dic271—22no—
98C2×C271—21no—
99C2×C271—42no—
100C2×C271D4211yescharacteristic
101D271—42no—
102D271—84no—
103C271#329211yescharacteristic, complemented, Sylow
104#3292—2711nomaximal, Sylow
105#576—2712no—
106#576—2712no—
107#493—2711no—
108#472—2712no—
109#472—2712no—
110#453—2714no—
111#443—2711no—
112#443D27111yescharacteristic, complemented
113H(5,𝔽2)—2714no—
114AGL(1,ℤ/8)—2714no—
115AGL(1,ℤ/8)—2714no—
116AGL(1,ℤ/8)—5424no—
117AGL(1,ℤ/8)—5424no—
118#135—2711no—
119#135—5422no—
120#135—5422no—
121#135C2×D27111yescharacteristic
122C2×D8—2712no—
123C2×D8—2712no—
124C2×D8—5422no—
125C2×D8—5422no—
126#131—2712no—
127#131C2×D27112yescomplemented
128#127—2712no—
129#108—2711no—
130#108C2×D27111yescharacteristic, complemented
131#104—2712no—
132#104C2×D27112yescomplemented
133#100—2714no—
134C4×C8C2×D27111yescharacteristic, complemented
135#41—2712no—
136#41—5424no—
137#41—5424no—
138#41C22×D27112yes—
139C2×D4—2714no—
140C2×D4—5424no—
141C2×D4—5424no—
142C2×D4—10842no—
143C2×D4—10844no—
144Q16—2711no—
145Q16D108411yescharacteristic, complemented
146SD16—2712no—
147SD16—5424no—
148SD16—5424no—
149SD16D108412yescomplemented
150D8—2711no—
151D8—5424no—
152D8—5424no—
153D8—5422no—
154D8—5422no—
155D8—5422no—
156D8—5422no—
157D8D108411yescharacteristic, complemented
158M16—22no—
159M16—2712no—
160M16—5422no—
161M16C22×D27112yes—
162C2×C8—22no—
163C2×C8—5422no—
164C2×C8C22×D27111yescharacteristic
165C2×C8C22×D27111yescharacteristic
166C42C22×D27111yescharacteristic
167C23—5424no—
168C23—10844no—
169Q8—2712no—
170Q8C2×D108412yes—
171D4—22no—
172D4—2712no—
173D4—5422no—
174D4—5424no—
175D4—5424no—
176D4—10844no—
177D4—10844no—
178D4—21684no—
179D4C2×D108412yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—5422no—
183C2×C4—10844no—
184C2×C4C23×D27111yescharacteristic
185C8—22no—
186C8—5422no—
187C8D4×D27111yescharacteristic
188C8D4×D27111yescharacteristic
189C8C2×D108411yescharacteristic
190C8C2×D108411yescharacteristic
191C22—22no—
192C22—5422no—
193C22—10844no—
194C22—21684no—
195C22—43364no—
196C22#2257620611yescharacteristic
197C4—22no—
198C4—22no—
199C4—5422no—
200C4C2×D4×D27111yescharacteristic
201C4C22×D108411yescharacteristic
202C2—21no—
203C2—42no—
204C2—10842no—
205C2—21684no—
206C2D4×D108411yescharacteristic
subgroupG/H
centerC2D4×D1084
commutator, cosocleC2×C4×C271C24
FrattiniC2×C4C23×D271
Fitting#443×C271C2
socleC2×C271D42
hypercenter#443D271
hypocenterC271#3292
LCSG ▷ C2×C4×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #31774702C2semidirect
2. #31774699C2semidirect
3. #31774274C2semidirect
4. #31774168C2semidirect
5. #31774167C2semidirect
6. #31774066C2semidirect
7. #31774015C2semidirect
8. #443×C271C2semidirect
9. #22576206C22semidirect
11. #22576188C22semidirect
13. #22576185C22semidirect
18. C2×D2168C22semidirect
19. C2×D2168C22semidirect
21. #22576166C22semidirect
22. #22576157C22semidirect
23. #22576096C22semidirect
24. #22576088C22semidirect
25. #22576076C22semidirect
27. #131×C271C22semidirect
28. #108×C271C22semidirect
29. #104×C271C22semidirect
30. C4×C8×C271C22semidirect
39. Dic1084D4semidirect
41. #16864643D4semidirect
48. D2168D4semidirect
54. Q16×C271D4semidirect
55. SD16×C271D4semidirect
56. D8×C271D4semidirect
103. C271#3292semidirect
112. #443D271semidirect
127. #131C2×D271semidirect
130. #108C2×D271semidirect
132. #104C2×D271semidirect
134. C4×C8C2×D271semidirect
145. Q16D1084semidirect
149. SD16D1084semidirect
157. D8D1084semidirect

Maximal subgroups

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

subgroupindex
#317747022
#317746992
#317742742
#317741682
#317741672
#317740662
#317740152
#443×C2712
#3292271