Finite Group Databasebeta

← #20286515

Subgroups — #20286515

Order 6784.

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#16811428C212yesmaximal, complemented
2#16811425C212yesmaximal, complemented
3#16811000C211yesmaximal, characteristic, complemented
4#16810894C212yesmaximal, complemented
5#16810893C212yesmaximal, complemented
6#16810792C214yesmaximal, complemented
7#16810741C211yesmaximal, characteristic, complemented
8C53×#443C211yesmaximal, characteristic, complemented
9#13133263C2214yescomplemented
10#13133245—24no—
11#13133245C2214yescomplemented
12#13133244—24no—
13#13133242C2214yescomplemented
14#13133240—22no—
15#13133239—22no—
16#13133239C2211yescharacteristic
17C2×D424—22no—
18C2×D424C2212yescomplemented
19C2×D424C2212yescomplemented
20D8×D53—22no—
21#13133223C2212yescomplemented
22#13133214C2212yescomplemented
23#13133153C2211yescharacteristic, complemented
24#13133145C2212yescomplemented
25#13133133C2214yescomplemented
26#135×C53C2211yescharacteristic
27#131×C53C2212yescomplemented
28#108×C53C2211yescharacteristic, complemented
29#104×C53C2212yescomplemented
30C4×C8×C53C2211yescharacteristic, complemented
31#41×C53C2312yes—
32#11719949—24no—
33D4×D53—24no—
34D4×D53—44no—
35#11719945—24no—
36#11719945C2312yes—
37C2×D212—24no—
38C2×D212—42no—
39C2×D212C2314yes—
40Q16×C53D411yescharacteristic, complemented
41SD16×C53D412yescomplemented
42D8×C53D411yescharacteristic, complemented
43M16×C53—22no—
44M16×C53C2312yes—
45C2×C8×C53—22no—
46C2×C8×C53C2311yescharacteristic
47C2×C8×C53C2311yescharacteristic
48C42×C53C2311yescharacteristic
49#11719924—24no—
50#11719922—24no—
51#11719917C2312yes—
52Dic212D411yescharacteristic, complemented
53D424—24no—
54D424—22no—
55D424—22no—
56D424—22no—
57D424—22no—
58D424D411yescharacteristic, complemented
59#11719913—24no—
60#11719913D412yescomplemented
61#11719912—22no—
62C8×D53—22no—
63C22×D53—24no—
64C22×D53—44no—
65Q8×C53C2×D412yes—
66D4×C53—22no—
67D4×C53C2×D412yes—
68C2×C4×C53—21no—
69C2×C4×C53—22no—
70C2×C4×C53C2411yescharacteristic
71C22⋊D53—22no—
72C22⋊D53—44no—
73D212—24no—
74D212—24no—
75D212—44no—
76D212—84no—
77D212C2×D412yes—
78C4×D53—22no—
79C4×D53—44no—
80Dic106C2×D412yes—
81C8×C53—22no—
82C8×C53C2×D411yescharacteristic
83C8×C53C2×D411yescharacteristic
84C8×C53C2×D411yescharacteristic
85C8×C53C2×D411yescharacteristic
86#88322—22no—
87C22×C53—22no—
88C22×C53H(5,𝔽2)11yescharacteristic
89C2×D53—22no—
90C2×D53—44no—
91C2×D53—84no—
92C2×D53—164no—
93C4×C53—22no—
94C4×C53—22no—
95C4×C53C22×D411yescharacteristic
96C4×C53C22×D411yescharacteristic
97Dic53—22no—
98#3292—531nomaximal, Sylow
99C2×C53—21no—
100C2×C53—42no—
101C2×C53D4211yescharacteristic
102D53—42no—
103D53—84no—
104#576—532no—
105#576—532no—
106#493—531no—
107#472—532no—
108#472—532no—
109#453—534no—
110#443—531no—
111#443D5311yescharacteristic, complemented
112C53#329211yescharacteristic, complemented, Sylow
113H(5,𝔽2)—534no—
114AGL(1,ℤ/8)—534no—
115AGL(1,ℤ/8)—534no—
116AGL(1,ℤ/8)—1064no—
117AGL(1,ℤ/8)—1064no—
118#135—531no—
119#135—1062no—
120#135—1062no—
121#135C2×D5311yescharacteristic
122C2×D8—532no—
123C2×D8—532no—
124C2×D8—1062no—
125C2×D8—1062no—
126#131—532no—
127#131C2×D5312yescomplemented
128#127—532no—
129#108—531no—
130#108C2×D5311yescharacteristic, complemented
131#104—532no—
132#104C2×D5312yescomplemented
133#100—534no—
134C4×C8C2×D5311yescharacteristic, complemented
135#41—532no—
136#41—1064no—
137#41—1064no—
138#41C22×D5312yes—
139C2×D4—534no—
140C2×D4—1064no—
141C2×D4—1064no—
142C2×D4—2122no—
143C2×D4—2124no—
144Q16—531no—
145Q16D21211yescharacteristic, complemented
146SD16—532no—
147SD16—1064no—
148SD16—1064no—
149SD16D21212yescomplemented
150D8—531no—
151D8—1064no—
152D8—1064no—
153D8—1062no—
154D8—1062no—
155D8—1062no—
156D8—1062no—
157D8D21211yescharacteristic, complemented
158M16—22no—
159M16—532no—
160M16—1062no—
161M16C22×D5312yes—
162C2×C8—22no—
163C2×C8—1062no—
164C2×C8C22×D5311yescharacteristic
165C2×C8C22×D5311yescharacteristic
166C42C22×D5311yescharacteristic
167C23—1064no—
168C23—2124no—
169Q8—532no—
170Q8C2×D21212yes—
171D4—22no—
172D4—532no—
173D4—1062no—
174D4—1064no—
175D4—1064no—
176D4—4244no—
177D4—2124no—
178D4—2124no—
179D4C2×D21212yes—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—1062no—
183C2×C4—2124no—
184C2×C4C23×D5311yescharacteristic
185C8—22no—
186C8—1062no—
187C8C2×D21211yescharacteristic
188C8C2×D21211yescharacteristic
189C8D4×D5311yescharacteristic
190C8D4×D5311yescharacteristic
191C22—22no—
192C22—8484no—
193C22—1062no—
194C22—4244no—
195C22—2124no—
196C22#1313326311yescharacteristic
197C4—22no—
198C4—22no—
199C4—1062no—
200C4C2×D4×D5311yescharacteristic
201C4C22×D21211yescharacteristic
202C2—21no—
203C2—42no—
204C2—4244no—
205C2—2122no—
206C2D4×D21211yescharacteristic
subgroupG/H
centerC2D4×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#443C2
socleC2×C53D42
hypercenter#443D53
hypocenterC53#3292
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×C4 ◁ #443

Complements

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

NH ≅ G/Ntype
1. #16811428C2semidirect
2. #16811425C2semidirect
3. #16811000C2semidirect
4. #16810894C2semidirect
5. #16810893C2semidirect
6. #16810792C2semidirect
7. #16810741C2semidirect
8. C53×#443C2semidirect
9. #13133263C22semidirect
11. #13133245C22semidirect
13. #13133242C22semidirect
18. C2×D424C22semidirect
19. C2×D424C22semidirect
21. #13133223C22semidirect
22. #13133214C22semidirect
23. #13133153C22semidirect
24. #13133145C22semidirect
25. #13133133C22semidirect
27. #131×C53C22semidirect
28. #108×C53C22semidirect
29. #104×C53C22semidirect
30. C4×C8×C53C22semidirect
40. Q16×C53D4semidirect
41. SD16×C53D4semidirect
42. D8×C53D4semidirect
52. Dic212D4semidirect
58. D424D4semidirect
60. #11719913D4semidirect
111. #443D53semidirect
112. C53#3292semidirect
127. #131C2×D53semidirect
130. #108C2×D53semidirect
132. #104C2×D53semidirect
134. C4×C8C2×D53semidirect
145. Q16D212semidirect
149. SD16D212semidirect
157. D8D212semidirect