Finite Group Databasebeta

← #20285088

Subgroups — #20285088

Order 6784.

484 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 121 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#16811179C211yesmaximal, characteristic, complemented
2#127×D53C211yesmaximal, characteristic
3#16810989C211yesmaximal, characteristic, complemented
4#16810988C211yesmaximal, characteristic, complemented
5#16810987C211yesmaximal, characteristic, complemented
6#16810968C214yesmaximal, complemented
7#16810626C214yesmaximal, complemented
8#16810575C211yesmaximal, characteristic, complemented
9C53×#490C211yesmaximal, characteristic, complemented
10C2×D4×D53—22no—
11C2×D4×D53C2212yes—
12C2×#11719912C2212yes—
13#13133216C2212yescomplemented
14#13133214C2211yescharacteristic
15#13133213—22no—
16#13133213C2212yes—
17#13133209C2211yescharacteristic, complemented
18#13133195—22no—
19#13133182—22no—
20#13133175C2212yes—
21C42×D53C2211yescharacteristic
22#13133139C2214yescomplemented
23#13133138C2214yescomplemented
24#13133137C2214yescomplemented
25#13133128C2211yescharacteristic, complemented
26C8×Dic53C2212yescomplemented
27#127×C53C2211yescharacteristic
28#125×C53C2211yescharacteristic, complemented
29#102×C53C2214yescomplemented
30#97×C53C2211yescharacteristic, complemented
31C23×D53—22no—
32C2×D4×C53—22no—
33C2×D4×C53C2312yes—
34C2×C22⋊D53—22no—
35C2×C22⋊D53—24no—
36D4×D53—28no—
37D4×D53—48no—
38C2×D212—22no—
39C2×D212C2312yes—
40C2×C4×D53D412yes—
41C2×C4×D53C2311yescharacteristic
42C2×Dic106—22no—
43C2×C8×C53C2312yes—
44C4⋊C4×C53—22no—
45C4⋊C4×C53C2312yes—
46C42×C53C2311yescharacteristic
47#11719921—22no—
48C4⋊Dic53C2312yes—
49#11719919—22no—
50#11719919—22no—
51C4×Dic53—22no—
52C4×Dic53D412yes—
53C4×Dic53C2311yescharacteristic
54C2×#88322C2312yes—
55#11719912—24no—
56C23×C53—22no—
57C22×D53—22no—
58C22×D53—48no—
59C22×D53—44no—
60C22×D53C2×D411yescharacteristic
61D4×C53—24no—
62D4×C53—44no—
63C2×C4×C53—22no—
64C2×C4×C53C2×D412yes—
65C2×C4×C53C2411yescharacteristic
66C22⋊D53—44no—
67C22⋊D53—48no—
68C2×Dic53—22no—
69C2×Dic53C2×D411yescharacteristic
70C2×Dic53C2×D412yes—
71D212—24no—
72D212—44no—
73C4×D53—24no—
74C4×D53#4114yes—
75Dic106—44no—
76C8×C53—22no—
77#88322—22no—
78C22×C53—42no—
79C22×C53—44no—
80C22×C53C22×D411yescharacteristic
81C2×D53—22no—
82C2×D53—42no—
83C2×D53—44no—
84C2×D53—88no—
85C2×D53#12412yes—
86C4×C53—22no—
87C4×C53—42no—
88C4×C53C2×#4112yes—
89Dic53—42no—
90Dic53#12412yes—
91Dic53AGL(1,ℤ/8)14yes—
92#3138—531nomaximal, Sylow
93C2×C53—82no—
94C2×C53C2×#12411yescharacteristic
95C2×C53C2×AGL(1,ℤ/8)12yes—
96D53—22no—
97D53—82no—
98#533—531no—
99C2×#127—531no—
100#490—531no—
101#490—531no—
102#490—531no—
103#490D5311yescharacteristic, complemented
104#486—534no—
105#418—534no—
106C4×M16—531no—
107C53#313811yescharacteristic, complemented, Sylow
108C22×D4—532no—
109C22×D4—1062no—
110C2×M16—532no—
111C4⋊Q8—532no—
112#127—532no—
113#127—531no—
114#127—1062no—
115#127C2×D5311yescharacteristic
116#125—531no—
117#125C2×D5311yescharacteristic, complemented
118C22⋊Q8—1062no—
119C4×Q8—1062no—
120#117—532no—
121C2×C42—531no—
122#102—534no—
123#102—534no—
124#102—534no—
125#102C2×D5314yescomplemented
126#97—531no—
127#97C2×D5311yescharacteristic, complemented
128C4×C8—532no—
129C24—1062no—
130C2×Q8—1062no—
131C2×D4—22no—
132C2×D4—532no—
133C2×D4—1068no—
134C2×D4—1064no—
135C2×D4—1062no—
136C2×D4—1062no—
137C2×D4—2128no—
138C2×D4C22×D5312yes—
139C22×C4—531no—
140C22×C4—532no—
141M16—1064no—
142C2×C8—532no—
143C2×C8C22×D5312yes—
144C4⋊C4—22no—
145C4⋊C4—532no—
146C4⋊C4—1062no—
147C4⋊C4—1062no—
148C4⋊C4C22×D5312yes—
149C22⋊C4—1062no—
150C42—532no—
151C42—531no—
152C42—1062no—
153C42C22×D5311yescharacteristic
154C23—22no—
155C23—531no—
156C23—1062no—
157C23—2124no—
158C23—2128no—
159Q8—2124no—
160D4—24no—
161D4—44no—
162D4—1064no—
163D4—2124no—
164D4—2124no—
165D4—2128no—
166C2×C4—22no—
167C2×C4—531no—
168C2×C4—532no—
169C2×C4—534no—
170C2×C4—1064no—
171C2×C4—1062no—
172C2×C4D4×D5312yes—
173C2×C4C23×D5311yescharacteristic
174C8—22no—
175C8—1062no—
176C22—42no—
177C22—44no—
178C22—532no—
179C22—1062no—
180C22—4248no—
181C22—2122no—
182C22—2124no—
183C22C2×D4×D5311yescharacteristic
184C4—22no—
185C4—42no—
186C4—532no—
187C4—534no—
188C4—2122no—
189C4#41×D5312yes—
190C2—82no—
191C2—1062no—
192C2—4242no—
193C2#124×D5311yescharacteristic
194C2AGL(1,ℤ/8)×D5312yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#490C2
socleC22×C53C22×D4
hypercenter#490D53
hypocenterC53#3138
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #16811179C2semidirect
3. #16810989C2semidirect
4. #16810988C2semidirect
5. #16810987C2semidirect
6. #16810968C2semidirect
7. #16810626C2semidirect
8. #16810575C2semidirect
9. C53×#490C2semidirect
13. #13133216C22semidirect
17. #13133209C22semidirect
22. #13133139C22semidirect
23. #13133138C22semidirect
24. #13133137C22semidirect
25. #13133128C22semidirect
26. C8×Dic53C22semidirect
28. #125×C53C22semidirect
29. #102×C53C22semidirect
30. #97×C53C22semidirect
103. #490D53semidirect
107. C53#3138semidirect
117. #125C2×D53semidirect
125. #102C2×D53semidirect
127. #97C2×D53semidirect