Finite Group Databasebeta

← #20284953

Subgroups — #20284953

Order 6784.

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

normal subgroups only

This group has 418 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#16811097C211yesmaximal, characteristic, complemented
2#16810944C212yesmaximal, complemented
3#16810920C212yesmaximal, complemented
4#16810815C212yesmaximal, complemented
5#16810756C212yesmaximal, complemented
6#16810733C212yesmaximal, complemented
7C4×#11719922C212yesmaximal, complemented
8#16810684C211yesmaximal, characteristic, complemented
9C53×#540C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C2×#11719922C2212yes—
12C2×#11719922C2212yes—
13C4×D212C2212yes—
14C4×D212C2212yes—
15#13133179—22no—
16#13133176C2211yescharacteristic
17C22⋊C4×D53—22no—
18#13133150C2212yescomplemented
19#13133148C2212yescomplemented
20C4⋊#88322C2212yescomplemented
21#13133138C2212yescomplemented
22#13133138C2212yescomplemented
23#13133137C2212yes—
24#13133137C2212yes—
25C22⋊#88322C2212yes—
26#13133128C2211yescharacteristic, complemented
27C4×#88322C2211yescharacteristic, complemented
28#127×C53C2211yescharacteristic, complemented
29#125×C53C2211yescharacteristic, complemented
30#123×C53—22no—
31#121×C53—22no—
32#121×C53C2212yes—
33C4×D4×C53C2212yescomplemented
34#117×C53C2211yescharacteristic
35C23×D53—21no—
36C2×D4×C53—22no—
37C2×D4×C53—22no—
38C2×D4×C53C2312yes—
39C22×C4×C53—22no—
40C22×C4×C53C2311yescharacteristic
41C2×D212—21no—
42C2×D212—24no—
43C2×D212C2312yes—
44C2×C4×D53—22no—
45C2×C4×D53—22no—
46C4⋊C4×C53—22no—
47C4⋊C4×C53D412yes—
48C4⋊C4×C53C2312yes—
49C22⋊C4×C53—22no—
50C22⋊C4×C53—22no—
51C22⋊C4×C53D412yes—
52C42×C53C2311yescharacteristic
53C42×C53C2311yescharacteristic
54C22⋊Dic53—21no—
55#11719922—24no—
56#11719922—24no—
57#11719921—22no—
58#11719921—22no—
59C4⋊Dic53C2312yes—
60C4×Dic53—21no—
61C2×#88322C2312yes—
62C2×#88322C2312yes—
63C23×C53—22no—
64C23×C53C2×D411yescharacteristic
65C22×D53—22no—
66C22×D53—24no—
67C22×D53—82no—
68D4×C53—22no—
69D4×C53—22no—
70D4×C53—82no—
71D4×C53—82no—
72C2×C4×C53—22no—
73C2×C4×C53—22no—
74C2×C4×C53—82no—
75C2×C4×C53C2×D412yes—
76C2×C4×C53C2×D412yes—
77C2×C4×C53C2×D411yescharacteristic
78C2×C4×C53C2411yescharacteristic
79C2×Dic53—22no—
80D212—24no—
81D212—22no—
82D212#4114yes—
83C4×D53—82no—
84C4×D53—82no—
85#88322—22no—
86#88322—22no—
87C22×C53—21no—
88C22×C53—22no—
89C22×C53—42no—
90C22×C53—42no—
91C22×C53—42no—
92C22×C53C22×D411yescharacteristic
93C2×D53—24no—
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—84no—
97C4×C53—21no—
98C4×C53—22no—
99C4×C53—22no—
100C4×C53—42no—
101C4×C53C2×#4112yes—
102Dic53—42no—
103#3122—531nomaximal, Sylow
104C2×C53—41no—
105C2×C53—82no—
106C2×C53#52511yescharacteristic
107C2×C53#57612yes—
108D53—44no—
109#540D5311yescharacteristic, complemented
110#518—531no—
111#482—532no—
112#478—532no—
113#459—532no—
114#447—532no—
115#442—532no—
116C4×D8—532no—
117#433—531no—
118C53#312211yescharacteristic, complemented, Sylow
119C22×D4—531no—
120C2×D8—532no—
121C2×D8—532no—
122#127C2×D5311yescharacteristic, complemented
123#125C2×D5311yescharacteristic, complemented
124#123—22no—
125#121—22no—
126#121C2×D5312yes—
127C4×D4—532no—
128C4×D4—532no—
129C4×D4—1062no—
130C4×D4C2×D5312yescomplemented
131#117—531no—
132#117C2×D5311yescharacteristic
133C2×C22⋊C4—1062no—
134#107—532no—
135#106—532no—
136C4⋊C8—532no—
137#102—532no—
138#102—532no—
139#102—532no—
140#102—532no—
141C22⋊C8—532no—
142#97—531no—
143C4×C8—531no—
144C24—1061no—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—532no—
148C2×D4—1064no—
149C2×D4—1061no—
150C2×D4C22×D5312yes—
151C22×C4—22no—
152C22×C4—1062no—
153C22×C4—1062no—
154C22×C4C22×D5311yescharacteristic
155D8—1064no—
156D8—1064no—
157C2×C8—532no—
158C2×C8—532no—
159C4⋊C4—22no—
160C4⋊C4—532no—
161C4⋊C4C22⋊D5312yes—
162C4⋊C4C22×D5312yes—
163C22⋊C4—22no—
164C22⋊C4—22no—
165C22⋊C4—1061no—
166C22⋊C4—1062no—
167C22⋊C4—1062no—
168C22⋊C4C22⋊D5312yes—
169C42—1061no—
170C42C22×D5311yescharacteristic
171C42C22×D5311yescharacteristic
172C23—22no—
173C23—1064no—
174C23—1062no—
175C23—4242no—
176C23C2×C22⋊D5311yescharacteristic
177D4—22no—
178D4—22no—
179D4—82no—
180D4—82no—
181D4—534no—
182D4—1064no—
183D4—1062no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—1062no—
188C2×C4—4242no—
189C2×C4—4242no—
190C2×C4C2×C22⋊D5311yescharacteristic
191C2×C4C2×C22⋊D5312yes—
192C2×C4C2×C22⋊D5312yes—
193C2×C4C23×D5311yescharacteristic
194C8—1062no—
195C8—1062no—
196C22—21no—
197C22—22no—
198C22—42no—
199C22—42no—
200C22—42no—
201C22—1064no—
202C22—4244no—
203C22—2122no—
204C22—2122no—
205C22C22×C22⋊D5311yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—2122no—
211C4#41×D5312yes—
212C2—41no—
213C2—82no—
214C2—2124no—
215C2#1681113011yescharacteristic
216C2#1681142612yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#540C2
socleC22×C53C22×D4
hypercenter#540D53
hypocenterC53#3122
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #16811097C2semidirect
2. #16810944C2semidirect
3. #16810920C2semidirect
4. #16810815C2semidirect
5. #16810756C2semidirect
6. #16810733C2semidirect
7. C4×#11719922C2semidirect
8. #16810684C2semidirect
9. C53×#540C2semidirect
18. #13133150C22semidirect
19. #13133148C22semidirect
20. C4⋊#88322C22semidirect
21. #13133138C22semidirect
22. #13133138C22semidirect
26. #13133128C22semidirect
27. C4×#88322C22semidirect
28. #127×C53C22semidirect
29. #125×C53C22semidirect
33. C4×D4×C53C22semidirect
109. #540D53semidirect
118. C53#3122semidirect
122. #127C2×D53semidirect
123. #125C2×D53semidirect
130. C4×D4C2×D53semidirect