Finite Group Databasebeta

← #20284967

Subgroups — #20284967

Order 6784.

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

normal subgroups only

This group has 394 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#16811235C211yesmaximal, characteristic, complemented
2#16810964C212yesmaximal, complemented
3#16810923C212yesmaximal, complemented
4#16810828C212yesmaximal, complemented
5#16810774C212yesmaximal
6#16810723C212yesmaximal, complemented
7C4×#11719923C212yesmaximal, complemented
8#16810684C211yesmaximal, characteristic, complemented
9C53×#518C211yesmaximal, characteristic
10C2×#11719923C2212yes—
11C2×#11719923C2212yes—
12#13133217C2211yescharacteristic, complemented
13#13133210—22no—
14#13133209C2211yescharacteristic, complemented
15#13133207—22no—
16#13133196C2212yes—
17C4×Dic106C2212yescomplemented
18#13133150C2212yescomplemented
19#13133148C2212yescomplemented
20C4⋊#88322C2212yescomplemented
21#13133140C2212yescomplemented
22#13133140C2212yescomplemented
23#13133138C2212yes—
24#13133138C2212yes—
25C22⋊#88322C2212yes—
26#13133128C2211yescharacteristic, complemented
27C4×#88322C2211yescharacteristic, complemented
28C22×D4×C53C2211yescharacteristic
29C4×D4×C53—22no—
30C4×D4×C53C2212yes—
31C4×D4×C53C2212yes—
32#117×C53C2211yescharacteristic
33#117×C53C2211yescharacteristic
34C2×C22⋊C4×C53—22no—
35C24×C53—21no—
36C2×D4×C53—24no—
37C2×D4×C53—21no—
38C2×D4×C53C2312yes—
39C22×C4×C53—22no—
40C22×C4×C53—22no—
41C22×C4×C53C2311yescharacteristic
42C2×Dic106C2312yes—
43C4⋊C4×C53D412yes—
44C4⋊C4×C53C2312yes—
45C22⋊C4×C53—21no—
46C22⋊C4×C53—22no—
47C22⋊C4×C53—22no—
48C22⋊C4×C53D412yes—
49C42×C53—21no—
50C42×C53C2311yescharacteristic
51C42×C53C2311yescharacteristic
52C22⋊Dic53—22no—
53#11719923—24no—
54#11719923—24no—
55C4⋊Dic53—22no—
56C4⋊Dic53C2312yes—
57#11719919—22no—
58#11719919—22no—
59#11719919—22no—
60C4×Dic53—22no—
61C2×#88322C2312yes—
62C2×#88322C2312yes—
63C23×C53—24no—
64C23×C53—22no—
65C23×C53—82no—
66C23×C53C2×D411yescharacteristic
67D4×C53—24no—
68D4×C53—22no—
69D4×C53#4114yes—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53—82no—
73C2×C4×C53—82no—
74C2×C4×C53C2×D412yes—
75C2×C4×C53C2×D412yes—
76C2×C4×C53C2×D411yescharacteristic
77C2×C4×C53C2411yescharacteristic
78C2×Dic53—22no—
79C2×Dic53—22no—
80Dic106—22no—
81Dic106—22no—
82#88322—22no—
83#88322—22no—
84C22×C53—21no—
85C22×C53—24no—
86C22×C53—22no—
87C22×C53—42no—
88C22×C53—42no—
89C22×C53—84no—
90C22×C53C22×D411yescharacteristic
91C4×C53—21no—
92C4×C53—22no—
93C4×C53—22no—
94C4×C53—42no—
95C4×C53C2×#4112yes—
96Dic53—42no—
97Dic53—42no—
98#3123—531nomaximal, Sylow
99C2×C53—41no—
100C2×C53—44no—
101C2×C53#52511yescharacteristic
102C2×C53#57712yes—
103#541—531no—
104#518D5311yescharacteristic
105#485—532no—
106#478—532no—
107#461—532no—
108#450—532no—
109#440—532no—
110C4×SD16—532no—
111#433—531no—
112C53#312311yescharacteristic, complemented, Sylow
113C22×D4C2×D5311yescharacteristic
114C2×SD16—532no—
115C2×SD16—532no—
116C4⋊Q8—531no—
117#126—1062no—
118#125—531no—
119#125—1062no—
120C22⋊Q8—532no—
121C4×Q8—532no—
122C4×D4—22no—
123C4×D4C2×D5312yes—
124C4×D4C2×D5312yes—
125#117C2×D5311yescharacteristic
126#117C2×D5311yescharacteristic
127C2×C22⋊C4—22no—
128#107—532no—
129#106—532no—
130C4⋊C8—532no—
131#103—532no—
132#103—532no—
133#102—532no—
134#102—532no—
135C22⋊C8—532no—
136#97—531no—
137C4×C8—531no—
138C24—21no—
139C2×Q8—532no—
140C2×D4—24no—
141C2×D4—21no—
142C2×D4C22×D5312yes—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4C22×D5311yescharacteristic
146SD16—1064no—
147SD16—1064no—
148C2×C8—532no—
149C2×C8—532no—
150C4⋊C4—532no—
151C4⋊C4—1062no—
152C4⋊C4—1062no—
153C4⋊C4—1062no—
154C4⋊C4—1062no—
155C4⋊C4C22⋊D5312yes—
156C4⋊C4C22×D5312yes—
157C22⋊C4—21no—
158C22⋊C4—22no—
159C22⋊C4—22no—
160C22⋊C4—1062no—
161C22⋊C4C22⋊D5312yes—
162C42—21no—
163C42—1062no—
164C42C22×D5311yescharacteristic
165C42C22×D5311yescharacteristic
166C23—24no—
167C23—22no—
168C23—82no—
169C23C2×C22⋊D5311yescharacteristic
170Q8—1062no—
171Q8—1062no—
172D4—24no—
173D4—22no—
174D4#1171994514yes—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—1062no—
180C2×C4—1062no—
181C2×C4C2×C22⋊D5311yescharacteristic
182C2×C4C2×C22⋊D5312yes—
183C2×C4C2×C22⋊D5312yes—
184C2×C4C23×D5311yescharacteristic
185C8—1062no—
186C8—1062no—
187C22—21no—
188C22—24no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—84no—
193C22C22×C22⋊D5311yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—2122no—
199C4—2122no—
200C4C2×#1171994512yes—
201C2—41no—
202C2—44no—
203C2#1681113411yescharacteristic
204C2#1681143412yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#518C2
socleC22×C53C22×D4
hypercenter#518D53
hypocenterC53#3123
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #16811235C2semidirect
2. #16810964C2semidirect
3. #16810923C2semidirect
4. #16810828C2semidirect
6. #16810723C2semidirect
7. C4×#11719923C2semidirect
8. #16810684C2semidirect
12. #13133217C22semidirect
14. #13133209C22semidirect
17. C4×Dic106C22semidirect
18. #13133150C22semidirect
19. #13133148C22semidirect
20. C4⋊#88322C22semidirect
21. #13133140C22semidirect
22. #13133140C22semidirect
26. #13133128C22semidirect
27. C4×#88322C22semidirect
112. C53#3123semidirect