Finite Group Databasebeta

← #20284962

Subgroups — #20284962

Order 6784.

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

normal subgroups only

This group has 394 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
2#16810961C212yesmaximal, complemented
3#16810920C212yesmaximal, complemented
4#16810825C212yesmaximal, complemented
5#16810773C212yesmaximal
6#16810721C212yesmaximal, complemented
7C4×#11719924C212yesmaximal, complemented
8#16810684C211yesmaximal, characteristic, complemented
9C53×#541C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C2×#11719924C2212yes—
12C2×#11719924C2212yes—
13C4×D212C2212yes—
14C4×D212C2212yes—
15#13133179—22no—
16#13133176C2211yescharacteristic
17C22⋊C4×D53—22no—
18#13133150C2212yescomplemented
19#13133148C2212yescomplemented
20C4⋊#88322C2212yescomplemented
21#13133141C2212yescomplemented
22#13133141C2212yescomplemented
23#13133137C2212yes—
24#13133137C2212yes—
25C22⋊#88322C2212yes—
26#13133128C2211yescharacteristic, complemented
27C4×#88322C2211yescharacteristic, complemented
28C4⋊Q8×C53C2211yescharacteristic, complemented
29#126×C53—22no—
30#125×C53—22no—
31#125×C53C2211yescharacteristic, complemented
32C22⋊Q8×C53C2212yes—
33C4×Q8×C53C2212yescomplemented
34#117×C53C2211yescharacteristic
35C23×D53—21no—
36C2×Q8×C53C2312yes—
37C22×C4×C53C2311yescharacteristic
38C2×D212—21no—
39C2×D212—24no—
40C2×D212C2312yes—
41C2×C4×D53—22no—
42C2×C4×D53—22no—
43C4⋊C4×C53—22no—
44C4⋊C4×C53—22no—
45C4⋊C4×C53—22no—
46C4⋊C4×C53—22no—
47C4⋊C4×C53D412yes—
48C4⋊C4×C53C2312yes—
49C22⋊C4×C53—22no—
50C22⋊C4×C53D412yes—
51C42×C53—22no—
52C42×C53C2311yescharacteristic
53C42×C53C2311yescharacteristic
54C22⋊Dic53—21no—
55#11719924—24no—
56#11719924—24no—
57#11719921—22no—
58#11719921—22no—
59C4⋊Dic53C2312yes—
60C4×Dic53—21no—
61C2×#88322C2312yes—
62C2×#88322C2312yes—
63C23×C53C2×D411yescharacteristic
64C22×D53—22no—
65C22×D53—24no—
66C22×D53—82no—
67Q8×C53—22no—
68Q8×C53—22no—
69C2×C4×C53—22no—
70C2×C4×C53—22no—
71C2×C4×C53—22no—
72C2×C4×C53C2×D412yes—
73C2×C4×C53C2×D412yes—
74C2×C4×C53C2×D411yescharacteristic
75C2×C4×C53C2411yescharacteristic
76C2×Dic53—22no—
77D212—24no—
78D212—22no—
79D212#4114yes—
80C4×D53—82no—
81C4×D53—82no—
82#88322—22no—
83#88322—22no—
84C22×C53—21no—
85C22×C53—22no—
86C22×C53C22×D411yescharacteristic
87C2×D53—24no—
88C2×D53—42no—
89C2×D53—42no—
90C2×D53—84no—
91C4×C53—21no—
92C4×C53—22no—
93C4×C53—22no—
94C4×C53—42no—
95C4×C53—42no—
96C4×C53C2×#4112yes—
97Dic53—42no—
98#3123—531nomaximal, Sylow
99C2×C53—41no—
100C2×C53#52511yescharacteristic
101C2×C53#57712yes—
102D53—44no—
103#541D5311yescharacteristic, complemented
104#518—531no—
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×D4—531no—
114C2×SD16—532no—
115C2×SD16—532no—
116C4⋊Q8C2×D5311yescharacteristic, complemented
117#126—22no—
118#125—22no—
119#125C2×D5311yescharacteristic, complemented
120C22⋊Q8C2×D5312yes—
121C4×Q8C2×D5312yescomplemented
122C4×D4—532no—
123C4×D4—532no—
124C4×D4—1062no—
125#117—531no—
126#117C2×D5311yescharacteristic
127C2×C22⋊C4—1062no—
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—1061no—
139C2×Q8C22×D5312yes—
140C2×D4—532no—
141C2×D4—1064no—
142C2×D4—1061no—
143C22×C4—1062no—
144C22×C4—1062no—
145C22×C4C22×D5311yescharacteristic
146SD16—1064no—
147SD16—1064no—
148C2×C8—532no—
149C2×C8—532no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—532no—
155C4⋊C4C22⋊D5312yes—
156C4⋊C4C22×D5312yes—
157C22⋊C4—22no—
158C22⋊C4—1061no—
159C22⋊C4—1062no—
160C22⋊C4—1062no—
161C22⋊C4C22⋊D5312yes—
162C42—22no—
163C42—1061no—
164C42C22×D5311yescharacteristic
165C42C22×D5311yescharacteristic
166C23—1064no—
167C23—1062no—
168C23—4242no—
169C23C2×C22⋊D5311yescharacteristic
170Q8—22no—
171Q8—22no—
172D4—534no—
173D4—1064no—
174D4—1062no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—1062no—
179C2×C4—4242no—
180C2×C4—4242no—
181C2×C4C2×C22⋊D5312yes—
182C2×C4C2×C22⋊D5311yescharacteristic
183C2×C4C2×C22⋊D5312yes—
184C2×C4C23×D5311yescharacteristic
185C8—1062no—
186C8—1062no—
187C22—21no—
188C22—22no—
189C22—1064no—
190C22—4244no—
191C22—2122no—
192C22—2122no—
193C22C22×C22⋊D5311yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—2122no—
200C4#41×D5312yes—
201C2—41no—
202C2—2124no—
203C2#1681113011yescharacteristic
204C2#1681143112yes—
subgroupG/H
centerC22C22×C22⋊D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#541C2
socleC22×C53C22×D4
hypercenter#541D53
hypocenterC53#3123
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #16810961C2semidirect
3. #16810920C2semidirect
4. #16810825C2semidirect
6. #16810721C2semidirect
7. C4×#11719924C2semidirect
8. #16810684C2semidirect
9. C53×#541C2semidirect
18. #13133150C22semidirect
19. #13133148C22semidirect
20. C4⋊#88322C22semidirect
21. #13133141C22semidirect
22. #13133141C22semidirect
26. #13133128C22semidirect
27. C4×#88322C22semidirect
28. C4⋊Q8×C53C22semidirect
31. #125×C53C22semidirect
33. C4×Q8×C53C22semidirect
103. #541D53semidirect
112. C53#3123semidirect
116. C4⋊Q8C2×D53semidirect
119. #125C2×D53semidirect
121. C4×Q8C2×D53semidirect