Finite Group Databasebeta

← #20284968

Subgroups — #20284968

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#16811235C211yesmaximal, characteristic, complemented
2#16811097C211yesmaximal, characteristic
3#16810965C212yesmaximal, complemented
4#16810924C212yesmaximal, complemented
5#16810829C212yesmaximal, complemented
6#16810776C212yesmaximal
7#16810725C212yesmaximal, complemented
8C4×#11719913C212yesmaximal, complemented
9C53×#433C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C2×#11719913C2212yes—
12C2×#11719913C2212yes—
13#13133217C2211yescharacteristic, complemented
14#13133210—22no—
15#13133209C2211yescharacteristic, complemented
16#13133207—22no—
17#13133196C2212yes—
18C4×Dic106C2212yescomplemented
19C4×D212C2212yes—
20C4×D212C2212yes—
21#13133179—22no—
22#13133176C2211yescharacteristic
23C22⋊C4×D53—22no—
24#13133151C2212yescomplemented
25#13133149C2212yescomplemented
26#13133142C2212yescomplemented
27#13133142C2212yescomplemented
28#13133139C2212yes—
29#13133139C2212yes—
30#117×C53C2211yescharacteristic
31C4⋊C8×C53C2212yescomplemented
32C22⋊C8×C53C2212yes—
33#97×C53C2211yescharacteristic, complemented
34C4×C8×C53C2211yescharacteristic, complemented
35C23×D53—21no—
36C22×C4×C53C2311yescharacteristic
37C2×D212—21no—
38C2×D212—24no—
39C2×D212C2312yes—
40C2×C4×D53—22no—
41C2×C4×D53—22no—
42C2×Dic106C2312yes—
43C2×C8×C53C2312yes—
44C2×C8×C53C2312yes—
45C4⋊C4×C53D412yes—
46C22⋊C4×C53D412yes—
47C42×C53C2311yescharacteristic
48C42×C53C2311yescharacteristic
49C22⋊Dic53—21no—
50C22⋊Dic53—22no—
51#11719921—22no—
52#11719921—22no—
53C4⋊Dic53—22no—
54C4⋊Dic53C2312yes—
55C4⋊Dic53C2312yes—
56#11719919—22no—
57#11719919—22no—
58#11719919—22no—
59C4×Dic53—21no—
60C4×Dic53—22no—
61#11719913—24no—
62#11719913—24no—
63C23×C53C2×D411yescharacteristic
64C22×D53—22no—
65C22×D53—24no—
66C22×D53—82no—
67C2×C4×C53—22no—
68C2×C4×C53C2×D412yes—
69C2×C4×C53C2×D412yes—
70C2×C4×C53C2×D411yescharacteristic
71C2×C4×C53C2411yescharacteristic
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—22no—
75D212—24no—
76D212—22no—
77D212#4114yes—
78C4×D53—82no—
79C4×D53—82no—
80Dic106—22no—
81Dic106—22no—
82C8×C53—22no—
83C8×C53—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×C53C2×#4112yes—
95Dic53—42no—
96Dic53—42no—
97Dic53—42no—
98#3123—531nomaximal, Sylow
99C2×C53—41no—
100C2×C53#52511yescharacteristic
101C2×C53#57712yes—
102D53—44no—
103#541—531no—
104#518—531no—
105#485—532no—
106#478—532no—
107#461—532no—
108#450—532no—
109#440—532no—
110C4×SD16—532no—
111#433D5311yescharacteristic, complemented
112C53#312311yescharacteristic, complemented, Sylow
113C22×D4—531no—
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—532no—
123C4×D4—532no—
124C4×D4—1062no—
125#117—531no—
126#117C2×D5311yescharacteristic
127C2×C22⋊C4—1062no—
128#107—532no—
129#106—532no—
130C4⋊C8C2×D5312yescomplemented
131#103—532no—
132#103—532no—
133#102—532no—
134#102—532no—
135C22⋊C8C2×D5312yes—
136#97C2×D5311yescharacteristic, complemented
137C4×C8C2×D5311yescharacteristic, complemented
138C24—1061no—
139C2×Q8—532no—
140C2×D4—532no—
141C2×D4—1064no—
142C2×D4—1061no—
143C22×C4—1062no—
144C22×C4—1062no—
145C22×C4C22×D5311yescharacteristic
146SD16—1064no—
147SD16—1064no—
148C2×C8C22×D5312yes—
149C2×C8C22×D5312yes—
150C4⋊C4—532no—
151C4⋊C4—532no—
152C4⋊C4—1062no—
153C4⋊C4—1062no—
154C4⋊C4—1062no—
155C4⋊C4—1062no—
156C4⋊C4D21212yes—
157C22⋊C4—1061no—
158C22⋊C4—1062no—
159C22⋊C4—1062no—
160C22⋊C4—1062no—
161C22⋊C4D21212yes—
162C42—1061no—
163C42—1062no—
164C42C22×D5311yescharacteristic
165C42C22×D5311yescharacteristic
166C23—1064no—
167C23—1062no—
168C23—4242no—
169C23C2×D21211yescharacteristic
170Q8—1062no—
171Q8—1062no—
172D4—534no—
173D4—1064no—
174D4—1062no—
175C2×C4—22no—
176C2×C4—1062no—
177C2×C4—1062no—
178C2×C4—1062no—
179C2×C4—4242no—
180C2×C4—4242no—
181C2×C4C2×D21212yes—
182C2×C4C2×D21211yescharacteristic
183C2×C4C2×D21212yes—
184C2×C4C23×D5311yescharacteristic
185C8—22no—
186C8—22no—
187C22—21no—
188C22—22no—
189C22—1064no—
190C22—4244no—
191C22—2122no—
192C22—2122no—
193C22C22×D21211yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—2122no—
198C4—2122no—
199C4—2122no—
200C4#41×D5312yes—
201C2—41no—
202C2—2124no—
203C2#1681113511yescharacteristic
204C2#1681143512yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#433C2
socleC22×C53C22×D4
hypercenter#433D53
hypocenterC53#3123
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ #117 ◁ #433

Complements

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

NH ≅ G/Ntype
1. #16811235C2semidirect
3. #16810965C2semidirect
4. #16810924C2semidirect
5. #16810829C2semidirect
7. #16810725C2semidirect
8. C4×#11719913C2semidirect
9. C53×#433C2semidirect
13. #13133217C22semidirect
15. #13133209C22semidirect
18. C4×Dic106C22semidirect
24. #13133151C22semidirect
25. #13133149C22semidirect
26. #13133142C22semidirect
27. #13133142C22semidirect
31. C4⋊C8×C53C22semidirect
33. #97×C53C22semidirect
34. C4×C8×C53C22semidirect
111. #433D53semidirect
112. C53#3123semidirect
130. C4⋊C8C2×D53semidirect
136. #97C2×D53semidirect
137. C4×C8C2×D53semidirect