Finite Group Databasebeta

← #20284959

Subgroups — #20284959

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#16811228C211yesmaximal, characteristic, complemented
2#16811097C211yesmaximal, characteristic, complemented
3#16810948C212yesmaximal, complemented
4#16810924C212yesmaximal, complemented
5#16810819C212yesmaximal, complemented
6#16810759C212yesmaximal, complemented
7#16810736C212yesmaximal, complemented
8C4×D424C212yesmaximal, complemented
9C53×#433C211yesmaximal, characteristic, complemented
10C22×D212C2211yescharacteristic
11C2×D424C2212yes—
12C2×D424C2212yes—
13#13133214C2211yescharacteristic, complemented
14#13133209C2211yescharacteristic, complemented
15#13133202—22no—
16#13133191C2212yes—
17#13133190—22no—
18C4×D212C2212yes—
19C4×D212C2212yes—
20C4×D212C2212yescomplemented
21#13133179—22no—
22#13133176C2211yescharacteristic
23C22⋊C4×D53—22no—
24#13133151C2212yescomplemented
25#13133149C2212yescomplemented
26#13133139C2212yes—
27#13133139C2212yes—
28#13133139C2212yescomplemented
29#13133139C2212yescomplemented
30#117×C53C2211yescharacteristic
31C4⋊C8×C53C2212yescomplemented
32C22⋊C8×C53C2212yes—
33#97×C53C2211yescharacteristic, complemented
34C4×C8×C53C2211yescharacteristic, complemented
35C23×D53—21no—
36C22×C4×C53C2311yescharacteristic
37C2×C22⋊D53—22no—
38C2×D212—21no—
39C2×D212—24no—
40C2×D212—22no—
41C2×D212C2312yes—
42C2×D212C2312yes—
43C2×C4×D53—22no—
44C2×C4×D53—22no—
45C2×C4×D53—22no—
46C2×C8×C53C2312yes—
47C2×C8×C53C2312yes—
48C4⋊C4×C53D412yes—
49C22⋊C4×C53D412yes—
50C42×C53C2311yescharacteristic
51C42×C53C2311yescharacteristic
52C22⋊Dic53—21no—
53#11719921—22no—
54#11719921—22no—
55#11719921—22no—
56#11719921—22no—
57C4⋊Dic53C2312yes—
58C4⋊Dic53C2312yes—
59#11719919—22no—
60C4×Dic53—21no—
61D424—24no—
62D424—24no—
63C23×C53C2×D411yescharacteristic
64C22×D53—22no—
65C22×D53—24no—
66C22×D53—22no—
67C22×D53—82no—
68C2×C4×C53—22no—
69C2×C4×C53C2×D412yes—
70C2×C4×C53C2×D412yes—
71C2×C4×C53C2×D411yescharacteristic
72C2×C4×C53C2411yescharacteristic
73C22⋊D53—82no—
74C2×Dic53—22no—
75C2×Dic53—22no—
76D212—24no—
77D212—22no—
78D212—22no—
79D212—22no—
80D212—82no—
81D212#4114yes—
82C4×D53—82no—
83C4×D53—82no—
84C4×D53—82no—
85C8×C53—22no—
86C8×C53—22no—
87C22×C53—21no—
88C22×C53—22no—
89C22×C53C22×D411yescharacteristic
90C2×D53—24no—
91C2×D53—42no—
92C2×D53—42no—
93C2×D53—42no—
94C2×D53—42no—
95C2×D53—42no—
96C2×D53—84no—
97C4×C53—21no—
98C4×C53—22no—
99C4×C53—22no—
100C4×C53C2×#4112yes—
101Dic53—42no—
102Dic53—42no—
103#3122—531nomaximal, Sylow
104C2×C53—41no—
105C2×C53#52511yescharacteristic
106C2×C53#57612yes—
107D53—44no—
108D53—82no—
109#540—531no—
110#518—531no—
111#482—532no—
112#478—532no—
113#459—532no—
114#447—532no—
115#442—532no—
116C4×D8—532no—
117#433D5311yescharacteristic, complemented
118C53#312211yescharacteristic, complemented, Sylow
119C22×D4—531no—
120C2×D8—532no—
121C2×D8—532no—
122#127—531no—
123#125—531no—
124#123—1062no—
125#121—532no—
126#121—1062no—
127C4×D4—532no—
128C4×D4—532no—
129C4×D4—532no—
130C4×D4—1062no—
131#117—531no—
132#117C2×D5311yescharacteristic
133C2×C22⋊C4—1062no—
134#107—532no—
135#106—532no—
136C4⋊C8C2×D5312yescomplemented
137#102—532no—
138#102—532no—
139#102—532no—
140#102—532no—
141C22⋊C8C2×D5312yes—
142#97C2×D5311yescharacteristic, complemented
143C4×C8C2×D5311yescharacteristic, complemented
144C24—1061no—
145C2×D4—532no—
146C2×D4—532no—
147C2×D4—1064no—
148C2×D4—1062no—
149C2×D4—1061no—
150C2×D4—1062no—
151C22×C4—1062no—
152C22×C4—1062no—
153C22×C4—1062no—
154C22×C4C22×D5311yescharacteristic
155D8—1064no—
156D8—1064no—
157C2×C8C22×D5312yes—
158C2×C8C22×D5312yes—
159C4⋊C4—532no—
160C4⋊C4—532no—
161C4⋊C4—1062no—
162C4⋊C4D21212yes—
163C22⋊C4—1061no—
164C22⋊C4—1062no—
165C22⋊C4—1062no—
166C22⋊C4—1062no—
167C22⋊C4—1062no—
168C22⋊C4D21212yes—
169C42—1061no—
170C42C22×D5311yescharacteristic
171C42C22×D5311yescharacteristic
172C23—1064no—
173C23—1062no—
174C23—1062no—
175C23—4242no—
176C23C2×D21211yescharacteristic
177D4—534no—
178D4—1064no—
179D4—1062no—
180D4—1062no—
181D4—1062no—
182D4—4242no—
183D4—4242no—
184C2×C4—22no—
185C2×C4—1062no—
186C2×C4—1062no—
187C2×C4—4242no—
188C2×C4—4242no—
189C2×C4—4242no—
190C2×C4C2×D21211yescharacteristic
191C2×C4C2×D21212yes—
192C2×C4C2×D21212yes—
193C2×C4C23×D5311yescharacteristic
194C8—22no—
195C8—22no—
196C22—21no—
197C22—22no—
198C22—1064no—
199C22—4244no—
200C22—2122no—
201C22—2122no—
202C22—2122no—
203C22—2122no—
204C22—2122no—
205C22C22×D21211yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—2122no—
210C4—2122no—
211C4#41×D5312yes—
212C2—41no—
213C2—4242no—
214C2—2124no—
215C2#1681113511yescharacteristic
216C2#1681142812yes—
subgroupG/H
centerC22C22×D212
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#433C2
socleC22×C53C22×D4
hypercenter#433D53
hypocenterC53#3122
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. #16811228C2semidirect
2. #16811097C2semidirect
3. #16810948C2semidirect
4. #16810924C2semidirect
5. #16810819C2semidirect
6. #16810759C2semidirect
7. #16810736C2semidirect
8. C4×D424C2semidirect
9. C53×#433C2semidirect
13. #13133214C22semidirect
14. #13133209C22semidirect
20. C4×D212C22semidirect
24. #13133151C22semidirect
25. #13133149C22semidirect
28. #13133139C22semidirect
29. #13133139C22semidirect
31. C4⋊C8×C53C22semidirect
33. #97×C53C22semidirect
34. C4×C8×C53C22semidirect
117. #433D53semidirect
118. C53#3122semidirect
136. C4⋊C8C2×D53semidirect
142. #97C2×D53semidirect
143. C4×C8C2×D53semidirect