Finite Group Databasebeta

← #20280427

Subgroups — #20280427

Order 6784.

1134 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1134 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C2×#13133200C211yesmaximal, characteristic, complemented
2C2×#13133186C212yesmaximal, complemented
3#16810537C214yesmaximal, complemented
4#16810531C214yesmaximal, complemented
5#16810486C211yesmaximal, characteristic, complemented
6#16810479C212yesmaximal, complemented
7C23×D4×C53C211yesmaximal, characteristic, complemented
8C22×C22⋊D53C2211yescharacteristic
9C22×C22⋊D53C2214yescomplemented
10#13133200—24no—
11#13133186—28no—
12C2×#11719919C2212yescomplemented
13C2×C22⋊Dic53C2212yes—
14C2×C22⋊Dic53C2211yescharacteristic
15C2×C22⋊Dic53C2214yescomplemented
16C2×C22⋊Dic53C2218yescomplemented
17C2×#11719921C2212yescomplemented
18#13133124C2214yescomplemented
19C25×C53C2212yes—
20C22×D4×C53—28no—
21C22×D4×C53—24no—
22C22×D4×C53C2214yes—
23C23×C4×C53C2211yescharacteristic
24C24×C53—24no—
25C24×C53—24no—
26C24×C53—48no—
27C24×C53D418yescomplemented
28C24×C53C2314yes—
29C24×C53C2311yescharacteristic
30C23×D53C2311yescharacteristic
31C2×D4×C53—232no—
32C2×D4×C53—216no—
33C2×D4×C53—28no—
34C22×C4×C53—24no—
35C22×C4×C53D414yescomplemented
36C22×C4×C53C2312yes—
37C2×C22⋊D53—24no—
38C2×C22⋊D53—28no—
39C2×C22⋊D53—22no—
40C2×C22⋊D53—48no—
41C22×Dic53C2314yes—
42C22×Dic53C2312yes—
43C22×Dic53C2311yescharacteristic
44C22⋊Dic53—24no—
45C22⋊Dic53—22no—
46C22⋊Dic53—28no—
47C22⋊Dic53—48no—
48#11719921—24no—
49#11719919—24no—
50C23×C53—24no—
51C23×C53—216no—
52C23×C53—24no—
53C23×C53—24no—
54C23×C53—28no—
55C23×C53—28no—
56C23×C53—416no—
57C23×C53—416no—
58C23×C53—416no—
59C23×C53C2×D418yes—
60C23×C53C2×D414yes—
61C23×C53C2×D412yes—
62C23×C53C2411yescharacteristic
63C22×D53—22no—
64C22×D53—21no—
65C22×D53—24no—
66D4×C53—232no—
67C2×C4×C53—28no—
68C2×C4×C53—24no—
69C2×C4×C53C2×D414yes—
70C22⋊D53—416no—
71C22⋊D53—44no—
72C2×Dic53—21no—
73C2×Dic53—22no—
74C2×Dic53—24no—
75C2×Dic53—22no—
76C2×Dic53—28no—
77C2×Dic53—24no—
78C22×C53—28no—
79C22×C53—24no—
80C22×C53—28no—
81C22×C53—28no—
82C22×C53—28no—
83C22×C53—432no—
84C22×C53—48no—
85C22×C53—48no—
86C22×C53#12014yes—
87C22×C53C22×D412yes—
88C22×C53C22×D411yescharacteristic
89C22×C53H(5,𝔽2)14yes—
90C2×D53—41no—
91C2×D53—44no—
92C2×D53—42no—
93C4×C53—24no—
94Dic53—41no—
95Dic53—44no—
96Dic53—42no—
97#2613—531nomaximal, Sylow
98C2×C53—24no—
99C2×C53—48no—
100C2×C53C2×#12011yescharacteristic
101C2×C53#53414yes—
102C2×C53#53512yes—
103D53—81no—
104C23×D4D5311yescharacteristic, complemented
105C2×#123—531no—
106C2×#120—532no—
107#394—534no—
108#392—534no—
109#381—531no—
110#379—532no—
111C53#261311yescharacteristic, complemented, Sylow
112C25C2×D5312yes—
113C22×D4—28no—
114C22×D4—24no—
115C22×D4—531no—
116C22×D4—534no—
117C22×D4C2×D5314yes—
118C23×C4C2×D5311yescharacteristic
119#123—1064no—
120#120—1068no—
121C2×C4⋊C4—532no—
122C2×C22⋊C4—532no—
123C2×C22⋊C4—534no—
124C2×C22⋊C4—531no—
125C2×C22⋊C4—532no—
126C2×C22⋊C4—538no—
127#95—534no—
128C24—24no—
129C24—24no—
130C24—48no—
131C24—531no—
132C24C22⋊D5318yescomplemented
133C24C22×D5314yes—
134C24C22×D5311yescharacteristic
135C2×D4—232no—
136C2×D4—216no—
137C2×D4—28no—
138C2×D4—1068no—
139C2×D4—1064no—
140C2×D4—1062no—
141C2×D4—2128no—
142C22×C4—24no—
143C22×C4—534no—
144C22×C4—532no—
145C22×C4—531no—
146C22×C4C22⋊D5314yescomplemented
147C22×C4C22×D5312yes—
148C4⋊C4—1064no—
149C22⋊C4—1068no—
150C22⋊C4—1064no—
151C22⋊C4—1062no—
152C22⋊C4—1064no—
153C22⋊C4—2128no—
154C23—24no—
155C23—216no—
156C23—24no—
157C23—24no—
158C23—28no—
159C23—28no—
160C23—416no—
161C23—416no—
162C23—416no—
163C23—1061no—
164C23—1062no—
165C23—1064no—
166C23C2×C22⋊D5318yes—
167C23C2×C22⋊D5314yes—
168C23C2×C22⋊D5312yes—
169C23C23×D5311yescharacteristic
170D4—232no—
171D4—21216no—
172D4—2124no—
173C2×C4—28no—
174C2×C4—24no—
175C2×C4—1062no—
176C2×C4—1062no—
177C2×C4—1064no—
178C2×C4—1061no—
179C2×C4—1064no—
180C2×C4—1068no—
181C2×C4C2×C22⋊D5314yes—
182C22—28no—
183C22—24no—
184C22—28no—
185C22—28no—
186C22—28no—
187C22—432no—
188C22—48no—
189C22—48no—
190C22—2121no—
191C22—2124no—
192C22—2122no—
193C22#1313318614yes—
194C22C22×C22⋊D5312yes—
195C22C22×C22⋊D5311yescharacteristic
196C22#1313326214yes—
197C4—24no—
198C4—2121no—
199C4—2124no—
200C4—2122no—
201C2—24no—
202C2—48no—
203C2—4241no—
204C2C2×#1313318611yescharacteristic
205C2#1681118214yes—
206C2#1681119012yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC23×D4×C53C2
hypercenterC23×D4D53
hypocenterC53#2613
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C2×#13133200C2semidirect
2. C2×#13133186C2semidirect
3. #16810537C2semidirect
4. #16810531C2semidirect
5. #16810486C2semidirect
6. #16810479C2semidirect
7. C23×D4×C53C2semidirect
9. C22×C22⋊D53C22semidirect
12. C2×#11719919C22semidirect
15. C2×C22⋊Dic53C22semidirect
16. C2×C22⋊Dic53C22semidirect
17. C2×#11719921C22semidirect
18. #13133124C22semidirect
27. C24×C53D4semidirect
35. C22×C4×C53D4semidirect
104. C23×D4D53semidirect
111. C53#2613semidirect
132. C24C22⋊D53semidirect
146. C22×C4C22⋊D53semidirect