Finite Group Databasebeta

← #20290462

Subgroups — #20290462

Order 6784.

412 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 94 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×#13133275C212yesmaximal
2C4×C22⋊#6157C212yesmaximal, complemented
3#16811501C212yesmaximal, complemented
4C2×#13133177C211yesmaximal, characteristic
5C2×C22⋊#6157C2214yes—
6C2×C4×#6157—22no—
7#13133275C418yescomplemented
8(C2×#6157)⋊C4—22no—
9C22×C22⋊D53C2211yescharacteristic
10C23×Dic53C411yescharacteristic, complemented
11C22×C4×D53C2211yescharacteristic
12#13133177—22no—
13#13133177—22no—
14C2×#11719919C411yescharacteristic, complemented
15C2×#11719921C2211yescharacteristic
16C2×C4×Dic53C411yescharacteristic, complemented
17C2×C22⋊C4×C53C411yescharacteristic, complemented
18C24×C53C2×C411yescharacteristic
19C23×D53C2311yescharacteristic
20C22×#6157—24no—
21C22×C4×C53C2×C411yescharacteristic
22C22×C4×C53C2×C411yescharacteristic
23C2×C22⋊D53—22no—
24C2×C22⋊D53—22no—
25C2×C22⋊D53C2×C414yes—
26C22×Dic53—21no—
27C22×Dic53—21no—
28C22×Dic53—42no—
29C22×Dic53C2×C411yescharacteristic
30C22×Dic53C2×C411yescharacteristic
31C22×Dic53C2×C411yescharacteristic
32C2×C4×D53—42no—
33C2×C4×D53D412yes—
34C2×C4×D53D412yes—
35C22⋊#6157—24no—
36C22⋊#6157C2×C418yescomplemented
37C4×#6157—44no—
38C22⋊C4×C53—22no—
39#11719921—22no—
40#11719919—22no—
41C4×Dic53—21no—
42C4×Dic53—21no—
43C23×C53—21no—
44C23×C53—21no—
45C23×C53—41no—
46C23×C53—41no—
47C23×C53C4212yescomplemented
48C23×C53C22×C411yescharacteristic
49C22×D53—41no—
50C22×D53—41no—
51C22×D53C22×C412yes—
52C22×D53C2×D412yes—
53C22×D53#4112yes—
54C2×#6157—28no—
55C2×#6157—44no—
56C2×#6157—44no—
57C2×C4×C53—21no—
58C2×C4×C53—41no—
59C2×C4×C53—41no—
60C2×C4×C53C22⋊C412yes—
61C22⋊D53—28no—
62C2×Dic53—21no—
63C2×Dic53—21no—
64C2×Dic53—22no—
65C2×Dic53—21no—
66C2×Dic53—42no—
67C2×Dic53—42no—
68C2×Dic53—41no—
69C2×Dic53C4212yescomplemented
70C2×Dic53C22⋊C411yescharacteristic
71C2×Dic53C22⋊C411yescharacteristic
72C4×D53—24no—
73C4×D53—42no—
74C22×C53—21no—
75C22×C53—21no—
76C22×C53—22no—
77C22×C53—42no—
78C22×C53—42no—
79C22×C53—41no—
80C22×C53—41no—
81C22×C53C2×C4211yescharacteristic
82C22×C53C2×C22⋊C411yescharacteristic
83C22×C53#11711yescharacteristic
84C2×D53—22no—
85C2×D53—22no—
86C2×D53—42no—
87C2×D53—42no—
88C2×D53C4×D414yes—
89#6157—48no—
90C4×C53—22no—
91C4×C53—41no—
92Dic53—22no—
93Dic53—21no—
94Dic53—41no—
95Dic53#9912yescomplemented
96C4×#99—531nomaximal, Sylow
97C2×C53—21no—
98C2×C53—21no—
99C2×C53—42no—
100C2×C53C4×C22⋊C411yescharacteristic
101C2×C53C2×#9911yescharacteristic
102C2×C53#41011yescharacteristic
103D53—24no—
104C2×C4×D4—531no—
105C2×#99—532no—
106C4×C22⋊C4—532no—
107#342—532no—
108C53C4×#9911yescharacteristic, complemented, Sylow
109C22×D4—531no—
110C23×C4—531no—
111C23×C4—531no—
112C4×D4—1062no—
113C4×D4—1062no—
114C2×C4⋊C4—531no—
115C2×C22⋊C4—534no—
116C2×C22⋊C4—531no—
117C2×C22⋊C4#615711yescharacteristic, complemented
118C2×C42—531no—
119C2×C42—1062no—
120#99—538no—
121#95—1062no—
122C24—531no—
123C24C2×#615711yescharacteristic
124C2×D4—534no—
125C2×D4—1062no—
126C2×D4—1062no—
127C22×C4—531no—
128C22×C4—531no—
129C22×C4—532no—
130C22×C4—532no—
131C22×C4—531no—
132C22×C4—1061no—
133C22×C4—1061no—
134C22×C4—1064no—
135C22×C4—2122no—
136C22×C4—2122no—
137C22×C4C2×#615711yescharacteristic
138C22×C4C2×#615711yescharacteristic
139C4⋊C4—1062no—
140C22⋊C4—22no—
141C22⋊C4—538no—
142C22⋊C4—1062no—
143C22⋊C4—1064no—
144C42—1061no—
145C42—1061no—
146C42—2124no—
147C23—21no—
148C23—21no—
149C23—41no—
150C23—41no—
151C23—532no—
152C23—532no—
153C23—532no—
154C23—2121no—
155C23—2121no—
156C23C4×#615712yescomplemented
157C23C22×#615711yescharacteristic
158D4—1068no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—531no—
163C2×C4—532no—
164C2×C4—531no—
165C2×C4—1064no—
166C2×C4—1062no—
167C2×C4—1061no—
168C2×C4—1061no—
169C2×C4—1068no—
170C2×C4—1061no—
171C2×C4—2122no—
172C2×C4—2122no—
173C2×C4—2122no—
174C2×C4—2124no—
175C2×C4—2124no—
176C2×C4—2121no—
177C2×C4C22⋊#615712yes—
178C22—21no—
179C22—21no—
180C22—22no—
181C22—42no—
182C22—42no—
183C22—41no—
184C22—41no—
185C22—534no—
186C22—1062no—
187C22—1062no—
188C22—2122no—
189C22—2122no—
190C22C2×C4×#615711yescharacteristic
191C22C2×C22⋊#615711yescharacteristic
192C22#1313329611yescharacteristic
193C4—22no—
194C4—41no—
195C4—532no—
196C4—1062no—
197C4—1061no—
198C4—2128no—
199C4—2121no—
200C2—21no—
201C2—21no—
202C2—42no—
203C2—1064no—
204C2C4×C22⋊#615711yescharacteristic
205C2D53⋊#9911yescharacteristic
206C2#1681160311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC22×C53C2×C42
FrattiniC23C22×#6157
FittingC2×C22⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C22⋊C4#6157
hypocenterC53C4×#99
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊#6157C2semidirect
3. #16811501C2semidirect
7. #13133275C4semidirect
10. C23×Dic53C4semidirect
14. C2×#11719919C4semidirect
16. C2×C4×Dic53C4semidirect
17. C2×C22⋊C4×C53C4semidirect
36. C22⋊#6157C2×C4semidirect
47. C23×C53C42semidirect
69. C2×Dic53C42semidirect
95. Dic53#99semidirect
108. C53C4×#99semidirect
117. C2×C22⋊C4#6157semidirect
156. C23C4×#6157semidirect