Finite Group Databasebeta

← #20289799

Subgroups — #20289799

Order 6784.

234 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 44 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×#13133277C211yesmaximal, characteristic
2C2×#13133275C211yesmaximal, characteristic
3#16810502C211yesmaximal, characteristic
4C2×#11719940—21no—
5C2×C22⋊#6157—21no—
6#13133277—21no—
7#13133277C412yescomplemented
8#13133275—21no—
9#13133275C412yescomplemented
10C22×C22⋊D53C2211yescharacteristic
11C2×C22⋊Dic53C411yescharacteristic, complemented
12C2×#11719921—21no—
13#13133124—21no—
14C23×C4×C53C411yescharacteristic, complemented
15C24×C53C2×C411yescharacteristic
16C23×D53D411yescharacteristic
17C22×#6157—41no—
18C22×C4×C53—21no—
19C22×C4×C53—42no—
20C22×C4×C53—41no—
21C2×C22⋊D53—41no—
22C2×C22⋊D53—41no—
23C2×C22⋊D53C2×C411yescharacteristic
24C2×C22⋊D53C2×C411yescharacteristic
25C2×C22⋊D53D411yescharacteristic
26C2×C22⋊D53Q811yescharacteristic
27C22×Dic53—21no—
28C22×Dic53D411yescharacteristic
29C22⋊#6157—22no—
30C22⋊#6157—41no—
31#11719940—22no—
32#11719940—41no—
33C2×#88324—41no—
34C22⋊Dic53—41no—
35#11719921—42no—
36C23×C53—21no—
37C23×C53—21no—
38C23×C53—41no—
39C23×C53—41no—
40C23×C53C4211yescharacteristic
41C23×C53C22⋊C411yescharacteristic
42C23×C53C4⋊C411yescharacteristic
43C22×D53—21no—
44C22×D53—21no—
45C22×D53—41no—
46C22×D53—41no—
47C22×D53C22⋊C411yescharacteristic
48C22×D53C4⋊C411yescharacteristic
49C2×#6157—42no—
50C2×#6157—81no—
51C2×#6157—81no—
52C2×C4×C53—22no—
53C2×C4×C53—41no—
54C2×C4×C53—42no—
55C2×C4×C53—41no—
56C2×C4×C53—42no—
57C22⋊D53—41no—
58C22⋊D53—41no—
59C22⋊D53—41no—
60C22⋊D53—41no—
61C2×Dic53—21no—
62C2×Dic53—21no—
63C2×Dic53—41no—
64C2×Dic53—41no—
65C2×Dic53—41no—
66C2×Dic53C22⋊C411yescharacteristic
67C2×Dic53C4⋊C411yescharacteristic
68#88324—42no—
69C22×C53—21no—
70C22×C53—21no—
71C22×C53—21no—
72C22×C53—21no—
73C22×C53—41no—
74C22×C53—41no—
75C22×C53—42no—
76C22×C53—41no—
77C22×C53—41no—
78C22×C53#9511yescharacteristic
79C22×C53#9911yescharacteristic
80C22×C53#9911yescharacteristic
81C2×D53—21no—
82C2×D53—21no—
83C2×D53—41no—
84C2×D53—41no—
85C2×D53—41no—
86C2×D53—41no—
87C2×D53—81no—
88#6157—82no—
89C4×C53—42no—
90Dic53—21no—
91Dic53—21no—
92Dic53—81no—
93#1390—531nomaximal, Sylow
94C2×C53—21no—
95C2×C53—21no—
96C2×C53—41no—
97C2×C53—41no—
98C2×C53#34211yescharacteristic
99C2×C53AΣL(1,𝔽16)11yescharacteristic
100C2×C53#35211yescharacteristic
101D53—41no—
102D53—41no—
103C2×#100—531no—
104C2×#99—531no—
105#386—531no—
106C53#139011yescharacteristic, complemented, Sylow
107C22×D4—531no—
108C23×C4#615711yescharacteristic, complemented
109C2×M16—1061no—
110C2×C22⋊C4—531no—
111C2×C22⋊C4—1061no—
112C2×C22⋊C4—1061no—
113#100—532no—
114#100—1061no—
115#99—532no—
116#99—1061no—
117#95—1061no—
118C24—531no—
119C24C2×#615711yescharacteristic
120C2×D4—531no—
121C2×D4—531no—
122C2×D4—531no—
123C2×D4—531no—
124C2×D4—2121no—
125C2×D4—2121no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—41no—
129C22×C4—531no—
130C22×C4—1061no—
131C22×C4—2121no—
132M16—1062no—
133M16—2121no—
134C2×C8—2121no—
135C22⋊C4—1062no—
136C22⋊C4—2122no—
137C22⋊C4—2121no—
138C22⋊C4—2121no—
139C23—21no—
140C23—21no—
141C23—41no—
142C23—41no—
143C23—531no—
144C23—531no—
145C23—1061no—
146C23—1061no—
147C23—2121no—
148C23—2121no—
149C23C4×#615711yescharacteristic
150C23C4⋊#615711yescharacteristic
151C23C22⋊#615711yescharacteristic
152D4—2121no—
153D4—2121no—
154D4—2121no—
155D4—2121no—
156C2×C4—22no—
157C2×C4—41no—
158C2×C4—42no—
159C2×C4—41no—
160C2×C4—42no—
161C2×C4—531no—
162C2×C4—531no—
163C2×C4—1061no—
164C2×C4—1061no—
165C2×C4—4241no—
166C2×C4—4241no—
167C2×C4—2121no—
168C2×C4—2121no—
169C2×C4—2122no—
170C2×C4—2121no—
171C8—2122no—
172C22—21no—
173C22—21no—
174C22—21no—
175C22—21no—
176C22—41no—
177C22—41no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—1061no—
182C22—1061no—
183C22—4241no—
184C22—2121no—
185C22—2121no—
186C22—2121no—
187C22—2121no—
188C22(C2×#6157)⋊C411yescharacteristic
189C22#1313327511yescharacteristic
190C22#1313327611yescharacteristic
191C4—42no—
192C4—1061no—
193C4—1061no—
194C4—4241no—
195C4—4242no—
196C2—21no—
197C2—21no—
198C2—41no—
199C2—41no—
200C2—2121no—
201C2—2121no—
202C2#1681150111yescharacteristic
203C2#1681152311yescharacteristic
204C2#1681152511yescharacteristic
subgroupG/H
centerC22#13133275
commutatorC23×C53C42
FrattiniC24C2×#6157
FittingC23×C4×C53C4
socleC22×C53#99
hypercenterC23×C4#6157
hypocenterC53#1390
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C24 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
7. #13133277C4semidirect
9. #13133275C4semidirect
11. C2×C22⋊Dic53C4semidirect
14. C23×C4×C53C4semidirect
106. C53#1390semidirect
108. C23×C4#6157semidirect