Finite Group Databasebeta

← #20289982

Subgroups — #20289982

Order 6784.

280 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 62 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#16811593C211yesmaximal, characteristic, complemented
2C2×C22⋊#88324C211yesmaximal, characteristic, complemented
3#16811468C212yesmaximal, complemented
4#16811467C212yesmaximal, complemented
5C2×#13133201C211yesmaximal, characteristic
6C2×#11719940—21no—
7C22×#88324—21no—
8C22⋊#88324—21no—
9C22⋊#88324—21no—
10C22⋊#88324C2212yes—
11C22⋊#88324C2212yes—
12C22×C22⋊D53C411yescharacteristic
13C23×Dic53C2211yescharacteristic
14#13133201—21no—
15#13133201—21no—
16#13133201—21no—
17#13133201—21no—
18C2×C4⋊Dic53C411yescharacteristic
19C2×C4⋊Dic53C2211yescharacteristic
20C2×#11719921C411yescharacteristic
21C2×#11719921C2211yescharacteristic
22C2×C22⋊C4×C53C411yescharacteristic
23C24×C53C2×C411yescharacteristic
24C23×D53C2×C411yescharacteristic
25C22×C4×C53C2×C411yescharacteristic
26C22×C4×C53C2×C411yescharacteristic
27C2×C22⋊D53—21no—
28C2×C22⋊D53—21no—
29C2×C22⋊D53—42no—
30C22×Dic53—42no—
31C22×Dic53C2×C411yescharacteristic
32C22×Dic53C2×C411yescharacteristic
33C22×Dic53D411yescharacteristic
34C22×Dic53D411yescharacteristic
35C22×Dic53D411yescharacteristic
36C22×Dic53D411yescharacteristic
37C22×Dic53C2311yescharacteristic
38#11719940—42no—
39C2×#88324—22no—
40C2×#88324—22no—
41C2×#88324—42no—
42C22⋊C4×C53—21no—
43C22⋊C4×C53—21no—
44#11719921—21no—
45#11719921—21no—
46#11719921—21no—
47#11719921—21no—
48C4⋊Dic53—21no—
49C4⋊Dic53—21no—
50C4⋊Dic53—21no—
51C4⋊Dic53—21no—
52C23×C53—22no—
53C23×C53—22no—
54C23×C53—41no—
55C23×C53C22⋊C411yescharacteristic
56C23×C53C22⋊C411yescharacteristic
57C23×C53C22×C411yescharacteristic
58C22×D53—21no—
59C22×D53—22no—
60C22×D53—41no—
61C22×D53—41no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—22no—
65C2×C4×C53—41no—
66C22⋊D53—42no—
67C22⋊D53—42no—
68C2×Dic53—22no—
69C2×Dic53—22no—
70C2×Dic53—21no—
71C2×Dic53—41no—
72C2×Dic53—41no—
73C2×Dic53—41no—
74C2×Dic53—41no—
75C2×Dic53—41no—
76C2×Dic53C22⋊C411yescharacteristic
77C2×Dic53C22⋊C411yescharacteristic
78C2×Dic53C2×D411yescharacteristic
79C2×Dic53C2×D411yescharacteristic
80#88324—42no—
81#88324—42no—
82C22×C53—21no—
83C22×C53—21no—
84C22×C53—21no—
85C22×C53—21no—
86C22×C53—21no—
87C22×C53—21no—
88C22×C53—21no—
89C22×C53—44no—
90C22×C53#9912yes—
91C22×C53C2×C22⋊C411yescharacteristic
92C22×C53#13112yes—
93C2×D53—41no—
94C2×D53—41no—
95C2×D53—41no—
96C2×D53—82no—
97C4×C53—41no—
98C4×C53—41no—
99Dic53—21no—
100Dic53—21no—
101Dic53—21no—
102Dic53—21no—
103Dic53—41no—
104#1468—531nomaximal, Sylow
105C2×C53—22no—
106C2×C53—22no—
107C2×C53—41no—
108C2×C53#40811yescharacteristic
109C2×C53C2×#9911yescharacteristic
110C2×C53#41311yescharacteristic
111D53—81no—
112C2×#123—531no—
113#407—531no—
114C2×C22⋊C8—531no—
115#324—532no—
116#323—532no—
117C53#146811yescharacteristic, complemented, Sylow
118C22×D4—531no—
119C23×C4—531no—
120C2×M16—1061no—
121C22×C8—1061no—
122#123—1061no—
123#123—1061no—
124#123—1061no—
125#123—1061no—
126C2×C4⋊C4—531no—
127C2×C4⋊C4—531no—
128C2×C22⋊C4—531no—
129C2×C22⋊C4—531no—
130C2×C22⋊C4#615711yescharacteristic
131C22⋊C8—532no—
132C22⋊C8—532no—
133C22⋊C8—1061no—
134C22⋊C8—1061no—
135C24—531no—
136C24C2×#615711yescharacteristic
137C2×D4—1061no—
138C2×D4—1061no—
139C2×D4—2122no—
140C22×C4—531no—
141C22×C4—531no—
142C22×C4—531no—
143C22×C4—531no—
144C22×C4—531no—
145C22×C4—531no—
146C22×C4—531no—
147C22×C4—2122no—
148C22×C4C2×#615711yescharacteristic
149C22×C4C2×#615711yescharacteristic
150M16—2122no—
151C2×C8—1062no—
152C2×C8—1062no—
153C2×C8—2122no—
154C4⋊C4—1061no—
155C4⋊C4—1061no—
156C4⋊C4—1061no—
157C4⋊C4—1061no—
158C22⋊C4—21no—
159C22⋊C4—21no—
160C22⋊C4—1061no—
161C22⋊C4—1061no—
162C22⋊C4—1061no—
163C22⋊C4—1061no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—1062no—
168C23—1061no—
169C23—2121no—
170C23—2121no—
171C23C22⋊#615711yescharacteristic
172C23C22⋊#615711yescharacteristic
173C23C22×#615711yescharacteristic
174D4—2122no—
175D4—2122no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—41no—
180C2×C4—531no—
181C2×C4—531no—
182C2×C4—531no—
183C2×C4—531no—
184C2×C4—1062no—
185C2×C4—1062no—
186C2×C4—1061no—
187C2×C4—2121no—
188C2×C4—2121no—
189C2×C4—2121no—
190C2×C4—2121no—
191C2×C4—2121no—
192C8—2122no—
193C8—2122no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—4242no—
203C22—2121no—
204C22—2121no—
205C22—2121no—
206C22#1313327612yes—
207C22C2×C22⋊#615711yescharacteristic
208C22#1313330412yes—
209C4—41no—
210C4—41no—
211C4—1061no—
212C4—1061no—
213C4—1061no—
214C4—1061no—
215C4—2121no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—4241no—
220C2#1681159611yescharacteristic
221C2C2×#1313327611yescharacteristic
222C2#1681161011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C22⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C22⋊C4#6157
hypocenterC53#1468
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #16811593C2semidirect
2. C2×C22⋊#88324C2semidirect
3. #16811468C2semidirect
4. #16811467C2semidirect
117. C53#1468semidirect