Finite Group Databasebeta

← #20290577

Subgroups — #20290577

Order 6784.

380 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 78 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
2#16811565C212yesmaximal, complemented
3#16811501C212yesmaximal, complemented
4C2×#13133177C211yesmaximal, characteristic
5C2×C4⋊#6157—22no—
6C2×C22⋊#6157C2214yes—
7#13133275—24no—
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⋊D53D412yes—
26C2×C22⋊D53Q812yes—
27C22×Dic53—21no—
28C22×Dic53—21no—
29C22×Dic53—42no—
30C22×Dic53C2×C411yescharacteristic
31C22×Dic53C2×C411yescharacteristic
32C22×Dic53C2×C411yescharacteristic
33C2×C4×D53—42no—
34C2×C4×D53D412yes—
35C2×C4×D53D412yes—
36C22⋊#6157—24no—
37C22⋊#6157—24no—
38C4⋊#6157—44no—
39C22⋊C4×C53—22no—
40#11719921—22no—
41#11719919—22no—
42C4×Dic53—21no—
43C4×Dic53—21no—
44C23×C53—21no—
45C23×C53—21no—
46C23×C53—41no—
47C23×C53—41no—
48C23×C53C4⋊C412yescomplemented
49C23×C53C22×C411yescharacteristic
50C22×D53—41no—
51C22×D53—41no—
52C22×D53C2×D412yes—
53C22×D53C2×D411yescharacteristic
54C22×D53C2×Q811yescharacteristic
55C22×D53#4112yes—
56C2×#6157—44no—
57C2×#6157—44no—
58C2×#6157—44no—
59C2×C4×C53—21no—
60C2×C4×C53—41no—
61C2×C4×C53—41no—
62C2×C4×C53C22⋊C412yes—
63C22⋊D53—24no—
64C22⋊D53—24no—
65C2×Dic53—21no—
66C2×Dic53—21no—
67C2×Dic53—22no—
68C2×Dic53—21no—
69C2×Dic53—42no—
70C2×Dic53—42no—
71C2×Dic53—41no—
72C2×Dic53C22⋊C411yescharacteristic
73C2×Dic53C22⋊C411yescharacteristic
74C2×Dic53C4⋊C412yescomplemented
75C4×D53—24no—
76C4×D53—42no—
77C22×C53—21no—
78C22×C53—21no—
79C22×C53—22no—
80C22×C53—42no—
81C22×C53—42no—
82C22×C53—41no—
83C22×C53—41no—
84C22×C53C2×C22⋊C411yescharacteristic
85C22×C53C2×C4⋊C411yescharacteristic
86C22×C53#11711yescharacteristic
87C2×D53—22no—
88C2×D53—22no—
89C2×D53—41no—
90C2×D53—41no—
91C2×D53—42no—
92C2×D53#12112yes—
93C2×D53C22⋊Q812yes—
94#6157—84no—
95C4×C53—22no—
96C4×C53—41no—
97Dic53—22no—
98Dic53—21no—
99Dic53—41no—
100Dic53#9912yescomplemented
101#1799—531nomaximal, Sylow
102C2×C53—21no—
103C2×C53—21no—
104C2×C53—42no—
105C2×C53#38011yescharacteristic
106C2×C53C2×#9911yescharacteristic
107C2×C53#41011yescharacteristic
108D53—22no—
109D53—22no—
110C2×C4×D4—531no—
111C2×#99—532no—
112#380—532no—
113#342—532no—
114C53#179911yescharacteristic, complemented, Sylow
115C22×D4—531no—
116C23×C4—531no—
117C23×C4—531no—
118C4×D4—1062no—
119C4×D4—1062no—
120C2×C4⋊C4—531no—
121C2×C4⋊C4—1062no—
122C2×C22⋊C4—531no—
123C2×C22⋊C4—534no—
124C2×C22⋊C4#615711yescharacteristic, complemented
125C2×C42—531no—
126#99—1064no—
127#95—1062no—
128C24—531no—
129C24C2×#615711yescharacteristic
130C2×D4—532no—
131C2×D4—532no—
132C2×D4—1062no—
133C2×D4—1062no—
134C22×C4—531no—
135C22×C4—531no—
136C22×C4—532no—
137C22×C4—532no—
138C22×C4—531no—
139C22×C4—1061no—
140C22×C4—1061no—
141C22×C4—1064no—
142C22×C4—2122no—
143C22×C4—2122no—
144C22×C4C2×#615711yescharacteristic
145C22×C4C2×#615711yescharacteristic
146C4⋊C4—1062no—
147C4⋊C4—2124no—
148C22⋊C4—22no—
149C22⋊C4—1062no—
150C22⋊C4—1064no—
151C22⋊C4—1064no—
152C42—1061no—
153C42—1061no—
154C23—21no—
155C23—21no—
156C23—41no—
157C23—41no—
158C23—532no—
159C23—531no—
160C23—531no—
161C23—532no—
162C23—2121no—
163C23—2121no—
164C23C4⋊#615712yescomplemented
165C23C22×#615711yescharacteristic
166D4—1064no—
167D4—1064no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—531no—
172C2×C4—531no—
173C2×C4—532no—
174C2×C4—1064no—
175C2×C4—1062no—
176C2×C4—1061no—
177C2×C4—1061no—
178C2×C4—1061no—
179C2×C4—2122no—
180C2×C4—2122no—
181C2×C4—2122no—
182C2×C4—2124no—
183C2×C4—2124no—
184C2×C4—2124no—
185C2×C4—2121no—
186C2×C4C22⋊#615712yes—
187C22—21no—
188C22—21no—
189C22—22no—
190C22—42no—
191C22—42no—
192C22—41no—
193C22—41no—
194C22—532no—
195C22—532no—
196C22—1062no—
197C22—1062no—
198C22—2121no—
199C22—2121no—
200C22—2122no—
201C22C2×C22⋊#615711yescharacteristic
202C22C2×C4⋊#615711yescharacteristic
203C22#1313329611yescharacteristic
204C4—22no—
205C4—41no—
206C4—532no—
207C4—1062no—
208C4—1061no—
209C4—4244no—
210C4—2121no—
211C2—21no—
212C2—21no—
213C2—42no—
214C2—1062no—
215C2—1062no—
216C2#1681156511yescharacteristic
217C2D53⋊#9911yescharacteristic
218C2#1681160311yescharacteristic
subgroupG/H
centerC22C2×C22⋊#6157
commutatorC23×C53C22×C4
FrattiniC23C22×#6157
FittingC2×C22⋊C4×C53C4
socleC22×C53C2×C22⋊C4
hypercenterC2×C22⋊C4#6157
hypocenterC53#1799
LCSG ▷ C23×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. #16811565C2semidirect
3. #16811501C2semidirect
10. C23×Dic53C4semidirect
14. C2×#11719919C4semidirect
16. C2×C4×Dic53C4semidirect
17. C2×C22⋊C4×C53C4semidirect
48. C23×C53C4⋊C4semidirect
74. C2×Dic53C4⋊C4semidirect
100. Dic53#99semidirect
114. C53#1799semidirect
124. C2×C22⋊C4#6157semidirect
164. C23C4⋊#6157semidirect