Finite Group Databasebeta

← #20274703

Subgroups — #20274703

Order 6784.

380 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 113 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×#13133132C212yesmaximal
2#16810483C212yesmaximal, complemented
3#16810390C212yesmaximal, complemented
4C2×C4×D4×C53C211yesmaximal, characteristic
5C2×C4⋊Dic53—22no—
6C2×C22⋊Dic53C2214yes—
7#13133132—24no—
8#13133124—22no—
9C22×D4×C53C2211yescharacteristic
10C23×C4×C53C411yescharacteristic, complemented
11C23×C4×C53C2211yescharacteristic
12C4×D4×C53—22no—
13C4×D4×C53—22no—
14C2×C4⋊C4×C53C411yescharacteristic, complemented
15C2×C22⋊C4×C53C411yescharacteristic, complemented
16C2×C22⋊C4×C53C2211yescharacteristic
17C2×C42×C53C411yescharacteristic, complemented
18C24×C53C2×C411yescharacteristic
19C24×C53C2311yescharacteristic
20C2×D4×C53—22no—
21C2×D4×C53—22no—
22C2×D4×C53D412yes—
23C2×D4×C53Q812yes—
24C22×C4×C53—21no—
25C22×C4×C53—21no—
26C22×C4×C53—42no—
27C22×C4×C53—42no—
28C22×C4×C53C2×C411yescharacteristic
29C22×C4×C53C2×C411yescharacteristic
30C22×C4×C53C2×C411yescharacteristic
31C22×C4×C53C2×C411yescharacteristic
32C22×C4×C53C2×C411yescharacteristic
33C22×C4×C53D412yes—
34C22×C4×C53D412yes—
35C22×Dic53—24no—
36C4⋊C4×C53—22no—
37C22⋊C4×C53—22no—
38C22⋊C4×C53—22no—
39C42×C53—21no—
40C42×C53—21no—
41C22⋊Dic53—24no—
42C22⋊Dic53—24no—
43C4⋊Dic53—44no—
44C23×C53—21no—
45C23×C53—21no—
46C23×C53—41no—
47C23×C53—41no—
48C23×C53—41no—
49C23×C53—41no—
50C23×C53C4⋊C412yescomplemented
51C23×C53C22×C411yescharacteristic
52C23×C53C2×D411yescharacteristic
53C23×C53C2×D412yes—
54C23×C53C2×Q811yescharacteristic
55C23×C53#4112yes—
56D4×C53—24no—
57D4×C53—24no—
58C2×C4×C53—21no—
59C2×C4×C53—21no—
60C2×C4×C53—22no—
61C2×C4×C53—24no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—41no—
65C2×C4×C53—42no—
66C2×C4×C53—42no—
67C2×C4×C53—42no—
68C2×C4×C53—41no—
69C2×C4×C53—41no—
70C2×C4×C53C22⋊C411yescharacteristic
71C2×C4×C53C22⋊C412yes—
72C2×C4×C53C22⋊C411yescharacteristic
73C2×C4×C53C4⋊C412yescomplemented
74C2×Dic53—44no—
75C2×Dic53—44no—
76C2×Dic53—44no—
77C22×C53—22no—
78C22×C53—21no—
79C22×C53—22no—
80C22×C53—21no—
81C22×C53—22no—
82C22×C53—41no—
83C22×C53—41no—
84C22×C53—42no—
85C22×C53—42no—
86C22×C53—42no—
87C22×C53—41no—
88C22×C53—41no—
89C22×C53C2×C22⋊C411yescharacteristic
90C22×C53C2×C4⋊C411yescharacteristic
91C22×C53#11711yescharacteristic
92C22×C53#12112yes—
93C22×C53C22⋊Q812yes—
94C4×C53—22no—
95C4×C53—21no—
96C4×C53—22no—
97C4×C53—41no—
98C4×C53—41no—
99C4×C53#9912yescomplemented
100Dic53—84no—
101#1799—531nomaximal, Sylow
102C2×C53—22no—
103C2×C53—21no—
104C2×C53—21no—
105C2×C53—22no—
106C2×C53—42no—
107C2×C53#38011yescharacteristic
108C2×C53C2×#9911yescharacteristic
109C2×C53#41011yescharacteristic
110C2×C4×D4D5311yescharacteristic
111C2×#99—532no—
112#380—532no—
113#342—532no—
114C53#179911yescharacteristic, complemented, Sylow
115C22×D4C2×D5311yescharacteristic
116C23×C4Dic5311yescharacteristic, complemented
117C23×C4C2×D5311yescharacteristic
118C4×D4—22no—
119C4×D4—22no—
120C2×C4⋊C4—1062no—
121C2×C4⋊C4Dic5311yescharacteristic, complemented
122C2×C22⋊C4—534no—
123C2×C22⋊C4Dic5311yescharacteristic, complemented
124C2×C22⋊C4C2×D5311yescharacteristic
125C2×C42Dic5311yescharacteristic, complemented
126#99—1064no—
127#95—1062no—
128C24C2×Dic5311yescharacteristic
129C24C22×D5311yescharacteristic
130C2×D4—22no—
131C2×D4—22no—
132C2×D4Dic10612yes—
133C2×D4D21212yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—42no—
137C22×C4—42no—
138C22×C4—1064no—
139C22×C4C2×Dic5311yescharacteristic
140C22×C4C2×Dic5311yescharacteristic
141C22×C4C2×Dic5311yescharacteristic
142C22×C4C2×Dic5311yescharacteristic
143C22×C4C2×Dic5311yescharacteristic
144C22×C4C22⋊D5312yes—
145C22×C4C22⋊D5312yes—
146C4⋊C4—22no—
147C4⋊C4—2124no—
148C22⋊C4—22no—
149C22⋊C4—22no—
150C22⋊C4—1064no—
151C22⋊C4—1064no—
152C42—21no—
153C42—21no—
154C23—21no—
155C23—21no—
156C23—41no—
157C23—41no—
158C23—41no—
159C23—41no—
160C23C4⋊Dic5312yescomplemented
161C23C2×Dic10611yescharacteristic
162C23C2×D21211yescharacteristic
163C23#1171994512yes—
164C23C22×Dic5311yescharacteristic
165C23C2×C22⋊D5312yes—
166D4—24no—
167D4—24no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—24no—
171C2×C4—22no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—42no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—2124no—
181C2×C4—2124no—
182C2×C4—2124no—
183C2×C4C4⋊Dic5312yescomplemented
184C2×C4C22⋊Dic5311yescharacteristic
185C2×C4C22⋊Dic5312yes—
186C2×C4C22⋊Dic5311yescharacteristic
187C22—21no—
188C22—22no—
189C22—21no—
190C22—22no—
191C22—22no—
192C22—41no—
193C22—41no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—41no—
198C22—41no—
199C22C2×C22⋊Dic5311yescharacteristic
200C22C2×C4⋊Dic5311yescharacteristic
201C22#1313317611yescharacteristic
202C22#1313319112yes—
203C22#1313319612yes—
204C4—21no—
205C4—22no—
206C4—22no—
207C4—41no—
208C4—41no—
209C4—4244no—
210C4#1313313212yescomplemented
211C2—22no—
212C2—21no—
213C2—21no—
214C2—22no—
215C2—42no—
216C2#1681048311yescharacteristic
217C2C2×#1313313211yescharacteristic
218C2#1681059711yescharacteristic
subgroupG/H
centerC22C2×C22⋊Dic53
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC2×C4×D4×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×C4×D4D53
hypocenterC53#1799
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #16810483C2semidirect
3. #16810390C2semidirect
10. C23×C4×C53C4semidirect
14. C2×C4⋊C4×C53C4semidirect
15. C2×C22⋊C4×C53C4semidirect
17. C2×C42×C53C4semidirect
50. C23×C53C4⋊C4semidirect
73. C2×C4×C53C4⋊C4semidirect
99. C4×C53#99semidirect
114. C53#1799semidirect
116. C23×C4Dic53semidirect
121. C2×C4⋊C4Dic53semidirect
123. C2×C22⋊C4Dic53semidirect
125. C2×C42Dic53semidirect
160. C23C4⋊Dic53semidirect
183. C2×C4C4⋊Dic53semidirect
210. C4#13133132semidirect