Finite Group Databasebeta

← #20275129

Subgroups — #20275129

Order 6784.

386 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 97 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#16811098C211yesmaximal, characteristic
2C2×#13133132C211yesmaximal, characteristic
3#16810486C212yesmaximal, complemented
4#16810389C212yesmaximal, complemented
5C2×#123×C53C211yesmaximal, characteristic
6D4×Dic53—24no—
7#13133176—21no—
8C2×C22⋊Dic53—21no—
9C2×C22⋊Dic53C2212yes—
10C2×C22⋊Dic53C2212yes—
11#13133132—22no—
12#13133124—24no—
13C22×D4×C53C2211yescharacteristic
14C23×C4×C53C2211yescharacteristic
15#123×C53—22no—
16#123×C53C414yescomplemented
17C2×C4⋊C4×C53—21no—
18C2×C22⋊C4×C53—21no—
19C2×C22⋊C4×C53C2211yescharacteristic
20C24×C53D411yescharacteristic
21C24×C53C2311yescharacteristic
22C2×D4×C53—22no—
23C2×D4×C53—22no—
24C2×D4×C53C2×C412yes—
25C2×D4×C53D412yes—
26C22×C4×C53—21no—
27C22×C4×C53—21no—
28C22×C4×C53—21no—
29C22×C4×C53—22no—
30C22×C4×C53—22no—
31C22×C4×C53C2×C412yescomplemented
32C22×C4×C53D411yescharacteristic
33C22×Dic53—22no—
34C22×Dic53—22no—
35C22×Dic53D414yes—
36C4⋊C4×C53—24no—
37C22⋊C4×C53—24no—
38C22⋊C4×C53—21no—
39C22⋊C4×C53C2×C412yescomplemented
40C22⋊Dic53—22no—
41C22⋊Dic53—22no—
42C22⋊Dic53—22no—
43C22⋊Dic53—22no—
44C22⋊Dic53—22no—
45C4⋊Dic53—22no—
46C4×Dic53—22no—
47C23×C53—21no—
48C23×C53—21no—
49C23×C53—42no—
50C23×C53—42no—
51C23×C53C22⋊C412yes—
52C23×C53C22×C411yescharacteristic
53C23×C53C2×D412yes—
54C23×C53C2×D411yescharacteristic
55C23×C53C2×D411yescharacteristic
56C23×C53#4112yes—
57D4×C53—24no—
58D4×C53—24no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—22no—
62C2×C4×C53—22no—
63C2×C4×C53—24no—
64C2×C4×C53—42no—
65C2×C4×C53—41no—
66C2×C4×C53—42no—
67C2×C4×C53—41no—
68C2×C4×C53C22⋊C412yes—
69C2×Dic53—24no—
70C2×Dic53—44no—
71C2×Dic53—44no—
72C2×Dic53—42no—
73C2×Dic53—82no—
74C22×C53—24no—
75C22×C53—22no—
76C22×C53—22no—
77C22×C53—41no—
78C22×C53—41no—
79C22×C53—42no—
80C22×C53—42no—
81C22×C53—42no—
82C22×C53—81no—
83C22×C53C2×C22⋊C411yescharacteristic
84C22×C53C4×D412yes—
85C22×C53#12011yescharacteristic
86C22×C53#12112yes—
87C22×C53#12311yescharacteristic
88C4×C53—22no—
89C4×C53—42no—
90C4×C53—42no—
91Dic53—44no—
92Dic53—82no—
93#1890—531nomaximal, Sylow
94C2×C53—22no—
95C2×C53—21no—
96C2×C53—21no—
97C2×C53—22no—
98C2×C53—42no—
99C2×C53#38611yescharacteristic
100C2×C53#45812yes—
101C2×#123D5311yescharacteristic
102#518—531no—
103C2×#99—531no—
104#381—532no—
105#342—532no—
106C53#189011yescharacteristic, complemented, Sylow
107C22×D4C2×D5311yescharacteristic
108C23×C4C2×D5311yescharacteristic
109#123—22no—
110#123Dic5314yescomplemented
111C4×D4—1064no—
112#117—1061no—
113C2×C4⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—532no—
116C2×C22⋊C4—532no—
117C2×C22⋊C4—1061no—
118C2×C22⋊C4C2×D5311yescharacteristic
119#99—1062no—
120#95—1064no—
121C24C22⋊D5311yescharacteristic
122C24C22×D5311yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C2×Dic5312yes—
126C2×D4C22⋊D5312yes—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—21no—
132C22×C4—534no—
133C22×C4—1062no—
134C22×C4—1062no—
135C22×C4C2×Dic5312yescomplemented
136C22×C4C22⋊D5311yescharacteristic
137C4⋊C4—24no—
138C4⋊C4—1062no—
139C22⋊C4—21no—
140C22⋊C4—24no—
141C22⋊C4—1062no—
142C22⋊C4—1062no—
143C22⋊C4—1062no—
144C22⋊C4—1062no—
145C22⋊C4—1062no—
146C22⋊C4C2×Dic5312yescomplemented
147C42—1062no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—42no—
152C23C22⋊Dic5312yes—
153C23D4×D5312yes—
154C23#1171994712yes—
155C23C22×Dic5311yescharacteristic
156C23C2×C22⋊D5311yescharacteristic
157C23C2×C22⋊D5311yescharacteristic
158D4—24no—
159D4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—42no—
166C2×C4—41no—
167C2×C4—42no—
168C2×C4—41no—
169C2×C4—1064no—
170C2×C4—4242no—
171C2×C4—2124no—
172C2×C4—2122no—
173C2×C4—2124no—
174C2×C4C22⋊Dic5312yes—
175C22—22no—
176C22—24no—
177C22—22no—
178C22—42no—
179C22—41no—
180C22—41no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22C2×C22⋊Dic5311yescharacteristic
185C22D4×Dic5312yes—
186C22#1313318511yescharacteristic
187C22#1313318912yes—
188C22#1313319811yescharacteristic
189C4—22no—
190C4—42no—
191C4—42no—
192C4—4242no—
193C4—2124no—
194C2—21no—
195C2—22no—
196C2—21no—
197C2—22no—
198C2—42no—
199C2#1681050311yescharacteristic
200C2#1681081212yes—
subgroupG/H
centerC22#13133185
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC2×#123×C53C2
socleC22×C53#120
hypercenterC2×#123D53
hypocenterC53#1890
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. #16810486C2semidirect
4. #16810389C2semidirect
16. #123×C53C4semidirect
31. C22×C4×C53C2×C4semidirect
39. C22⋊C4×C53C2×C4semidirect
106. C53#1890semidirect
110. #123Dic53semidirect
135. C22×C4C2×Dic53semidirect
146. C22⋊C4C2×Dic53semidirect