Finite Group Databasebeta

← #20275128

Subgroups — #20275128

Order 6784.

386 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 93 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×#13133198C211yesmaximal, characteristic
2C2×#13133132C211yesmaximal, characteristic
3#16810484C212yesmaximal, complemented
4#16810390C212yesmaximal, complemented
5C53×#518C211yesmaximal, characteristic
6C23×Dic53C2211yescharacteristic
7#13133198—22no—
8#13133198C414yescomplemented
9C2×#11719919—21no—
10C2×C22⋊Dic53—21no—
11C2×C22⋊Dic53C2212yes—
12C2×C22⋊Dic53C2211yescharacteristic
13#13133132—22no—
14#13133124—24no—
15C22×D4×C53C2211yescharacteristic
16C4×D4×C53—24no—
17#117×C53—21no—
18C2×C22⋊C4×C53—21no—
19C2×C22⋊C4×C53C2212yes—
20C24×C53D411yescharacteristic
21C24×C53C2311yescharacteristic
22C2×D4×C53—22no—
23C2×D4×C53—22no—
24C2×D4×C53C2×C412yes—
25C2×D4×C53D412yes—
26C22×C4×C53—22no—
27C22×C4×C53D411yescharacteristic
28C22×C4×C53D414yes—
29C22×Dic53—22no—
30C22×Dic53—21no—
31C22×Dic53—21no—
32C22×Dic53—21no—
33C22×Dic53—22no—
34C22×Dic53—22no—
35C22×Dic53C2×C412yescomplemented
36C4⋊C4×C53—22no—
37C22⋊C4×C53—22no—
38C22⋊C4×C53—22no—
39C22⋊C4×C53—22no—
40C42×C53—22no—
41C22⋊Dic53—24no—
42C22⋊Dic53—21no—
43C22⋊Dic53—22no—
44C22⋊Dic53—22no—
45C22⋊Dic53C2×C412yescomplemented
46#11719919—24no—
47C23×C53—21no—
48C23×C53—21no—
49C23×C53—42no—
50C23×C53—42no—
51C23×C53C22⋊C412yes—
52C23×C53C22×C411yescharacteristic
53C23×C53C2×D411yescharacteristic
54C23×C53C2×D411yescharacteristic
55C23×C53C2×D412yes—
56C23×C53#4112yes—
57D4×C53—24no—
58D4×C53—24no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—24no—
62C2×C4×C53—44no—
63C2×C4×C53—82no—
64C2×C4×C53C22⋊C412yes—
65C2×Dic53—22no—
66C2×Dic53—22no—
67C2×Dic53—24no—
68C2×Dic53—42no—
69C2×Dic53—41no—
70C2×Dic53—42no—
71C2×Dic53—41no—
72C2×Dic53—44no—
73C2×Dic53—42no—
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—44no—
90Dic53—42no—
91Dic53—42no—
92Dic53—82no—
93#1890—531nomaximal, Sylow
94C2×C53—21no—
95C2×C53—22no—
96C2×C53—21no—
97C2×C53—22no—
98C2×C53—42no—
99C2×C53#38611yescharacteristic
100C2×C53#45812yes—
101C2×#123—531no—
102#518D5311yescharacteristic
103C2×#99—531no—
104#381—532no—
105#342—532no—
106C53#189011yescharacteristic, complemented, Sylow
107C22×D4C2×D5311yescharacteristic
108C23×C4—531no—
109#123—534no—
110#123—1062no—
111C4×D4—24no—
112#117—21no—
113C2×C4⋊C4—1061no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—531no—
116C2×C22⋊C4—532no—
117C2×C22⋊C4—1061no—
118C2×C22⋊C4C2×D5312yes—
119#99—1062no—
120#95—1064no—
121C24C22⋊D5311yescharacteristic
122C24C22×D5311yescharacteristic
123C2×D4—22no—
124C2×D4—22no—
125C2×D4C4×D5312yes—
126C2×D4D21212yes—
127C22×C4—22no—
128C22×C4—532no—
129C22×C4—1061no—
130C22×C4—1062no—
131C22×C4—1062no—
132C22×C4—1061no—
133C22×C4—1061no—
134C22×C4—1062no—
135C22×C4C22⋊D5311yescharacteristic
136C22×C4C22⋊D5314yes—
137C4⋊C4—22no—
138C4⋊C4—1064no—
139C22⋊C4—22no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—532no—
143C22⋊C4—1061no—
144C22⋊C4—1064no—
145C22⋊C4—1062no—
146C22⋊C4—1062no—
147C42—22no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—42no—
152C23#1171992112yes—
153C23C2×C4×D5311yescharacteristic
154C23C2×D21211yescharacteristic
155C23#1171994512yes—
156C23C2×C22⋊D5311yescharacteristic
157C23C2×C22⋊D5312yes—
158D4—24no—
159D4—24no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—24no—
163C2×C4—44no—
164C2×C4—82no—
165C2×C4—1062no—
166C2×C4—1062no—
167C2×C4—1064no—
168C2×C4—2124no—
169C2×C4—2122no—
170C2×C4—2122no—
171C2×C4—2121no—
172C2×C4—2122no—
173C2×C4—2121no—
174C2×C4#1171992112yes—
175C22—24no—
176C22—22no—
177C22—22no—
178C22—41no—
179C22—41no—
180C22—42no—
181C22—42no—
182C22—42no—
183C22—81no—
184C22C2×#1171992111yescharacteristic
185C22C4×C22⋊D5312yes—
186C22#1313318611yescharacteristic
187C22#1313319112yes—
188C22#1313320011yescharacteristic
189C4—22no—
190C4—44no—
191C4—4242no—
192C4—2122no—
193C4—2122no—
194C2—22no—
195C2—21no—
196C2—21no—
197C2—22no—
198C2—42no—
199C2#1681050211yescharacteristic
200C2#1681081312yes—
subgroupG/H
centerC22#13133186
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#518C2
socleC22×C53#120
hypercenter#518D53
hypocenterC53#1890
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
3. #16810484C2semidirect
4. #16810390C2semidirect
8. #13133198C4semidirect
35. C22×Dic53C2×C4semidirect
45. C22⋊Dic53C2×C4semidirect
106. C53#1890semidirect