Finite Group Databasebeta

← #20275173

Subgroups — #20275173

Order 6784.

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

normal subgroups only

This group has 378 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×#13133200C211yesmaximal, characteristic
2C2×#13133131C212yesmaximal
3#16810486C211yesmaximal, characteristic, complemented
4#16810389C212yesmaximal, complemented
5C2×#117×C53C211yesmaximal, characteristic, complemented
6C22×C22⋊D53C2211yescharacteristic
7#13133200—22no—
8#13133200—21no—
9#13133200—21no—
10C2×#11719919C412yescomplemented
11C2×C22⋊Dic53C2212yes—
12C2×C22⋊Dic53C2211yescharacteristic
13C2×#11719921C412yescomplemented
14#13133131—24no—
15#13133124—22no—
16C23×C4×C53C2211yescharacteristic
17#117×C53—24no—
18C2×C4⋊C4×C53—21no—
19C2×C22⋊C4×C53C2212yes—
20C2×C42×C53—21no—
21C24×C53C2311yescharacteristic
22C23×D53C2×C411yescharacteristic
23C22×C4×C53—22no—
24C22×C4×C53—22no—
25C22×C4×C53—22no—
26C22×C4×C53C2×C412yes—
27C22×C4×C53D412yes—
28C22×C4×C53D411yescharacteristic
29C22×C4×C53D411yescharacteristic
30C2×C22⋊D53—22no—
31C2×C22⋊D53—21no—
32C2×C22⋊D53—21no—
33C2×C22⋊D53—22no—
34C22×Dic53—22no—
35C22×Dic53C2×C411yescharacteristic
36C22×Dic53C2×C412yes—
37C4⋊C4×C53—24no—
38C22⋊C4×C53—22no—
39C22⋊C4×C53D414yes—
40C42×C53—24no—
41C22⋊Dic53—21no—
42C22⋊Dic53—21no—
43C22⋊Dic53—22no—
44C22⋊Dic53—22no—
45#11719921—22no—
46#11719921—22no—
47#11719919—22no—
48#11719919—22no—
49C23×C53—42no—
50C23×C53C22×C411yescharacteristic
51C23×C53C2×D411yescharacteristic
52C23×C53C2×D411yescharacteristic
53C23×C53C2×D412yes—
54C23×C53#4112yes—
55C22×D53—21no—
56C22×D53—22no—
57C22×D53—42no—
58C2×C4×C53—24no—
59C2×C4×C53—21no—
60C2×C4×C53—21no—
61C2×C4×C53—24no—
62C2×C4×C53—21no—
63C2×C4×C53—21no—
64C2×C4×C53—24no—
65C2×C4×C53—44no—
66C2×C4×C53C22⋊C412yes—
67C2×C4×C53C22⋊C412yes—
68C22⋊D53—41no—
69C22⋊D53—42no—
70C22⋊D53—41no—
71C2×Dic53—21no—
72C2×Dic53—22no—
73C2×Dic53—22no—
74C2×Dic53—42no—
75C2×Dic53—44no—
76C2×Dic53—42no—
77C22×C53—22no—
78C22×C53—22no—
79C22×C53—22no—
80C22×C53—42no—
81C22×C53—42no—
82C22×C53C2×C22⋊C411yescharacteristic
83C22×C53C4×D412yes—
84C22×C53#12011yescharacteristic
85C22×C53#12112yes—
86C22×C53#12311yescharacteristic
87C2×D53—41no—
88C2×D53—42no—
89C2×D53—82no—
90C4×C53—22no—
91C4×C53—22no—
92C4×C53—44no—
93Dic53—41no—
94Dic53—42no—
95Dic53—82no—
96#1899—531nomaximal, Sylow
97C2×C53—22no—
98C2×C53—21no—
99C2×C53—22no—
100C2×C53—21no—
101C2×C53#38611yescharacteristic
102C2×C53#41012yes—
103D53—81no—
104C2×#123—531no—
105C2×#117D5311yescharacteristic, complemented
106C2×#99—532no—
107#381—531no—
108#342—532no—
109C53#189911yescharacteristic, complemented, Sylow
110C22×D4—531no—
111C23×C4C2×D5311yescharacteristic
112#123—1062no—
113#123—1061no—
114#123—1061no—
115#117—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—532no—
118C2×C22⋊C4—531no—
119C2×C22⋊C4—532no—
120C2×C22⋊C4—532no—
121C2×C22⋊C4C2×D5312yes—
122C2×C42—21no—
123#99—1064no—
124#95—1062no—
125C24—531no—
126C24C22×D5311yescharacteristic
127C2×D4—1061no—
128C2×D4—1062no—
129C2×D4—1062no—
130C2×D4—1061no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—531no—
135C22×C4—532no—
136C22×C4—1062no—
137C22×C4C4×D5312yes—
138C22×C4D21212yes—
139C22×C4C22⋊D5311yescharacteristic
140C22×C4C22⋊D5311yescharacteristic
141C4⋊C4—24no—
142C4⋊C4—1062no—
143C4⋊C4—1062no—
144C22⋊C4—22no—
145C22⋊C4—1061no—
146C22⋊C4—1061no—
147C22⋊C4—1062no—
148C22⋊C4—1062no—
149C22⋊C4—1062no—
150C22⋊C4—1062no—
151C22⋊C4C22⋊D5314yes—
152C42—24no—
153C23—42no—
154C23—1062no—
155C23—1061no—
156C23—2122no—
157C23C2×C4×D5311yescharacteristic
158C23C2×D21211yescharacteristic
159C23#1171994512yes—
160C23C2×C22⋊D5311yescharacteristic
161C23C2×C22⋊D5312yes—
162D4—2121no—
163D4—2122no—
164D4—2121no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—24no—
172C2×C4—44no—
173C2×C4—1062no—
174C2×C4—1061no—
175C2×C4—1062no—
176C2×C4—2124no—
177C2×C4—2122no—
178C2×C4—2122no—
179C2×C4#1171992112yes—
180C2×C4#1171992112yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—4242no—
187C22—2121no—
188C22—2122no—
189C22C2×#1171992111yescharacteristic
190C22C4×C22⋊D5312yes—
191C22#1313318611yescharacteristic
192C22#1313319112yes—
193C22#1313320011yescharacteristic
194C4—22no—
195C4—22no—
196C4—44no—
197C4—4242no—
198C4—2121no—
199C4—2122no—
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2—4241no—
205C2#1681050211yescharacteristic
206C2#1681059612yes—
subgroupG/H
centerC22C2×#11719921
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC2×#117×C53C2
socleC22×C53C2×C22⋊C4
hypercenterC2×#117D53
hypocenterC53#1899
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #16810486C2semidirect
4. #16810389C2semidirect
5. C2×#117×C53C2semidirect
10. C2×#11719919C4semidirect
13. C2×#11719921C4semidirect
105. C2×#117D53semidirect
109. C53#1899semidirect