Finite Group Databasebeta

← #20275257

Subgroups — #20275257

Order 6784.

380 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 89 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#16811182C211yesmaximal, characteristic
2C2×#13133132C211yesmaximal, characteristic, complemented
3C2×#13133131C212yesmaximal, complemented
4#16810389C212yesmaximal, complemented
5C53×#518C211yesmaximal, characteristic, complemented
6C22×C22⋊D53C2211yescharacteristic
7#13133200—21no—
8#13133198C412yescomplemented
9#13133189—22no—
10#13133186—21no—
11#13133185C412yescomplemented
12C2×C22⋊Dic53C2212yes—
13C2×C22⋊Dic53C2211yescharacteristic
14#13133132—22no—
15#13133131—24no—
16C22×D4×C53C2211yescharacteristic
17C4×D4×C53—24no—
18#117×C53—21no—
19C2×C22⋊C4×C53—21no—
20C2×C22⋊C4×C53C2212yes—
21C24×C53D411yescharacteristic
22C24×C53C2311yescharacteristic
23C23×D53C2×C411yescharacteristic, complemented
24C2×D4×C53—22no—
25C2×D4×C53—22no—
26C2×D4×C53C2×C412yes—
27C2×D4×C53D412yes—
28C22×C4×C53—22no—
29C22×C4×C53—22no—
30C22×C4×C53D411yescharacteristic
31C2×C22⋊D53—22no—
32C2×C22⋊D53—21no—
33C2×C22⋊D53—21no—
34C2×C22⋊D53—22no—
35C2×C22⋊D53—22no—
36C22×Dic53—21no—
37C22×Dic53—22no—
38C22×Dic53C2×C411yescharacteristic, complemented
39C4⋊C4×C53—22no—
40C22⋊C4×C53—22no—
41C22⋊C4×C53—22no—
42C22⋊C4×C53D414yescomplemented
43C42×C53—22no—
44C22⋊Dic53—22no—
45C22⋊Dic53—21no—
46C22⋊Dic53—22no—
47C22⋊Dic53—22no—
48C22⋊Dic53C2×C412yescomplemented
49#11719921—22no—
50#11719919—22no—
51C23×C53—21no—
52C23×C53—21no—
53C23×C53—81no—
54C23×C53—81no—
55C23×C53C22⋊C412yes—
56C23×C53C22×C411yescharacteristic
57C23×C53C2×D411yescharacteristic
58C23×C53C2×D411yescharacteristic
59C23×C53C2×D412yes—
60C23×C53#4112yes—
61C22×D53—21no—
62C22×D53—22no—
63C22×D53—81no—
64D4×C53—24no—
65D4×C53—24no—
66C2×C4×C53—21no—
67C2×C4×C53—21no—
68C2×C4×C53—24no—
69C2×C4×C53—82no—
70C2×C4×C53—82no—
71C2×C4×C53C22⋊C412yes—
72C22⋊D53—41no—
73C22⋊D53—42no—
74C22⋊D53—41no—
75C22⋊D53—82no—
76C2×Dic53—22no—
77C2×Dic53—21no—
78C2×Dic53—22no—
79C2×Dic53—42no—
80C2×Dic53—81no—
81C2×Dic53—82no—
82C22×C53—22no—
83C22×C53—41no—
84C22×C53—42no—
85C22×C53—42no—
86C22×C53—41no—
87C22×C53—41no—
88C22×C53—42no—
89C22×C53—42no—
90C22×C53—81no—
91C22×C53C2×C22⋊C411yescharacteristic
92C22×C53C4×D412yes—
93C22×C53#12011yescharacteristic
94C22×C53#12112yes—
95C22×C53#12311yescharacteristic
96C2×D53—42no—
97C2×D53—41no—
98C2×D53—82no—
99C4×C53—22no—
100C4×C53—44no—
101Dic53—42no—
102Dic53—41no—
103Dic53—82no—
104#1913—531nomaximal, Sylow
105C2×C53—21no—
106C2×C53—22no—
107C2×C53—21no—
108C2×C53—22no—
109C2×C53—21no—
110C2×C53—42no—
111C2×C53#38611yescharacteristic
112D53—81no—
113#534—531no—
114#518D5311yescharacteristic, complemented
115C2×#99—531no—
116C2×#99—532no—
117#342—532no—
118C53#191311yescharacteristic, complemented, Sylow
119C22×D4—531no—
120C22×D4C2×D5311yescharacteristic
121#123—532no—
122#123—1061no—
123#121—1062no—
124#120—532no—
125#120—1061no—
126C4×D4—24no—
127#117—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—531no—
130C2×C22⋊C4—532no—
131C2×C22⋊C4C2×D5312yes—
132#99—1062no—
133#99—1064no—
134C24—531no—
135C24C22⋊D5311yescharacteristic
136C24C22×D5311yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—1062no—
140C2×D4—1061no—
141C2×D4—1062no—
142C2×D4—1062no—
143C2×D4—1061no—
144C2×D4C4×D5312yes—
145C2×D4D21212yes—
146C22×C4—22no—
147C22×C4—22no—
148C22×C4—531no—
149C22×C4—1061no—
150C22×C4—1062no—
151C22×C4C22⋊D5311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—1062no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—532no—
157C22⋊C4—1061no—
158C22⋊C4—1062no—
159C22⋊C4—1062no—
160C22⋊C4—1062no—
161C22⋊C4—1062no—
162C22⋊C4C22⋊D5314yescomplemented
163C42—22no—
164C23—21no—
165C23—21no—
166C23—81no—
167C23—81no—
168C23—1062no—
169C23—1061no—
170C23—4241no—
171C23#1171992112yes—
172C23C2×C4×D5311yescharacteristic
173C23C2×D21211yescharacteristic
174C23#1171994512yes—
175C23C2×C22⋊D5311yescharacteristic
176C23C2×C22⋊D5312yes—
177D4—24no—
178D4—24no—
179D4—4242no—
180D4—2121no—
181D4—2122no—
182D4—2121no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—24no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—1062no—
189C2×C4—1061no—
190C2×C4—1062no—
191C2×C4—4242no—
192C2×C4—4241no—
193C2×C4—2122no—
194C2×C4#1171992112yes—
195C22—22no—
196C22—42no—
197C22—41no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—81no—
204C22—4242no—
205C22—2122no—
206C22—2121no—
207C22C2×#1171992111yescharacteristic
208C22C4×C22⋊D5312yes—
209C22#1313318611yescharacteristic
210C22#1313319112yes—
211C22#1313320011yescharacteristic
212C4—22no—
213C4—44no—
214C4—4242no—
215C4—2122no—
216C4—2121no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—21no—
221C2—22no—
222C2—42no—
223C2—4241no—
224C2#1681050211yescharacteristic
subgroupG/H
centerC2#16810502
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#518C2
socleC2×C53#386
hypercenter#518D53
hypocenterC53#1913
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#13133132C2semidirect
3. C2×#13133131C2semidirect
4. #16810389C2semidirect
5. C53×#518C2semidirect
8. #13133198C4semidirect
11. #13133185C4semidirect
23. C23×D53C2×C4semidirect
38. C22×Dic53C2×C4semidirect
42. C22⋊C4×C53D4semidirect
48. C22⋊Dic53C2×C4semidirect
114. #518D53semidirect
118. C53#1913semidirect
162. C22⋊C4C22⋊D53semidirect