Finite Group Databasebeta

← #20275258

Subgroups — #20275258

Order 6784.

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

normal subgroups only

This group has 380 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
1#16811098C211yesmaximal, characteristic, complemented
2C2×#13133132C212yesmaximal, complemented
3C2×#13133132C211yesmaximal, characteristic, complemented
4#16810389C212yesmaximal, complemented
5C53×#534C211yesmaximal, characteristic
6D4×Dic53—24no—
7#13133176—21no—
8C2×C22⋊Dic53—21no—
9C2×C22⋊Dic53C2212yes—
10C2×C22⋊Dic53C2212yes—
11#13133132—24no—
12#13133132—22no—
13C22×D4×C53C2211yescharacteristic
14C22×D4×C53C2211yescharacteristic
15#123×C53—21no—
16#123×C53C412yescomplemented
17#121×C53—22no—
18#120×C53—21no—
19#120×C53C412yescomplemented
20C2×C22⋊C4×C53C2211yescharacteristic
21C24×C53C2×C411yescharacteristic, complemented
22C24×C53D411yescharacteristic
23C24×C53C2311yescharacteristic
24C2×D4×C53—22no—
25C2×D4×C53—22no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—22no—
29C2×D4×C53—22no—
30C2×D4×C53—22no—
31C2×D4×C53C2×C412yes—
32C2×D4×C53D412yes—
33C22×C4×C53—21no—
34C22×C4×C53C2×C411yescharacteristic, complemented
35C22×C4×C53D411yescharacteristic
36C22×Dic53—22no—
37C22×Dic53—22no—
38C22×Dic53—22no—
39C4⋊C4×C53—22no—
40C22⋊C4×C53—22no—
41C22⋊C4×C53—21no—
42C22⋊C4×C53—22no—
43C22⋊C4×C53C2×C412yescomplemented
44C22⋊Dic53—22no—
45C22⋊Dic53—22no—
46C22⋊Dic53—22no—
47C22⋊Dic53—22no—
48C22⋊Dic53D414yescomplemented
49C4⋊Dic53—22no—
50C4×Dic53—22no—
51C23×C53—21no—
52C23×C53—21no—
53C23×C53—21no—
54C23×C53—22no—
55C23×C53—81no—
56C23×C53—81no—
57C23×C53—81no—
58C23×C53C22⋊C412yes—
59C23×C53C22×C411yescharacteristic
60C23×C53C2×D412yes—
61C23×C53C2×D411yescharacteristic
62C23×C53C2×D411yescharacteristic
63C23×C53#4112yes—
64D4×C53—24no—
65D4×C53—24no—
66D4×C53—41no—
67D4×C53—42no—
68D4×C53—41no—
69D4×C53—82no—
70C2×C4×C53—21no—
71C2×C4×C53—21no—
72C2×C4×C53—22no—
73C2×C4×C53—21no—
74C2×C4×C53—22no—
75C2×C4×C53—81no—
76C2×C4×C53C22⋊C412yes—
77C2×Dic53—24no—
78C2×Dic53—42no—
79C2×Dic53—82no—
80C2×Dic53—82no—
81C2×Dic53—82no—
82C22×C53—22no—
83C22×C53—42no—
84C22×C53—42no—
85C22×C53—41no—
86C22×C53—41no—
87C22×C53—41no—
88C22×C53—42no—
89C22×C53—42no—
90C22×C53—42no—
91C22×C53—41no—
92C22×C53—81no—
93C22×C53—82no—
94C22×C53C2×C22⋊C411yescharacteristic
95C22×C53C4×D412yes—
96C22×C53#12011yescharacteristic
97C22×C53#12112yes—
98C22×C53#12311yescharacteristic
99C4×C53—22no—
100C4×C53—42no—
101C4×C53—41no—
102Dic53—44no—
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—81no—
112C2×C53#38611yescharacteristic
113#534D5311yescharacteristic
114#518—531no—
115C2×#99—532no—
116C2×#99—531no—
117#342—532no—
118C53#191311yescharacteristic, complemented, Sylow
119C22×D4C2×D5311yescharacteristic
120C22×D4C2×D5311yescharacteristic
121#123—21no—
122#123Dic5312yescomplemented
123#121—22no—
124#120—21no—
125#120Dic5312yescomplemented
126C4×D4—1064no—
127#117—1061no—
128C2×C22⋊C4—532no—
129C2×C22⋊C4—532no—
130C2×C22⋊C4—1061no—
131C2×C22⋊C4C2×D5311yescharacteristic
132#99—1064no—
133#99—1062no—
134C24C2×Dic5311yescharacteristic, complemented
135C24C22⋊D5311yescharacteristic
136C24C22×D5311yescharacteristic
137C2×D4—22no—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4—22no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—22no—
144C2×D4C2×Dic5312yes—
145C2×D4C22⋊D5312yes—
146C22×C4—21no—
147C22×C4—1062no—
148C22×C4—1062no—
149C22×C4—1062no—
150C22×C4C2×Dic5311yescharacteristic, complemented
151C22×C4C22⋊D5311yescharacteristic
152C4⋊C4—22no—
153C4⋊C4—1062no—
154C22⋊C4—21no—
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—534no—
158C22⋊C4—1062no—
159C22⋊C4—1062no—
160C22⋊C4—1062no—
161C22⋊C4—1062no—
162C22⋊C4C2×Dic5312yescomplemented
163C42—1062no—
164C23—21no—
165C23—21no—
166C23—21no—
167C23—22no—
168C23—81no—
169C23—81no—
170C23—81no—
171C23C22⋊Dic5312yes—
172C23D4×D5312yes—
173C23#1171994712yes—
174C23C22×Dic5311yescharacteristic
175C23C2×C22⋊D5311yescharacteristic
176C23C2×C22⋊D5311yescharacteristic
177D4—24no—
178D4—24no—
179D4—41no—
180D4—42no—
181D4—41no—
182D4—82no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—22no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—81no—
189C2×C4—1064no—
190C2×C4—4242no—
191C2×C4—4242no—
192C2×C4—4242no—
193C2×C4—2122no—
194C2×C4C22⋊Dic5312yes—
195C22—22no—
196C22—41no—
197C22—42no—
198C22—42no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—42no—
204C22—41no—
205C22—81no—
206C22—82no—
207C22C2×C22⋊Dic5311yescharacteristic
208C22D4×Dic5312yes—
209C22#1313318511yescharacteristic
210C22#1313318912yes—
211C22#1313319811yescharacteristic
212C4—22no—
213C4—42no—
214C4—41no—
215C4—4242no—
216C4—2124no—
217C2—21no—
218C2—22no—
219C2—21no—
220C2—22no—
221C2—21no—
222C2—42no—
223C2—81no—
224C2#1681050311yescharacteristic
subgroupG/H
centerC2#16810503
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#534C2
socleC2×C53#386
hypercenter#534D53
hypocenterC53#1913
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C24 ◁ #534

Complements

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

NH ≅ G/Ntype
1. #16811098C2semidirect
2. C2×#13133132C2semidirect
3. C2×#13133132C2semidirect
4. #16810389C2semidirect
16. #123×C53C4semidirect
19. #120×C53C4semidirect
21. C24×C53C2×C4semidirect
34. C22×C4×C53C2×C4semidirect
43. C22⋊C4×C53C2×C4semidirect
48. C22⋊Dic53D4semidirect
118. C53#1913semidirect
122. #123Dic53semidirect
125. #120Dic53semidirect
134. C24C2×Dic53semidirect
150. C22×C4C2×Dic53semidirect
162. C22⋊C4C2×Dic53semidirect