Finite Group Databasebeta

← #20275123

Subgroups — #20275123

Order 6784.

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

normal subgroups only

This group has 578 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×#13133185C211yesmaximal, characteristic
2C2×#13133132C211yesmaximal, characteristic, complemented
3#16810478C212yesmaximal, complemented
4#16810390C212yesmaximal, complemented
5C53×#518C211yesmaximal, characteristic, complemented
6C24×D53C2211yescharacteristic
7C22×C22⋊D53—21no—
8#13133185—22no—
9#13133185C414yescomplemented
10C2×C22⋊Dic53C2212yes—
11C2×C22⋊Dic53C2211yescharacteristic
12C2×#11719921—21no—
13C2×#11719921—24no—
14#13133132—22no—
15C22×D4×C53C2211yescharacteristic
16C4×D4×C53—24no—
17#117×C53—21no—
18C2×C22⋊C4×C53—21no—
19C2×C22⋊C4×C53C2212yes—
20C24×C53D411yescharacteristic
21C24×C53C2311yescharacteristic
22C23×D53—22no—
23C23×D53—21no—
24C23×D53—21no—
25C23×D53—22no—
26C23×D53—44no—
27C23×D53C2×C412yescomplemented
28C2×D4×C53—22no—
29C2×D4×C53—22no—
30C2×D4×C53C2×C412yes—
31C2×D4×C53D412yes—
32C22×C4×C53—22no—
33C22×C4×C53D411yescharacteristic
34C22×C4×C53D414yescomplemented
35C2×C22⋊D53—24no—
36C2×C22⋊D53—44no—
37C22×Dic53—21no—
38C22×Dic53—22no—
39C4⋊C4×C53—22no—
40C22⋊C4×C53—22no—
41C22⋊C4×C53—22no—
42C22⋊C4×C53—22no—
43C42×C53—22no—
44C22⋊Dic53—21no—
45C22⋊Dic53—22no—
46C22⋊Dic53—22no—
47C22⋊Dic53C2×C412yescomplemented
48#11719921—24no—
49#11719921—48no—
50C23×C53—21no—
51C23×C53—21no—
52C23×C53—42no—
53C23×C53—42no—
54C23×C53C22⋊C412yescomplemented
55C23×C53C22×C411yescharacteristic
56C23×C53C2×D411yescharacteristic
57C23×C53C2×D411yescharacteristic
58C23×C53C2×D412yes—
59C23×C53#4112yes—
60C22×D53—24no—
61C22×D53—28no—
62C22×D53—22no—
63C22×D53—24no—
64C22×D53—44no—
65C22×D53—44no—
66C22×D53—48no—
67C22×D53—44no—
68C22×D53—42no—
69C22×D53—41no—
70C22×D53—42no—
71C22×D53—41no—
72D4×C53—24no—
73D4×C53—24no—
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—24no—
77C2×C4×C53—44no—
78C2×C4×C53—82no—
79C2×C4×C53C22⋊C412yescomplemented
80C22⋊D53—48no—
81C2×Dic53—22no—
82C2×Dic53—42no—
83C2×Dic53—44no—
84C2×Dic53—42no—
85C22×C53—24no—
86C22×C53—22no—
87C22×C53—22no—
88C22×C53—41no—
89C22×C53—41no—
90C22×C53—42no—
91C22×C53—42no—
92C22×C53—42no—
93C22×C53—81no—
94C22×C53C2×C22⋊C411yescharacteristic
95C22×C53C4×D412yes—
96C22×C53#12011yescharacteristic
97C22×C53#12112yes—
98C22×C53#12311yescharacteristic
99C2×D53—24no—
100C2×D53—28no—
101C2×D53—42no—
102C2×D53—48no—
103C2×D53—44no—
104C2×D53—44no—
105C2×D53—42no—
106C2×D53—44no—
107C4×C53—22no—
108C4×C53—44no—
109Dic53—42no—
110Dic53—82no—
111#1889—531nomaximal, Sylow
112C2×C53—21no—
113C2×C53—22no—
114C2×C53—22no—
115C2×C53—21no—
116C2×C53—42no—
117C2×C53#38611yescharacteristic
118C2×C53B(4,𝔽2)12yes—
119D53—44no—
120C2×#120—531no—
121#518D5311yescharacteristic, complemented
122C2×#99—531no—
123#379—532no—
124#342—532no—
125C53#188911yescharacteristic, complemented, Sylow
126C25—531no—
127C22×D4—1061no—
128C22×D4C2×D5311yescharacteristic
129#120—534no—
130#120—1062no—
131C4×D4—24no—
132#117—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—531no—
135C2×C22⋊C4—532no—
136C2×C22⋊C4—1061no—
137C2×C22⋊C4—1064no—
138C2×C22⋊C4C2×D5312yes—
139#99—1062no—
140C24—532no—
141C24—1062no—
142C24—1061no—
143C24—1061no—
144C24—1062no—
145C24—2124no—
146C24C22⋊D5311yescharacteristic
147C24C22×D5311yescharacteristic
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—1064no—
151C2×D4—2124no—
152C2×D4C4×D5312yes—
153C2×D4D21212yes—
154C22×C4—22no—
155C22×C4—1061no—
156C22×C4—1062no—
157C22×C4C22⋊D5311yescharacteristic
158C22×C4C22⋊D5314yescomplemented
159C4⋊C4—22no—
160C22⋊C4—22no—
161C22⋊C4—22no—
162C22⋊C4—22no—
163C22⋊C4—532no—
164C22⋊C4—1061no—
165C22⋊C4—1064no—
166C22⋊C4—1062no—
167C22⋊C4—1062no—
168C22⋊C4—2128no—
169C42—22no—
170C23—21no—
171C23—21no—
172C23—42no—
173C23—42no—
174C23—1068no—
175C23—1064no—
176C23—1062no—
177C23—1064no—
178C23—2124no—
179C23—2122no—
180C23—2124no—
181C23—2124no—
182C23—2128no—
183C23—2121no—
184C23—2122no—
185C23—2121no—
186C23#1171992112yescomplemented
187C23C2×C4×D5311yescharacteristic
188C23C2×D21211yescharacteristic
189C23#1171994512yes—
190C23C2×C22⋊D5311yescharacteristic
191C23C2×C22⋊D5312yes—
192D4—24no—
193D4—24no—
194D4—2128no—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—24no—
198C2×C4—44no—
199C2×C4—82no—
200C2×C4—1062no—
201C2×C4—2124no—
202C2×C4—2122no—
203C2×C4—2122no—
204C2×C4#1171992112yescomplemented
205C22—24no—
206C22—22no—
207C22—22no—
208C22—41no—
209C22—41no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—81no—
214C22—1064no—
215C22—1068no—
216C22—2128no—
217C22—2122no—
218C22—2124no—
219C22—2124no—
220C22—2122no—
221C22—2124no—
222C22C2×#1171992111yescharacteristic
223C22C4×C22⋊D5312yes—
224C22#1313318611yescharacteristic
225C22#1313319112yes—
226C22#1313320011yescharacteristic
227C4—22no—
228C4—44no—
229C4—4242no—
230C4—2122no—
231C2—22no—
232C2—21no—
233C2—21no—
234C2—22no—
235C2—42no—
236C2—2124no—
237C2#1681050211yescharacteristic
238C2#1681081012yes—
subgroupG/H
centerC22#13133186
commutatorC23×C53C22×C4
FrattiniC24C22×D53
FittingC53×#518C2
socleC22×C53#120
hypercenter#518D53
hypocenterC53#1889
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C24 ◁ #518

Complements

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

NH ≅ G/Ntype
2. C2×#13133132C2semidirect
3. #16810478C2semidirect
4. #16810390C2semidirect
5. C53×#518C2semidirect
9. #13133185C4semidirect
27. C23×D53C2×C4semidirect
34. C22×C4×C53D4semidirect
47. C22⋊Dic53C2×C4semidirect
54. C23×C53C22⋊C4semidirect
79. C2×C4×C53C22⋊C4semidirect
121. #518D53semidirect
125. C53#1889semidirect
158. C22×C4C22⋊D53semidirect
186. C23#11719921semidirect
204. C2×C4#11719921semidirect