Finite Group Databasebeta

← #20282770

Subgroups — #20282770

Order 6784.

464 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 125 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×#13133207C211yesmaximal, characteristic, complemented
2C2×#13133206C211yesmaximal, characteristic
3#16810558C212yesmaximal, complemented
4#16810552C212yesmaximal, complemented
5#16810547C212yesmaximal, complemented
6#16810545C212yesmaximal, complemented
7#16810542C211yesmaximal, characteristic, complemented
8#16810541C211yesmaximal, characteristic
9#16810540C211yesmaximal, characteristic
10#16810487C211yesmaximal, characteristic, complemented
11C53×#395C211yesmaximal, characteristic, complemented
12C22×D212C2211yescharacteristic
13#13133207—24no—
14#13133206—24no—
15C2×C4⋊Dic53C2211yescharacteristic
16C2×#11719919C2211yescharacteristic
17C2×#11719919C2211yescharacteristic
18C2×#11719919C2211yescharacteristic
19C2×#11719919C2211yescharacteristic
20C2×#11719919C2212yes—
21C2×#11719919C2212yes—
22C2×#11719921C2212yes—
23C2×#11719921C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2212yes—
26C2×#11719921C2212yes—
27C2×#11719921C2212yes—
28C2×C4×Dic53C2211yescharacteristic
29#13133124C2211yescharacteristic
30#13133124C2212yes—
31#13133124C2212yes—
32C22×Q8×C53C2211yescharacteristic
33C2×C4⋊C4×C53C2211yescharacteristic
34C2×C4⋊C4×C53C2212yes—
35#95×C53C2211yescharacteristic
36#95×C53C2212yes—
37C23×D53C2312yes—
38C2×Q8×C53—22no—
39C2×Q8×C53—42no—
40C22×C4×C53C2312yes—
41C22×C4×C53C2311yescharacteristic
42C22×C4×C53C2312yes—
43C22×C4×C53C2311yescharacteristic
44C22×C4×C53C2311yescharacteristic
45C22×Dic53C2312yes—
46C22×Dic53C2311yescharacteristic
47C22×Dic53C2311yescharacteristic
48C22×Dic53C2311yescharacteristic
49C22×Dic53C2311yescharacteristic
50C2×D212—22no—
51C2×D212—42no—
52C4⋊C4×C53—22no—
53C4⋊C4×C53—42no—
54#11719921—24no—
55#11719921—24no—
56#11719921—42no—
57#11719921—42no—
58#11719921—42no—
59#11719921—42no—
60C4⋊Dic53—22no—
61#11719919—22no—
62#11719919—22no—
63#11719919—22no—
64#11719919—22no—
65#11719919—42no—
66#11719919—42no—
67C4×Dic53D414yes—
68C23×C53C2411yescharacteristic
69C22×D53—22no—
70C22×D53—22no—
71C22×D53—22no—
72C22×D53—22no—
73C22×D53—22no—
74C22×D53—22no—
75C22×D53—22no—
76Q8×C53—44no—
77C2×C4×C53—22no—
78C2×C4×C53—22no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—21no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—21no—
88C2×C4×C53—22no—
89C2×C4×C53C2×D412yes—
90C2×Dic53—21no—
91C2×Dic53—22no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—21no—
99C2×Dic53—21no—
100C2×Dic53—22no—
101C2×Dic53C2×D412yes—
102C2×Dic53C2×D412yes—
103D212—44no—
104C22×C53C22×D411yescharacteristic
105C22×C53H(5,𝔽2)12yes—
106C22×C53H(5,𝔽2)11yescharacteristic
107C22×C532−1+412yes—
108C22×C532−1+411yescharacteristic
109C2×D53—42no—
110C2×D53—42no—
111C2×D53—42no—
112C2×D53—42no—
113C2×D53—42no—
114C2×D53—42no—
115C2×D53—42no—
116C4×C53—22no—
117C4×C53—41no—
118C4×C53—42no—
119C4×C53—41no—
120C4×C53—42no—
121Dic53—22no—
122Dic53—22no—
123Dic53—41no—
124Dic53—41no—
125Dic53—42no—
126#2820—531nomaximal, Sylow
127C2×C53#53511yescharacteristic
128C2×C53#53611yescharacteristic
129C2×C53#53711yescharacteristic
130C2×C53#56212yes—
131C2×C53#56312yes—
132D53—82no—
133C2×#125—531no—
134C2×#124—531no—
135#399—532no—
136#397—532no—
137#396—532no—
138#396—532no—
139#395—531no—
140#395D5311yescharacteristic, complemented
141#394—531no—
142#394—531no—
143#382—531no—
144C53#282011yescharacteristic, complemented, Sylow
145C22×Q8C2×D5311yescharacteristic
146C22×D4—531no—
147#125—1064no—
148#124—1064no—
149C2×C4⋊C4—531no—
150C2×C4⋊C4—531no—
151C2×C4⋊C4—531no—
152C2×C4⋊C4—531no—
153C2×C4⋊C4—531no—
154C2×C4⋊C4—532no—
155C2×C4⋊C4—532no—
156C2×C4⋊C4C2×D5311yescharacteristic
157C2×C4⋊C4C2×D5312yes—
158C2×C22⋊C4—532no—
159C2×C22⋊C4—532no—
160C2×C22⋊C4—532no—
161C2×C22⋊C4—532no—
162C2×C22⋊C4—532no—
163C2×C22⋊C4—532no—
164C2×C42—531no—
165#95—531no—
166#95—532no—
167#95—532no—
168#95C2×D5311yescharacteristic
169#95C2×D5312yes—
170C24—532no—
171C2×Q8—22no—
172C2×Q8—42no—
173C2×D4—1062no—
174C2×D4—2122no—
175C22×C4—531no—
176C22×C4—531no—
177C22×C4—532no—
178C22×C4—531no—
179C22×C4—531no—
180C22×C4C22×D5312yes—
181C22×C4C22×D5312yes—
182C22×C4C22×D5311yescharacteristic
183C22×C4C22×D5311yescharacteristic
184C22×C4C22×D5311yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—42no—
187C4⋊C4—1062no—
188C4⋊C4—1062no—
189C4⋊C4—1062no—
190C4⋊C4—1062no—
191C4⋊C4—1062no—
192C4⋊C4—2122no—
193C4⋊C4—2122no—
194C22⋊C4—1064no—
195C22⋊C4—1064no—
196C22⋊C4—2122no—
197C22⋊C4—2122no—
198C22⋊C4—2122no—
199C22⋊C4—2122no—
200C42—534no—
201C23—1062no—
202C23—1062no—
203C23—1062no—
204C23—1062no—
205C23—1062no—
206C23—1062no—
207C23—1062no—
208C23C23×D5311yescharacteristic
209Q8—44no—
210D4—2124no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—532no—
224C2×C4—532no—
225C2×C4—1061no—
226C2×C4—1062no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1062no—
231C2×C4—1062no—
232C2×C4—1061no—
233C2×C4—1062no—
234C2×C4—1062no—
235C2×C4—1061no—
236C2×C4D4×D5312yes—
237C22—2122no—
238C22—2122no—
239C22—2122no—
240C22—2122no—
241C22—2122no—
242C22—2122no—
243C22—2122no—
244C22C2×D4×D5311yescharacteristic
245C22#1313326212yes—
246C22#1313326311yescharacteristic
247C22#1313326512yes—
248C22#1313326511yescharacteristic
249C4—22no—
250C4—42no—
251C4—41no—
252C4—42no—
253C4—41no—
254C4—1062no—
255C4—1062no—
256C4—2122no—
257C4—2121no—
258C4—2121no—
259C2—4242no—
260C2#1681118811yescharacteristic
261C2#1681119811yescharacteristic
262C2#1681120311yescharacteristic
263C2#1681136912yes—
264C2#1681137212yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#395C2
hypercenter#395D53
hypocenterC53#2820
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
1. C2×#13133207C2semidirect
3. #16810558C2semidirect
4. #16810552C2semidirect
5. #16810547C2semidirect
6. #16810545C2semidirect
7. #16810542C2semidirect
10. #16810487C2semidirect
11. C53×#395C2semidirect
140. #395D53semidirect
144. C53#2820semidirect