Finite Group Databasebeta

← #20280588

Subgroups — #20280588

Order 6784.

512 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 131 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×#13133208C211yesmaximal, characteristic, complemented
2C2×#13133177C211yesmaximal, characteristic
3#16810560C211yesmaximal, characteristic, complemented
4#16810559C211yesmaximal, characteristic, complemented
5#16810557C212yesmaximal, complemented
6#16810549C211yesmaximal, characteristic, complemented
7#16810538C212yesmaximal
8#16810498C212yesmaximal, complemented
9#16810497C212yesmaximal, complemented
10#16810490C211yesmaximal, characteristic, complemented
11C53×#397C211yesmaximal, characteristic, complemented
12C22×C22⋊D53C2211yescharacteristic
13C23×Dic53C2211yescharacteristic
14C22×C4×D53C2211yescharacteristic
15#13133208—24no—
16#13133177—24no—
17C2×C4⋊Dic53C2212yes—
18C2×C4⋊Dic53C2211yescharacteristic
19C2×C4⋊Dic53C2212yes—
20C2×#11719919C2212yes—
21C2×#11719919C2211yescharacteristic
22C2×#11719919C2212yes—
23C2×C22⋊Dic53C2212yes—
24C2×#11719921C2212yes—
25C2×#11719921C2211yescharacteristic
26C2×#11719921C2212yes—
27C2×C4×Dic53C2211yescharacteristic
28#13133124C2212yes—
29#13133124C2212yes—
30#13133124C2212yes—
31#13133124C2211yescharacteristic
32C2×C4⋊C4×C53C2211yescharacteristic
33C2×C22⋊C4×C53C2212yes—
34C2×C22⋊C4×C53C2211yescharacteristic
35#95×C53C2212yes—
36#95×C53C2211yescharacteristic
37C24×C53C2311yescharacteristic
38C23×D53C2311yescharacteristic
39C22×C4×C53C2312yes—
40C22×C4×C53C2311yescharacteristic
41C22×C4×C53C2312yes—
42C22×C4×C53C2311yescharacteristic
43C2×C22⋊D53—22no—
44C2×C22⋊D53—22no—
45C2×C22⋊D53—22no—
46C22×Dic53—22no—
47C22×Dic53—42no—
48C22×Dic53C2312yes—
49C22×Dic53C2311yescharacteristic
50C22×Dic53C2312yes—
51C22×Dic53C2311yescharacteristic
52C22×Dic53C2311yescharacteristic
53C2×C4×D53—22no—
54C2×C4×D53—42no—
55C4⋊C4×C53—22no—
56C22⋊C4×C53—24no—
57C22⋊C4×C53—22no—
58C22⋊Dic53—42no—
59#11719921—24no—
60#11719921—22no—
61#11719921—42no—
62C4⋊Dic53—22no—
63C4⋊Dic53—24no—
64C4⋊Dic53—42no—
65#11719919—22no—
66#11719919—24no—
67#11719919—42no—
68C4×Dic53D414yes—
69C23×C53—21no—
70C23×C53—21no—
71C23×C53—22no—
72C23×C53—22no—
73C23×C53#4112yes—
74C23×C53C2411yescharacteristic
75C22×D53—21no—
76C22×D53—21no—
77C22×D53—22no—
78C22×D53—22no—
79C22×D53#4112yes—
80C2×C4×C53—22no—
81C2×C4×C53—21no—
82C2×C4×C53—22no—
83C2×C4×C53—22no—
84C2×C4×C53—22no—
85C2×C4×C53—22no—
86C2×C4×C53—22no—
87C2×C4×C53—22no—
88C2×C4×C53—22no—
89C2×C4×C53C2×D412yes—
90C22⋊D53—28no—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—22no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—22no—
97C2×Dic53—22no—
98C2×Dic53—22no—
99C2×Dic53—22no—
100C2×Dic53—22no—
101C2×Dic53—22no—
102C2×Dic53—44no—
103C2×Dic53C2×D412yes—
104C2×Dic53C2×D412yes—
105C4×D53—44no—
106C22×C53—22no—
107C22×C53—24no—
108C22×C53—41no—
109C22×C53—41no—
110C22×C53—42no—
111C22×C53C22×D411yescharacteristic
112C22×C53C2×#4111yescharacteristic
113C22×C53C2×#4111yescharacteristic
114C22×C53H(5,𝔽2)12yes—
115C22×C532−1+412yes—
116C2×D53—22no—
117C2×D53—24no—
118C2×D53—41no—
119C2×D53—41no—
120C2×D53—42no—
121C4×C53—22no—
122C4×C53—42no—
123C4×C53—41no—
124C4×C53—42no—
125Dic53—22no—
126Dic53—22no—
127Dic53—42no—
128Dic53—41no—
129Dic53—42no—
130#2633—531nomaximal, Sylow
131C2×C53—42no—
132C2×C53#52511yescharacteristic
133C2×C53#53511yescharacteristic
134C2×C53#53611yescharacteristic
135C2×C53#53912yes—
136C2×C53#53912yes—
137D53—42no—
138C2×#125—531no—
139C2×C4×D4—531no—
140#399—531no—
141#399—531no—
142#398—532no—
143#397—531no—
144#397D5311yescharacteristic, complemented
145#394—532no—
146#385—532no—
147#385—532no—
148#383—531no—
149C53#263311yescharacteristic, complemented, Sylow
150C22×D4—531no—
151C23×C4—531no—
152C23×C4—531no—
153#125—1064no—
154C4×D4—1064no—
155C2×C4⋊C4—531no—
156C2×C4⋊C4—532no—
157C2×C4⋊C4—532no—
158C2×C4⋊C4—531no—
159C2×C4⋊C4—532no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4C2×D5311yescharacteristic
162C2×C22⋊C4—531no—
163C2×C22⋊C4—532no—
164C2×C22⋊C4—532no—
165C2×C22⋊C4—532no—
166C2×C22⋊C4C2×D5311yescharacteristic
167C2×C22⋊C4C2×D5312yes—
168C2×C42—531no—
169#95—532no—
170#95—532no—
171#95—531no—
172#95—532no—
173#95C2×D5311yescharacteristic
174#95C2×D5312yes—
175C24—531no—
176C24C22×D5311yescharacteristic
177C2×D4—1062no—
178C2×D4—1062no—
179C2×D4—1062no—
180C22×C4—531no—
181C22×C4—531no—
182C22×C4—532no—
183C22×C4—531no—
184C22×C4—532no—
185C22×C4—1062no—
186C22×C4—1062no—
187C22×C4—2122no—
188C22×C4—2122no—
189C22×C4C22×D5312yes—
190C22×C4C22×D5312yes—
191C22×C4C22×D5311yescharacteristic
192C22×C4C22×D5311yescharacteristic
193C4⋊C4—22no—
194C4⋊C4—1062no—
195C4⋊C4—1064no—
196C4⋊C4—1064no—
197C4⋊C4—1062no—
198C4⋊C4—2122no—
199C4⋊C4—2122no—
200C22⋊C4—24no—
201C22⋊C4—22no—
202C22⋊C4—1064no—
203C22⋊C4—1062no—
204C22⋊C4—2122no—
205C22⋊C4—2122no—
206C42—534no—
207C23—21no—
208C23—21no—
209C23—22no—
210C23—22no—
211C23—532no—
212C23—1061no—
213C23—1061no—
214C23—1062no—
215C23—1062no—
216C23#1171994712yes—
217C23C23×D5311yescharacteristic
218D4—1068no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—532no—
229C2×C4—532no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1062no—
233C2×C4—1061no—
234C2×C4—1062no—
235C2×C4—1062no—
236C2×C4—1062no—
237C2×C4—1062no—
238C2×C4—1062no—
239C2×C4—1062no—
240C2×C4—1062no—
241C2×C4—2124no—
242C2×C4—2124no—
243C2×C4D4×D5312yes—
244C22—22no—
245C22—24no—
246C22—41no—
247C22—41no—
248C22—42no—
249C22—1062no—
250C22—1064no—
251C22—2121no—
252C22—2121no—
253C22—2122no—
254C22C2×D4×D5311yescharacteristic
255C22#41×D5311yescharacteristic
256C22C2×#1171994711yescharacteristic
257C22#1313326312yes—
258C22#1313326412yes—
259C4—22no—
260C4—42no—
261C4—41no—
262C4—42no—
263C4—1062no—
264C4—1062no—
265C4—2121no—
266C4—2122no—
267C4—2122no—
268C2—42no—
269C2—2122no—
270C2#1681113311yescharacteristic
271C2#1681119111yescharacteristic
272C2#1681119511yescharacteristic
273C2#1681121412yes—
274C2#1681121712yes—
subgroupG/H
center, FrattiniC23C23×D53
commutator, socleC23×C53C24
FittingC53×#397C2
hypercenter#397D53
hypocenterC53#2633
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#13133208C2semidirect
3. #16810560C2semidirect
4. #16810559C2semidirect
5. #16810557C2semidirect
6. #16810549C2semidirect
8. #16810498C2semidirect
9. #16810497C2semidirect
10. #16810490C2semidirect
11. C53×#397C2semidirect
144. #397D53semidirect
149. C53#2633semidirect