Finite Group Databasebeta

← #20285758

Subgroups — #20285758

Order 6784.

420 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 109 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#16811227C211yesmaximal, characteristic, complemented
2#16811186C211yesmaximal, characteristic, complemented
3#16811010C211yesmaximal, characteristic, complemented
4#16810987C211yesmaximal, characteristic, complemented
5#16810978C211yesmaximal, characteristic, complemented
6#16810836C212yesmaximal, complemented
7#16810816C212yesmaximal, complemented
8#16810765C212yesmaximal, complemented
9#16810755C212yesmaximal, complemented
10#16810682C211yesmaximal, characteristic, complemented
11C53×#493C211yesmaximal, characteristic, complemented
12C2×#11719947C2211yescharacteristic
13C2×D4×D53C2211yescharacteristic
14C2×#11719923C2212yescomplemented
15C2×#11719922C2212yes—
16#13133213C2211yescharacteristic
17#13133209C2211yescharacteristic, complemented
18#13133205C2211yescharacteristic
19#13133204C2211yescharacteristic
20#13133198—22no—
21#13133189—22no—
22#13133189—22no—
23#13133188C2212yes—
24#13133185—22no—
25D4×Dic53C2212yescomplemented
26#13133174C2211yescharacteristic
27#13133146C2212yescomplemented
28#13133142C2212yescomplemented
29#13133139C2212yes—
30#13133138C2212yes—
31#13133138C2212yescomplemented
32#13133129C2212yes—
33#13133128C2211yescharacteristic, complemented
34C2×D8×C53C2212yes—
35C2×D8×C53C2212yescomplemented
36#127×C53C2211yescharacteristic
37#127×C53C2211yescharacteristic, complemented
38C4×C8×C53C2211yescharacteristic, complemented
39C23×D53—21no—
40C2×D4×C53—22no—
41C2×D4×C53—22no—
42C2×D4×C53C2311yescharacteristic
43C2×D4×C53C2311yescharacteristic
44C2×D4×C53C2312yes—
45C2×C22⋊D53—21no—
46C2×C22⋊D53—22no—
47C2×C22⋊D53—21no—
48C2×C22⋊D53—22no—
49C22×Dic53—21no—
50C22×Dic53—22no—
51#11719947—42no—
52D4×D53—42no—
53C2×D212C2311yescharacteristic
54C2×C4×D53C2311yescharacteristic
55C2×Dic106C2311yescharacteristic
56D8×C53—42no—
57D8×C53—42no—
58C2×C8×C53C2312yes—
59C42×C53C2311yescharacteristic
60C22⋊Dic53—22no—
61C22⋊Dic53—22no—
62C22⋊Dic53—22no—
63#11719923—42no—
64#11719922—42no—
65#11719921—22no—
66#11719921D412yes—
67C4⋊Dic53C2312yes—
68#11719919—22no—
69#11719919D412yes—
70C4×Dic53C2311yescharacteristic
71C2×#88322C2312yes—
72C23×C53—21no—
73C23×C53—22no—
74C23×C53—21no—
75C22×D53—21no—
76C22×D53—42no—
77C22×D53—82no—
78C22×D53C2×D411yescharacteristic
79D4×C53—22no—
80D4×C53—22no—
81D4×C53—22no—
82D4×C53—22no—
83D4×C53—82no—
84D4×C53—82no—
85C2×C4×C53C2×D412yes—
86C2×C4×C53C2411yescharacteristic
87C22⋊D53—42no—
88C22⋊D53—42no—
89C22⋊D53—82no—
90C22⋊D53—82no—
91C2×Dic53—21no—
92C2×Dic53—22no—
93C2×Dic53—42no—
94C2×Dic53—82no—
95C2×Dic53C2×D411yescharacteristic
96C2×Dic53C2×D412yes—
97D212—22no—
98C4×D53—22no—
99Dic106—22no—
100C8×C53—22no—
101#88322—22no—
102C22×C53—41no—
103C22×C53—41no—
104C22×C53—42no—
105C22×C53—42no—
106C22×C53—42no—
107C22×C53—42no—
108C22×C53—42no—
109C22×C53C22×D411yescharacteristic
110C2×D53—21no—
111C2×D53—22no—
112C2×D53—41no—
113C2×D53—42no—
114C2×D53—84no—
115C4×C53—22no—
116C4×C53H(5,𝔽2)12yes—
117Dic53—21no—
118Dic53—22no—
119Dic53—41no—
120Dic53—42no—
121#3220—531nomaximal, Sylow
122C2×C53—81no—
123C2×C53—81no—
124C2×C53—82no—
125C2×C53#53511yescharacteristic
126C2×C53#57612yes—
127D53—41no—
128D53—81no—
129#540—531no—
130#535—531no—
131#496—531no—
132#493D5311yescharacteristic, complemented
133#490—531no—
134#488—531no—
135#463—532no—
136#459—532no—
137#449—532no—
138#447—532no—
139#433—531no—
140C53#322011yescharacteristic, complemented, Sylow
141C2×#41—531no—
142C22×D4—531no—
143C2×SD16—532no—
144C2×D8—532no—
145C2×D8C2×D5312yes—
146C2×D8C2×D5312yescomplemented
147#127—531no—
148#127C2×D5311yescharacteristic
149#127C2×D5311yescharacteristic, complemented
150#125—531no—
151#124—531no—
152#124—531no—
153#123—1062no—
154#121—532no—
155#121—1062no—
156#121—1062no—
157#120—1062no—
158C4×D4—532no—
159#117—531no—
160C4⋊C8—532no—
161#103—532no—
162#102—532no—
163#102—532no—
164#102—532no—
165C22⋊C8—532no—
166#97—531no—
167C4×C8C2×D5311yescharacteristic, complemented
168C24—1061no—
169#41—2122no—
170C2×Q8—531no—
171C2×D4—22no—
172C2×D4—22no—
173C2×D4—531no—
174C2×D4—1062no—
175C2×D4—1061no—
176C2×D4—1062no—
177C2×D4—1061no—
178C2×D4—2122no—
179C2×D4C22×D5311yescharacteristic
180C2×D4C22×D5311yescharacteristic
181C2×D4C22×D5312yes—
182C22×C4—531no—
183C22×C4—1062no—
184C22×C4—1061no—
185SD16—2122no—
186D8—42no—
187D8—42no—
188D8—2122no—
189C2×C8—532no—
190C2×C8C22×D5312yes—
191C4⋊C4—532no—
192C4⋊C4—532no—
193C4⋊C4—1062no—
194C22⋊C4—532no—
195C22⋊C4—1062no—
196C22⋊C4—1062no—
197C22⋊C4—1062no—
198C22⋊C4—1062no—
199C42—531no—
200C42C22×D5311yescharacteristic
201C23—22no—
202C23—21no—
203C23—21no—
204C23—531no—
205C23—1061no—
206C23—4242no—
207C23—2122no—
208Q8—1062no—
209D4—22no—
210D4—22no—
211D4—22no—
212D4—22no—
213D4—82no—
214D4—82no—
215D4—1062no—
216D4—4242no—
217D4—4242no—
218D4—2122no—
219D4—2122no—
220C2×C4—532no—
221C2×C4—531no—
222C2×C4—1062no—
223C2×C4—1061no—
224C2×C4—1062no—
225C2×C4—4242no—
226C2×C4—2122no—
227C2×C4D4×D5312yes—
228C2×C4C23×D5311yescharacteristic
229C8—22no—
230C8—1062no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—42no—
235C22—41no—
236C22—41no—
237C22—42no—
238C22—1062no—
239C22—1061no—
240C22—4244no—
241C22—2122no—
242C22—2121no—
243C22C2×D4×D5311yescharacteristic
244C4—22no—
245C4—1062no—
246C4—1061no—
247C4—2122no—
248C4—2121no—
249C4#1313326212yes—
250C2—81no—
251C2—82no—
252C2—81no—
253C2—4241no—
254C2—2121no—
255C2#1681118611yescharacteristic
256C2#1681142412yes—
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#493C2
socleC22×C53C22×D4
hypercenter#493D53
hypocenterC53#3220
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. #16811227C2semidirect
2. #16811186C2semidirect
3. #16811010C2semidirect
4. #16810987C2semidirect
5. #16810978C2semidirect
6. #16810836C2semidirect
7. #16810816C2semidirect
8. #16810765C2semidirect
9. #16810755C2semidirect
10. #16810682C2semidirect
11. C53×#493C2semidirect
14. C2×#11719923C22semidirect
17. #13133209C22semidirect
25. D4×Dic53C22semidirect
27. #13133146C22semidirect
28. #13133142C22semidirect
31. #13133138C22semidirect
33. #13133128C22semidirect
35. C2×D8×C53C22semidirect
37. #127×C53C22semidirect
38. C4×C8×C53C22semidirect
132. #493D53semidirect
140. C53#3220semidirect
146. C2×D8C2×D53semidirect
149. #127C2×D53semidirect
167. C4×C8C2×D53semidirect