Finite Group Databasebeta

← #20285502

Subgroups — #20285502

Order 6784.

424 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 113 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#16811133C212yesmaximal, complemented
2#16810966C212yesmaximal, complemented
3#16810945C212yesmaximal, complemented
4#16810864C211yesmaximal, characteristic, complemented
5#16810850C212yesmaximal, complemented
6#16810786C212yesmaximal, complemented
7#16810765C212yesmaximal, complemented
8C2×#13133129C211yesmaximal, characteristic, complemented
9C53×#466C211yesmaximal, characteristic, complemented
10C2×#11719947C2212yes—
11C22×C4×D53C2211yescharacteristic
12C2×Dic212C2211yescharacteristic, complemented
13C2×#11719923C2212yescomplemented
14C22×#88322C2211yescharacteristic
15#13133199—22no—
16#13133197C2212yes—
17#13133196C2212yes—
18#13133188C2212yes—
19#13133185—22no—
20D4×Dic53—22no—
21#13133177—22no—
22#13133175C2212yes—
23#13133151C2211yescharacteristic, complemented
24#13133148C2212yescomplemented
25#13133142C2212yescomplemented
26#13133140C2212yescomplemented
27#13133138C2212yescomplemented
28#13133129C2212yes—
29#13133129C2212yes—
30C2×D8×C53C2211yescharacteristic, complemented
31C22×C8×C53C2211yescharacteristic
32#121×C53C2212yes—
33#107×C53C2211yescharacteristic, complemented
34#102×C53C2212yescomplemented
35C23×D53D411yescharacteristic
36C2×D4×C53—22no—
37C2×D4×C53C2312yes—
38C22×C4×C53C2311yescharacteristic
39C2×C22⋊D53—22no—
40C2×C22⋊D53—22no—
41C22×Dic53—22no—
42C22×Dic53D411yescharacteristic
43#11719947—42no—
44#11719947—42no—
45C2×C4×D53—41no—
46C2×C4×D53—41no—
47C2×C4×D53D412yes—
48C2×C4×D53C2312yes—
49C2×Dic106C2312yes—
50D8×C53—41no—
51C2×C8×C53—41no—
52C2×C8×C53C2311yescharacteristic
53C2×C8×C53C2311yescharacteristic
54C4⋊C4×C53C2312yes—
55C22⋊C4×C53—22no—
56C22⋊Dic53—22no—
57C22⋊Dic53—22no—
58#11719923—42no—
59#11719921—22no—
60#11719921—22no—
61C4⋊Dic53—22no—
62C4⋊Dic53C2312yes—
63#11719919—22no—
64C4×Dic53—22no—
65C2×#88322—41no—
66C2×#88322C2312yes—
67Dic212—41no—
68C23×C53—22no—
69C23×C53C2×D411yescharacteristic
70C22×D53—22no—
71C22×D53—41no—
72C22×D53—41no—
73C22×D53C2×D412yes—
74D4×C53—22no—
75D4×C53—22no—
76D4×C53—44no—
77C2×C4×C53—21no—
78C2×C4×C53—21no—
79C2×C4×C53—22no—
80C2×C4×C53C2×D411yescharacteristic
81C2×C4×C53C2411yescharacteristic
82C22⋊D53—44no—
83C22⋊D53—44no—
84C2×Dic53—22no—
85C2×Dic53—22no—
86C2×Dic53—41no—
87C2×Dic53—44no—
88C2×Dic53C2×D412yes—
89C4×D53—22no—
90C4×D53—22no—
91C4×D53—42no—
92Dic106—22no—
93Dic106—22no—
94C8×C53—21no—
95C8×C53—21no—
96#88322—22no—
97C22×C53—21no—
98C22×C53—21no—
99C22×C53—42no—
100C22×C53—42no—
101C22×C53—42no—
102C22×C53#13512yes—
103C22×C53C22×D411yescharacteristic
104C2×D53—22no—
105C2×D53—22no—
106C2×D53—22no—
107C2×D53—44no—
108C4×C53—21no—
109C4×C53—42no—
110C4×C53H(5,𝔽2)11yescharacteristic
111C4×C53H(5,𝔽2)11yescharacteristic
112Dic53—22no—
113Dic53—42no—
114Dic53—42no—
115#3191—531nomaximal, Sylow
116C2×C53—22no—
117C2×C53—82no—
118C2×C53#53411yescharacteristic
119C2×C53C2×#13511yescharacteristic
120C2×C53#57511yescharacteristic
121D53—42no—
122#525—532no—
123#485—532no—
124#482—532no—
125#467—531no—
126#466D5311yescharacteristic, complemented
127#465—532no—
128#452—532no—
129#449—532no—
130C2×C22⋊C8—531no—
131C53#319111yescharacteristic, complemented, Sylow
132C2×#41—532no—
133C23×C4—531no—
134C2×Q16—531no—
135C2×SD16—532no—
136C2×D8C2×D5311yescharacteristic, complemented
137C22×C8—531no—
138C22×C8C2×D5311yescharacteristic
139#123—1062no—
140C22⋊Q8—532no—
141C22⋊Q8—532no—
142#121—532no—
143#121C2×D5312yes—
144#120—1062no—
145C4×D4—1062no—
146C4×D4—1062no—
147#117—532no—
148#107—531no—
149#107C2×D5311yescharacteristic, complemented
150#106—532no—
151#103—532no—
152#103—532no—
153#102—532no—
154#102C2×D5312yescomplemented
155C22⋊C8—532no—
156C22⋊C8—532no—
157C24—531no—
158#41—2122no—
159#41—2122no—
160C2×Q8—532no—
161C2×D4—22no—
162C2×D4—1062no—
163C2×D4—1062no—
164C2×D4C22×D5312yes—
165C22×C4—531no—
166C22×C4—532no—
167C22×C4—532no—
168C22×C4—1062no—
169C22×C4—2121no—
170C22×C4—2121no—
171C22×C4C22×D5311yescharacteristic
172Q16—2121no—
173SD16—2122no—
174D8—41no—
175C2×C8—41no—
176C2×C8—532no—
177C2×C8—2121no—
178C2×C8C22×D5311yescharacteristic
179C2×C8C22×D5311yescharacteristic
180C4⋊C4—532no—
181C4⋊C4—1062no—
182C4⋊C4—1062no—
183C4⋊C4C22×D5312yes—
184C22⋊C4—22no—
185C22⋊C4—1062no—
186C22⋊C4—1062no—
187C22⋊C4—1062no—
188C22⋊C4—1062no—
189C42—1062no—
190C23—22no—
191C23—532no—
192C23—1062no—
193C23—2121no—
194C23—2121no—
195C23D4×D5311yescharacteristic
196Q8—1062no—
197Q8—1062no—
198D4—22no—
199D4—22no—
200D4—44no—
201D4—2124no—
202D4—2124no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—532no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—1062no—
210C2×C4—1062no—
211C2×C4—2122no—
212C2×C4—2121no—
213C2×C4—2124no—
214C2×C4D4×D5311yescharacteristic
215C2×C4C23×D5311yescharacteristic
216C8—21no—
217C8—21no—
218C8—1062no—
219C22—21no—
220C22—21no—
221C22—42no—
222C22—42no—
223C22—42no—
224C22—1062no—
225C22—1062no—
226C22—1062no—
227C22—2124no—
228C22#1313323612yes—
229C22C2×D4×D5311yescharacteristic
230C4—21no—
231C4—42no—
232C4—1062no—
233C4—2122no—
234C4—2122no—
235C4#1313326211yescharacteristic
236C4#1313326311yescharacteristic
237C2—22no—
238C2—82no—
239C2—2122no—
240C2#1681118111yescharacteristic
241C2C2×#1313323611yescharacteristic
242C2#1681142011yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#466C2
socleC22×C53C22×D4
hypercenter#466D53
hypocenterC53#3191
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. #16811133C2semidirect
2. #16810966C2semidirect
3. #16810945C2semidirect
4. #16810864C2semidirect
5. #16810850C2semidirect
6. #16810786C2semidirect
7. #16810765C2semidirect
8. C2×#13133129C2semidirect
9. C53×#466C2semidirect
12. C2×Dic212C22semidirect
13. C2×#11719923C22semidirect
23. #13133151C22semidirect
24. #13133148C22semidirect
25. #13133142C22semidirect
26. #13133140C22semidirect
27. #13133138C22semidirect
30. C2×D8×C53C22semidirect
33. #107×C53C22semidirect
34. #102×C53C22semidirect
126. #466D53semidirect
131. C53#3191semidirect
136. C2×D8C2×D53semidirect
149. #107C2×D53semidirect
154. #102C2×D53semidirect