Finite Group Databasebeta

← #20286324

Subgroups — #20286324

Order 6784.

520 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 121 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×#13133244C212yesmaximal, complemented
2#16811266C212yesmaximal, complemented
3#16810866C212yesmaximal, complemented
4#16810835C212yesmaximal, complemented
5#16810784C212yesmaximal, complemented
6#16810755C212yesmaximal, complemented
7C4×D424C211yesmaximal, characteristic, complemented
8#16810702C211yesmaximal, characteristic, complemented
9C53×#495C211yesmaximal, characteristic, complemented
10C2×#11719949C2212yes—
11C2×D4×D53C2212yescomplemented
12#13133244—28no—
13C2×#11719924C2212yescomplemented
14C2×D424C2211yescharacteristic
15C2×#11719922C2212yes—
16C2×#11719912C2212yescomplemented
17#13133199—22no—
18#13133193C2212yescomplemented
19#13133190—22no—
20#13133188C2212yes—
21#13133187—22no—
22C4×D212C2211yescharacteristic
23C4×D212C2212yescomplemented
24C22⋊C4×D53—22no—
25#13133151C2211yescharacteristic, complemented
26C4⋊#88322C2211yescharacteristic, complemented
27#13133141C2212yescomplemented
28#13133139C2212yescomplemented
29#13133138C2212yescomplemented
30#13133129C2212yescomplemented
31C2×Q16×C53C2211yescharacteristic, complemented
32C2×SD16×C53C2212yescomplemented
33C2×D8×C53C2211yescharacteristic
34#124×C53C2212yescomplemented
35C4×C8×C53C2211yescharacteristic, complemented
36C23×D53—22no—
37C2×Q8×C53C2312yes—
38C2×D4×C53C2312yes—
39C2×C22⋊D53—22no—
40C2×C22⋊D53—22no—
41#11719949—24no—
42#11719949—42no—
43D4×D53—24no—
44D4×D53—42no—
45C2×D212—22no—
46C2×D212D411yescharacteristic
47C2×D212C2312yes—
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C4×D53C2312yes—
51Q16×C53—41no—
52SD16×C53—24no—
53D8×C53—41no—
54C2×C8×C53C2311yescharacteristic
55C2×C8×C53C2311yescharacteristic
56C22⋊C4×C53—22no—
57C22⋊C4×C53—22no—
58C42×C53C2311yescharacteristic
59C22⋊Dic53—22no—
60#11719924—24no—
61#11719922—24no—
62#11719921—22no—
63#11719921—22no—
64#11719921D412yes—
65C4⋊Dic53D411yescharacteristic
66C4⋊Dic53C2312yes—
67#11719919—22no—
68C2×#88322C2312yes—
69D424D414yescomplemented
70#11719912—24no—
71C23×C53—22no—
72C22×D53—22no—
73C22×D53—44no—
74C22×D53—44no—
75C22×D53—82no—
76C22×D53C2×D412yes—
77Q8×C53—22no—
78Q8×C53—22no—
79D4×C53—22no—
80D4×C53—22no—
81C2×C4×C53—22no—
82C2×C4×C53C2×D411yescharacteristic
83C2×C4×C53C2×D411yescharacteristic
84C2×C4×C53C2411yescharacteristic
85C22⋊D53—44no—
86C22⋊D53—82no—
87C2×Dic53—22no—
88C2×Dic53C2×D412yes—
89D212—22no—
90D212—41no—
91D212—44no—
92D212C2×D414yes—
93C4×D53—22no—
94C4×D53—22no—
95C4×D53—44no—
96C4×D53—82no—
97C8×C53—21no—
98C8×C53C2×D412yes—
99#88322—22no—
100C22×C53—42no—
101C22×C53—42no—
102C22×C53—42no—
103C22×C53C22×D411yescharacteristic
104C2×D53—22no—
105C2×D53—24no—
106C2×D53—22no—
107C2×D53—22no—
108C2×D53—42no—
109C2×D53—42no—
110C2×D53—88no—
111C4×C53—21no—
112C4×C53—21no—
113C4×C53—42no—
114C4×C53C22×D411yescharacteristic
115C4×C53H(5,𝔽2)11yescharacteristic
116Dic53—22no—
117Dic53—42no—
118#3273—531nomaximal, Sylow
119C2×C53—82no—
120C2×C53D4211yescharacteristic
121C2×C53#57511yescharacteristic
122C2×C53#57611yescharacteristic
123D53—42no—
124D53—44no—
125C2×AGL(1,ℤ/8)—532no—
126#546—532no—
127#495D5311yescharacteristic, complemented
128#468—532no—
129#463—532no—
130#452—532no—
131#447—532no—
132C4×D8—531no—
133#436—531no—
134C53#327311yescharacteristic, complemented, Sylow
135C2×#41—532no—
136C22×D4—532no—
137AGL(1,ℤ/8)—1068no—
138C2×Q16C2×D5311yescharacteristic, complemented
139C2×SD16—532no—
140C2×SD16C2×D5312yescomplemented
141C2×D8—532no—
142C2×D8—531no—
143C2×D8C2×D5311yescharacteristic
144C2×M16—532no—
145#124C2×D5312yescomplemented
146#123—1062no—
147C22⋊Q8—532no—
148#121—532no—
149#121—1062no—
150#120—1062no—
151C4×D4—531no—
152C4×D4—532no—
153C2×C22⋊C4—1062no—
154#107—531no—
155C4⋊C8—531no—
156#103—532no—
157#102—532no—
158#102—532no—
159C22⋊C8—532no—
160C4×C8C2×D5311yescharacteristic, complemented
161C24—1062no—
162#41—1064no—
163#41—2122no—
164C2×Q8C22×D5312yes—
165C2×D4—531no—
166C2×D4—532no—
167C2×D4—1062no—
168C2×D4—1062no—
169C2×D4—1064no—
170C2×D4—1062no—
171C2×D4—2122no—
172C2×D4C22×D5312yes—
173C22×C4—532no—
174C22×C4—1062no—
175C22×C4—1062no—
176Q16—41no—
177SD16—24no—
178SD16—1064no—
179D8—41no—
180D8—534no—
181D8—1064no—
182M16—1064no—
183C2×C8—532no—
184C2×C8C22×D5311yescharacteristic
185C2×C8C22×D5311yescharacteristic
186C4⋊C4—531no—
187C4⋊C4—532no—
188C4⋊C4—1062no—
189C22⋊C4—22no—
190C22⋊C4—22no—
191C22⋊C4—532no—
192C22⋊C4—1062no—
193C22⋊C4—1062no—
194C22⋊C4—1062no—
195C42C22×D5311yescharacteristic
196C23—22no—
197C23—532no—
198C23—1062no—
199C23—4242no—
200C23—2124no—
201C23—2124no—
202Q8—22no—
203Q8—22no—
204D4—22no—
205D4—22no—
206D4—534no—
207D4—1062no—
208D4—4242no—
209D4—2121no—
210D4—2124no—
211D4—2124no—
212C2×C4—22no—
213C2×C4—532no—
214C2×C4—1062no—
215C2×C4—1062no—
216C2×C4—1062no—
217C2×C4—4242no—
218C2×C4—2124no—
219C2×C4D4×D5311yescharacteristic
220C2×C4D4×D5311yescharacteristic
221C2×C4C23×D5311yescharacteristic
222C8—21no—
223C8—1062no—
224C8D4×D5312yes—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—1062no—
229C22—1062no—
230C22—1062no—
231C22—1064no—
232C22—4248no—
233C22—2122no—
234C22—2122no—
235C22C2×D4×D5311yescharacteristic
236C4—21no—
237C4—21no—
238C4—42no—
239C4—1062no—
240C4—2122no—
241C4C2×D4×D5311yescharacteristic
242C4#1313326211yescharacteristic
243C2—82no—
244C2—2122no—
245C2—2124no—
246C2#1681125811yescharacteristic
247C2#1681142011yescharacteristic
248C2#1681142511yescharacteristic
subgroupG/H
centerC22C2×D4×D53
commutatorC2×C4×C53C24
FrattiniC2×C4C23×D53
FittingC53×#495C2
socleC22×C53C22×D4
hypercenter#495D53
hypocenterC53#3273
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. C2×#13133244C2semidirect
2. #16811266C2semidirect
3. #16810866C2semidirect
4. #16810835C2semidirect
5. #16810784C2semidirect
6. #16810755C2semidirect
7. C4×D424C2semidirect
8. #16810702C2semidirect
9. C53×#495C2semidirect
11. C2×D4×D53C22semidirect
13. C2×#11719924C22semidirect
16. C2×#11719912C22semidirect
18. #13133193C22semidirect
23. C4×D212C22semidirect
25. #13133151C22semidirect
26. C4⋊#88322C22semidirect
27. #13133141C22semidirect
28. #13133139C22semidirect
29. #13133138C22semidirect
30. #13133129C22semidirect
31. C2×Q16×C53C22semidirect
32. C2×SD16×C53C22semidirect
34. #124×C53C22semidirect
35. C4×C8×C53C22semidirect
69. D424D4semidirect
127. #495D53semidirect
134. C53#3273semidirect
138. C2×Q16C2×D53semidirect
140. C2×SD16C2×D53semidirect
145. #124C2×D53semidirect
160. C4×C8C2×D53semidirect