Finite Group Databasebeta

← #20289494

Subgroups — #20289494

Order 6784.

1464 conjugacy classes of subgroups; only the 545 normal subgroups are shown.

normal subgroups only

This group has 1464 conjugacy classes of subgroups — too many to draw. Only its 545 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×#13133263C211yesmaximal, characteristic, complemented
2#16811431C218yesmaximal, complemented
3#16811426C218yesmaximal, complemented
4#16811421C216yesmaximal, complemented
5C2×#13133242C216yesmaximal, complemented
6#16811390C211yesmaximal, characteristic, complemented
7C53×#585C211yesmaximal, characteristic, complemented
8#13133263—24no—
9#13133263C2218yescomplemented
10C2×#11719949—21no—
11C2×#11719945C2213yescomplemented
12C22×D212C2213yescomplemented
13C2×D4×D53—23no—
14#13133248C22112yescomplemented
15#13133247C2214yescomplemented
16#13133243C22112yescomplemented
17#13133242C22112yescomplemented
18#13133242C22124yescomplemented
19#13133237C22124yescomplemented
20C2×#11719924C22112yescomplemented
21C2×#11719922C22112yescomplemented
22#13133224C2218yescomplemented
23C2×#11719917C2216yescomplemented
242−1+4×C53—21no—
252−1+4×C53C2214yescomplemented
26H(5,𝔽2)×C53—23no—
27H(5,𝔽2)×C53C2214yescomplemented
28C2×#41×C53—24no—
29C2×#41×C53C2211yescharacteristic
30C2×#41×C53C2216yescomplemented
31C23×D53—23no—
32#41×C53—24no—
33#41×C53—212no—
34#41×C53—26no—
35#41×C53—26no—
36#41×C53D414yes—
37#41×C53C2314yes—
38#41×C53C2314yes—
39#41×C53C23112yescomplemented
40C2×Q8×C53—24no—
41C2×Q8×C53D411yescharacteristic
42C2×Q8×C53C2316yescomplemented
43C2×D4×C53—26no—
44C2×D4×C53—212no—
45C2×D4×C53D413yes—
46C2×D4×C53C2316yescomplemented
47C22×C4×C53—26no—
48C22×C4×C53C2313yes—
49C2×C22⋊D53—23no—
50#11719949—28no—
51D4×D53—224no—
52#11719945—26no—
53#11719945C23112yes—
54C2×D212—23no—
55C2×D212—26no—
56C2×D212C23112yes—
57C2×D212C2313yes—
58C2×C4×D53—23no—
59C2×Dic106C2311yescharacteristic
60#11719925C2318yescomplemented
61#11719924C23124yescomplemented
62#11719923C2318yescomplemented
63#11719922C23124yescomplemented
64#11719917C2314yes—
65#11719917C23112yescomplemented
66C2×#88322C23112yescomplemented
67C23×C53—26no—
68C23×C53C2×D413yes—
69C22×D53—23no—
70C22×D53—212no—
71C22×D53—86no—
72Q8×C53—26no—
73Q8×C53C2×D414yes—
74Q8×C53C2414yes—
75D4×C53—24no—
76D4×C53—212no—
77D4×C53—26no—
78D4×C53C2×D4112yes—
79D4×C53C2414yes—
80C2×C4×C53—26no—
81C2×C4×C53—24no—
82C2×C4×C53—212no—
83C2×C4×C53C2×D413yes—
84C2×C4×C53C2×D416yes—
85C2×C4×C53C2411yescharacteristic
86C2×C4×C53C2416yes—
87C22⋊D53—212no—
88C2×Dic53—21no—
89D212—23no—
90D212—212no—
91D212C2416yes—
92C4×D53—212no—
93Dic106—21no—
94Dic106C2412yes—
95#88322C2418yes—
96C22×C53—24no—
97C22×C53—43no—
98C22×C53—86no—
99C22×C53C22×D411yescharacteristic
100C22×C53C22×D416yes—
101C2×D53—26no—
102C2×D53—46no—
103C2×D53—812no—
104C4×C53—24no—
105C4×C53C22×D411yescharacteristic
106C4×C53C22×D416yes—
107C4×C53C2511yescharacteristic
108Dic53—22no—
109#3585—531nomaximal, Sylow
110C2×C53—21no—
111C2×C53—26no—
112C2×C53—44no—
113C2×C53C23×D411yescharacteristic
114D53—46no—
115#585D5311yescharacteristic, complemented
116C2×H(5,𝔽2)—531no—
117#577—538no—
118#576—538no—
119#575—536no—
120C2×AGL(1,ℤ/8)—536no—
121#568—531no—
122C53#358511yescharacteristic, complemented, Sylow
1232−1+4—21no—
1242−1+4C2×D5314yescomplemented
125H(5,𝔽2)—23no—
126H(5,𝔽2)—538no—
127H(5,𝔽2)—1064no—
128H(5,𝔽2)C2×D5314yescomplemented
129C2×#41—24no—
130C2×#41—533no—
131C2×#41—1061no—
132C2×#41C2×D5311yescharacteristic
133C2×#41C2×D5316yescomplemented
134C22×D4—533no—
135C22×D4—1063no—
136#137—534no—
137#137—5312no—
138AGL(1,ℤ/8)—5312no—
139AGL(1,ℤ/8)—5324no—
140AGL(1,ℤ/8)—5312no—
141#135—5324no—
142C2×SD16—5312no—
143C2×D8—5312no—
144#131—538no—
145C2×M16—536no—
146C24—1063no—
147#41—24no—
148#41—212no—
149#41—26no—
150#41—26no—
151#41—5312no—
152#41—1068no—
153#41—1066no—
154#41C22⋊D5314yes—
155#41C22×D5314yes—
156#41C22×D5314yes—
157#41C22×D53112yescomplemented
158C2×Q8—24no—
159C2×Q8—531no—
160C2×Q8C22⋊D5311yescharacteristic
161C2×Q8C22×D5316yescomplemented
162C2×D4—26no—
163C2×D4—212no—
164C2×D4—5312no—
165C2×D4—533no—
166C2×D4—1063no—
167C2×D4—10624no—
168C2×D4—1063no—
169C2×D4—1066no—
170C2×D4C22⋊D5313yes—
171C2×D4C22×D5316yescomplemented
172C22×C4—26no—
173C22×C4—1063no—
174C22×C4C22×D5313yes—
175Q16—538no—
176SD16—5324no—
177SD16—538no—
178D8—5324no—
179M16—534no—
180M16—5312no—
181C2×C8—5312no—
182C23—26no—
183C23—10612no—
184C23—1063no—
185C23—4246no—
186C23C2×C22⋊D5313yes—
187Q8—26no—
188Q8—532no—
189Q8—1061no—
190Q8C2×C22⋊D5314yes—
191Q8C23×D5314yes—
192D4—24no—
193D4—212no—
194D4—26no—
195D4—536no—
196D4—10612no—
197D4—1063no—
198D4—10612no—
199D4C2×C22⋊D53112yes—
200D4C23×D5314yes—
201C2×C4—26no—
202C2×C4—24no—
203C2×C4—212no—
204C2×C4—1061no—
205C2×C4—10612no—
206C2×C4C2×C22⋊D5316yes—
207C2×C4C2×C22⋊D5313yes—
208C2×C4C23×D5316yes—
209C2×C4C23×D5311yescharacteristic
210C8—538no—
211C22—24no—
212C22—43no—
213C22—86no—
214C22—1066no—
215C22—42412no—
216C22—2126no—
217C22C22×C22⋊D5316yes—
218C22C22×C22⋊D5311yescharacteristic
219C4—24no—
220C4—1062no—
221C4C22×C22⋊D5311yescharacteristic
222C4C22×C22⋊D5316yes—
223C4C24×D5311yescharacteristic
224C2—21no—
225C2—26no—
226C2—44no—
227C2—2126no—
228C2C23×C22⋊D5311yescharacteristic
subgroupG/H
centerC2C23×C22⋊D53
commutatorC4×C53C25
FrattiniC4C24×D53
FittingC53×#585C2
socleC2×C53C23×D4
hypercenter#585D53
hypocenterC53#3585
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×#41 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#13133263C2semidirect
2. #16811431C2semidirect
3. #16811426C2semidirect
4. #16811421C2semidirect
5. C2×#13133242C2semidirect
6. #16811390C2semidirect
7. C53×#585C2semidirect
9. #13133263C22semidirect
11. C2×#11719945C22semidirect
12. C22×D212C22semidirect
14. #13133248C22semidirect
15. #13133247C22semidirect
16. #13133243C22semidirect
17. #13133242C22semidirect
18. #13133242C22semidirect
19. #13133237C22semidirect
20. C2×#11719924C22semidirect
21. C2×#11719922C22semidirect
22. #13133224C22semidirect
23. C2×#11719917C22semidirect
25. 2−1+4×C53C22semidirect
27. H(5,𝔽2)×C53C22semidirect
30. C2×#41×C53C22semidirect
39. #41×C53C23semidirect
42. C2×Q8×C53C23semidirect
46. C2×D4×C53C23semidirect
60. #11719925C23semidirect
61. #11719924C23semidirect
62. #11719923C23semidirect
63. #11719922C23semidirect
65. #11719917C23semidirect
66. C2×#88322C23semidirect
115. #585D53semidirect
122. C53#3585semidirect
124. 2−1+4C2×D53semidirect
128. H(5,𝔽2)C2×D53semidirect
133. C2×#41C2×D53semidirect
157. #41C22×D53semidirect
161. C2×Q8C22×D53semidirect
171. C2×D4C22×D53semidirect