Finite Group Databasebeta

← #20289496

Subgroups — #20289496

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
1#16811459C211yesmaximal, characteristic, complemented
2C2×#13133263C211yesmaximal, characteristic, complemented
3#16811435C218yesmaximal, complemented
4#16811428C218yesmaximal, complemented
5#16811423C216yesmaximal, complemented
6C2×#13133245C216yesmaximal, complemented
7C53×#568C211yesmaximal, characteristic, complemented
8#13133265—21no—
9#13133264C2214yescomplemented
10#13133263—24no—
11#13133263C2218yescomplemented
12#13133263C2214yescomplemented
13#13133262—23no—
14C2×#11719949—21no—
15C2×#11719945C2213yescomplemented
16C2×#11719945C2216yescomplemented
17#41×D53—24no—
18C22×D212C2213yescomplemented
19C2×D4×D53—23no—
20#13133250C2214yescomplemented
21#13133250C22112yescomplemented
22#13133245C22112yescomplemented
23#13133245C22124yescomplemented
24#13133245C22112yescomplemented
25#13133239C22124yescomplemented
26C2×#11719913C22112yescomplemented
27C2×D424C22112yescomplemented
28C2×#41×C53C2211yescharacteristic
29#131×C53C2218yescomplemented
30C2×M16×C53C2216yescomplemented
31C23×D53—23no—
32#41×C53D414yes—
33#41×C53C2314yes—
34C2×Q8×C53D411yescharacteristic
35C2×D4×C53D413yes—
36C22×C4×C53C2313yes—
37C2×C22⋊D53—23no—
38C2×C22⋊D53—26no—
39#11719949—24no—
40#11719949—28no—
41Q8×D53—24no—
42#11719947—212no—
43D4×D53—224no—
44D4×D53—212no—
45#11719945—26no—
46#11719945—26no—
47#11719945—26no—
48#11719945C23112yes—
49#11719945C2314yes—
50#11719945C23112yescomplemented
51C2×D212—23no—
52C2×D212—26no—
53C2×D212C23112yes—
54C2×D212C2313yes—
55C2×D212C2316yescomplemented
56C2×C4×D53—23no—
57C2×C4×D53—26no—
58C2×Dic106C2311yescharacteristic
59C2×Dic106C2316yescomplemented
60M16×C53C2314yes—
61M16×C53C23112yescomplemented
62C2×C8×C53C23112yescomplemented
63Dic212C2318yescomplemented
64D424C23124yescomplemented
65#11719913C23124yescomplemented
66#11719913C2318yescomplemented
67C23×C53C2×D413yes—
68C22×D53—23no—
69C22×D53—26no—
70C22×D53—212no—
71C22×D53—86no—
72Q8×C53C2×D414yes—
73D4×C53C2×D4112yes—
74C2×C4×C53C2×D413yes—
75C2×C4×C53C2×D416yes—
76C2×C4×C53C2411yescharacteristic
77C2×C4×C53C2416yes—
78C22⋊D53—24no—
79C22⋊D53—212no—
80C22⋊D53—212no—
81C2×Dic53—21no—
82C2×Dic53—26no—
83D212—23no—
84D212—212no—
85D212—26no—
86D212C2416yes—
87D212C2414yes—
88C4×D53—24no—
89C4×D53—212no—
90C4×D53—212no—
91Dic106—21no—
92Dic106—26no—
93Dic106C2412yes—
94Dic106C2414yes—
95C8×C53C2418yes—
96C22×C53—43no—
97C22×C53C22×D411yescharacteristic
98C22×C53C22×D416yes—
99C2×D53—24no—
100C2×D53—26no—
101C2×D53—46no—
102C2×D53—86no—
103C2×D53—812no—
104C4×C53C22×D411yescharacteristic
105C4×C53C22×D416yes—
106C4×C53C2511yescharacteristic
107Dic53—24no—
108Dic53—22no—
109#3585—531nomaximal, Sylow
110C2×C53—21no—
111C2×C53—26no—
112C2×C53C23×D411yescharacteristic
113D53—46no—
114D53—44no—
115#585—531no—
116C2×H(5,𝔽2)—531no—
117#577—538no—
118#576—538no—
119#575—536no—
120C2×AGL(1,ℤ/8)—536no—
121#568D5311yescharacteristic, complemented
122C53#358511yescharacteristic, complemented, Sylow
1232−1+4—534no—
1242−1+4—1061no—
125H(5,𝔽2)—534no—
126H(5,𝔽2)—538no—
127H(5,𝔽2)—1064no—
128H(5,𝔽2)—1063no—
129C2×#41—536no—
130C2×#41—533no—
131C2×#41—1064no—
132C2×#41—1061no—
133C2×#41C2×D5311yescharacteristic
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#131C2×D5318yescomplemented
145C2×M16C2×D5316yescomplemented
146C24—1063no—
147#41—5312no—
148#41—534no—
149#41—5312no—
150#41—1068no—
151#41—1064no—
152#41—10612no—
153#41—1066no—
154#41—1066no—
155#41—1066no—
156#41D21214yes—
157#41C22×D5314yes—
158C2×Q8—531no—
159C2×Q8—536no—
160C2×Q8—1064no—
161C2×Q8D21211yescharacteristic
162C2×D4—5312no—
163C2×D4—533no—
164C2×D4—536no—
165C2×D4—1063no—
166C2×D4—1066no—
167C2×D4—10624no—
168C2×D4—10612no—
169C2×D4—1063no—
170C2×D4—1066no—
171C2×D4D21213yes—
172C22×C4—1063no—
173C22×C4—1066no—
174C22×C4C22×D5313yes—
175Q16—538no—
176SD16—5324no—
177SD16—538no—
178D8—5324no—
179M16C22×D5314yes—
180M16C22×D53112yescomplemented
181C2×C8C22×D53112yescomplemented
182C23—1066no—
183C23—10612no—
184C23—1063no—
185C23—4246no—
186C23C2×D21213yes—
187Q8—534no—
188Q8—532no—
189Q8—1066no—
190Q8—1061no—
191Q8C2×D21214yes—
192D4—534no—
193D4—536no—
194D4—10612no—
195D4—1064no—
196D4—10612no—
197D4—1063no—
198D4—10612no—
199D4—1066no—
200D4C2×D212112yes—
201C2×C4—1061no—
202C2×C4—1066no—
203C2×C4—1064no—
204C2×C4—10612no—
205C2×C4—10612no—
206C2×C4C2×D21213yes—
207C2×C4C2×D21216yes—
208C2×C4C23×D5316yes—
209C2×C4C23×D5311yescharacteristic
210C8C23×D5318yes—
211C22—43no—
212C22—1064no—
213C22—1066no—
214C22—42412no—
215C22—4246no—
216C22—2126no—
217C22C22×D21211yescharacteristic
218C22C22×D21216yes—
219C4—1062no—
220C4—1064no—
221C4C22×D21211yescharacteristic
222C4C22×D21216yes—
223C4C24×D5311yescharacteristic
224C2—21no—
225C2—26no—
226C2—2126no—
227C2—2124no—
228C2C23×D21211yescharacteristic
subgroupG/H
centerC2C23×D212
commutatorC4×C53C25
FrattiniC4C24×D53
FittingC53×#568C2
socleC2×C53C23×D4
hypercenter#568D53
hypocenterC53#3585
LCSG ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #16811459C2semidirect
2. C2×#13133263C2semidirect
3. #16811435C2semidirect
4. #16811428C2semidirect
5. #16811423C2semidirect
6. C2×#13133245C2semidirect
7. C53×#568C2semidirect
9. #13133264C22semidirect
11. #13133263C22semidirect
12. #13133263C22semidirect
15. C2×#11719945C22semidirect
16. C2×#11719945C22semidirect
18. C22×D212C22semidirect
20. #13133250C22semidirect
21. #13133250C22semidirect
22. #13133245C22semidirect
23. #13133245C22semidirect
24. #13133245C22semidirect
25. #13133239C22semidirect
26. C2×#11719913C22semidirect
27. C2×D424C22semidirect
29. #131×C53C22semidirect
30. C2×M16×C53C22semidirect
50. #11719945C23semidirect
55. C2×D212C23semidirect
59. C2×Dic106C23semidirect
61. M16×C53C23semidirect
62. C2×C8×C53C23semidirect
63. Dic212C23semidirect
64. D424C23semidirect
65. #11719913C23semidirect
66. #11719913C23semidirect
121. #568D53semidirect
122. C53#3585semidirect
144. #131C2×D53semidirect
145. C2×M16C2×D53semidirect
180. M16C22×D53semidirect
181. C2×C8C22×D53semidirect