Finite Group Databasebeta

← #27209925

Subgroups — #27209925

Order 13184.

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#20268627C211yesmaximal, characteristic, complemented
2C2×#15693067C211yesmaximal, characteristic, complemented
3#20268603C218yesmaximal, complemented
4#20268596C218yesmaximal, complemented
5#20268591C216yesmaximal, complemented
6C2×#15693049C216yesmaximal, complemented
7#568×C103C211yesmaximal, characteristic, complemented
8#15693069—21no—
9#15693068C2214yescomplemented
10#15693067—24no—
11#15693067C2218yescomplemented
12#15693067C2214yescomplemented
13#15693066—23no—
14C2×#13110106—21no—
15C2×#13110105C2213yescomplemented
16C2×#13110105C2216yescomplemented
17#41×D103—24no—
18C22×D412C2213yescomplemented
19C2×D4×D103—23no—
20#15693054C2214yescomplemented
21#15693054C22112yescomplemented
22#15693049C22112yescomplemented
23#15693049C22124yescomplemented
24#15693049C22112yescomplemented
25#15693043C22124yescomplemented
26C2×#13110094C22112yescomplemented
27C2×D824C22112yescomplemented
28C2×#41×C103C2211yescharacteristic
29#131×C103C2218yescomplemented
30C2×M16×C103C2216yescomplemented
31C23×D103—23no—
32#13110106—24no—
33#13110106—28no—
34#13110105—26no—
35#13110105—26no—
36#13110105—26no—
37#13110105C2314yes—
38#13110105C23112yes—
39#13110105C23112yescomplemented
40#13110104—212no—
41C2×Dic206C2311yescharacteristic
42C2×Dic206C2316yescomplemented
43Q8×D103—24no—
44C2×D412—23no—
45C2×D412—26no—
46C2×D412C2313yes—
47C2×D412C23112yes—
48C2×D412C2316yescomplemented
49C2×C22⋊D103—23no—
50C2×C22⋊D103—26no—
51D4×D103—224no—
52D4×D103—212no—
53C2×C4×D103—23no—
54C2×C4×D103—26no—
55Dic412C2318yescomplemented
56#13110094C23124yescomplemented
57#13110094C2318yescomplemented
58D824C23124yescomplemented
59#41×C103D414yes—
60#41×C103C2314yes—
61C2×Q8×C103D411yescharacteristic
62C2×D4×C103D413yes—
63C22×C4×C103C2313yes—
64M16×C103C2314yes—
65M16×C103C23112yescomplemented
66C2×C8×C103C23112yescomplemented
67C23×C103C2×D413yes—
68C22×D103—23no—
69C22×D103—26no—
70C22×D103—212no—
71C22×D103—86no—
72Q8×C103C2×D414yes—
73D4×C103C2×D4112yes—
74C2×C4×C103C2×D413yes—
75C2×C4×C103C2×D416yes—
76C2×C4×C103C2416yes—
77C2×C4×C103C2411yescharacteristic
78C22⋊D103—212no—
79C22⋊D103—212no—
80C22⋊D103—24no—
81C2×Dic103—21no—
82C2×Dic103—26no—
83D412—212no—
84D412—26no—
85D412—23no—
86D412C2414yes—
87D412C2416yes—
88C4×D103—24no—
89C4×D103—212no—
90C4×D103—212no—
91Dic206—26no—
92Dic206—21no—
93Dic206C2414yes—
94Dic206C2412yes—
95C8×C103C2418yes—
96C22×C103—43no—
97C22×C103C22×D411yescharacteristic
98C22×C103C22×D416yes—
99C2×D103—24no—
100C2×D103—26no—
101C2×D103—46no—
102C2×D103—86no—
103C2×D103—812no—
104C4×C103C22×D411yescharacteristic
105C4×C103C22×D416yes—
106C4×C103C2511yescharacteristic
107Dic103—24no—
108Dic103—22no—
109C2×C103—21no—
110C2×C103—26no—
111C2×C103C23×D411yescharacteristic
112D103—46no—
113D103—44no—
114#3585—1031nomaximal, Sylow
115C103#358511yescharacteristic, complemented, Sylow
116#585—1031no—
117C2×H(5,𝔽2)—1031no—
118#577—1038no—
119#576—1038no—
120#575—1036no—
121C2×AGL(1,ℤ/8)—1036no—
122#568D10311yescharacteristic, complemented
1232−1+4—1034no—
1242−1+4—2061no—
125H(5,𝔽2)—1034no—
126H(5,𝔽2)—1038no—
127H(5,𝔽2)—2064no—
128H(5,𝔽2)—2063no—
129C2×#41—1036no—
130C2×#41—1033no—
131C2×#41—2064no—
132C2×#41—2061no—
133C2×#41C2×D10311yescharacteristic
134C22×D4—1033no—
135C22×D4—2063no—
136#137—1034no—
137#137—10312no—
138AGL(1,ℤ/8)—10312no—
139AGL(1,ℤ/8)—10324no—
140AGL(1,ℤ/8)—10312no—
141#135—10324no—
142C2×SD16—10312no—
143C2×D8—10312no—
144#131C2×D10318yescomplemented
145C2×M16C2×D10316yescomplemented
146C24—2063no—
147#41—10312no—
148#41—1034no—
149#41—10312no—
150#41—2068no—
151#41—2064no—
152#41—20612no—
153#41—2066no—
154#41—2066no—
155#41—2066no—
156#41D41214yes—
157#41C22×D10314yes—
158C2×Q8—1031no—
159C2×Q8—1036no—
160C2×Q8—2064no—
161C2×Q8D41211yescharacteristic
162C2×D4—10312no—
163C2×D4—1033no—
164C2×D4—1036no—
165C2×D4—2063no—
166C2×D4—2066no—
167C2×D4—20624no—
168C2×D4—20612no—
169C2×D4—2063no—
170C2×D4—2066no—
171C2×D4D41213yes—
172C22×C4—2063no—
173C22×C4—2066no—
174C22×C4C22×D10313yes—
175Q16—1038no—
176SD16—10324no—
177SD16—1038no—
178D8—10324no—
179M16C22×D10314yes—
180M16C22×D103112yescomplemented
181C2×C8C22×D103112yescomplemented
182C23—8246no—
183C23—2066no—
184C23—20612no—
185C23—2063no—
186C23C2×D41213yes—
187Q8—1034no—
188Q8—1032no—
189Q8—2066no—
190Q8—2061no—
191Q8C2×D41214yes—
192D4—1034no—
193D4—1036no—
194D4—20612no—
195D4—2064no—
196D4—20612no—
197D4—2063no—
198D4—20612no—
199D4—2066no—
200D4C2×D412112yes—
201C2×C4—2061no—
202C2×C4—2066no—
203C2×C4—2064no—
204C2×C4—20612no—
205C2×C4—20612no—
206C2×C4C2×D41213yes—
207C2×C4C2×D41216yes—
208C2×C4C23×D10316yes—
209C2×C4C23×D10311yescharacteristic
210C8C23×D10318yes—
211C22—43no—
212C22—82412no—
213C22—8246no—
214C22—4126no—
215C22—2064no—
216C22—2066no—
217C22C22×D41211yescharacteristic
218C22C22×D41216yes—
219C4—2062no—
220C4—2064no—
221C4C22×D41211yescharacteristic
222C4C22×D41216yes—
223C4C24×D10311yescharacteristic
224C2—21no—
225C2—26no—
226C2—4126no—
227C2—4124no—
228C2C23×D41211yescharacteristic
subgroupG/H
centerC2C23×D412
commutator, cosocleC4×C103C25
FrattiniC4C24×D103
Fitting#568×C103C2
socleC2×C103C23×D4
hypercenter#568D103
hypocenterC103#3585
LCSG ▷ C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #20268627C2semidirect
2. C2×#15693067C2semidirect
3. #20268603C2semidirect
4. #20268596C2semidirect
5. #20268591C2semidirect
6. C2×#15693049C2semidirect
7. #568×C103C2semidirect
9. #15693068C22semidirect
11. #15693067C22semidirect
12. #15693067C22semidirect
15. C2×#13110105C22semidirect
16. C2×#13110105C22semidirect
18. C22×D412C22semidirect
20. #15693054C22semidirect
21. #15693054C22semidirect
22. #15693049C22semidirect
23. #15693049C22semidirect
24. #15693049C22semidirect
25. #15693043C22semidirect
26. C2×#13110094C22semidirect
27. C2×D824C22semidirect
29. #131×C103C22semidirect
30. C2×M16×C103C22semidirect
39. #13110105C23semidirect
42. C2×Dic206C23semidirect
48. C2×D412C23semidirect
55. Dic412C23semidirect
56. #13110094C23semidirect
57. #13110094C23semidirect
58. D824C23semidirect
65. M16×C103C23semidirect
66. C2×C8×C103C23semidirect
115. C103#3585semidirect
122. #568D103semidirect
144. #131C2×D103semidirect
145. C2×M16C2×D103semidirect
180. M16C22×D103semidirect
181. C2×C8C22×D103semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202686272
C2×#156930672
#202686032
#202685962
#202685912
C2×#156930492
#568×C1032
#3585103