Finite Group Databasebeta

← #27209923

Subgroups — #27209923

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
1C2×#15693067C211yesmaximal, characteristic, complemented
2#20268599C218yesmaximal, complemented
3#20268594C218yesmaximal, complemented
4#20268589C216yesmaximal, complemented
5C2×#15693046C216yesmaximal, complemented
6#20268558C211yesmaximal, characteristic, complemented
7#585×C103C211yesmaximal, characteristic, complemented
8#15693067—24no—
9#15693067C2218yescomplemented
10C2×#13110106—21no—
11C2×#13110105C2213yescomplemented
12C22×D412C2213yescomplemented
13C2×D4×D103—23no—
14#15693052C22112yescomplemented
15#15693051C2214yescomplemented
16#15693047C22112yescomplemented
17#15693046C22112yescomplemented
18#15693046C22124yescomplemented
19#15693041C22124yescomplemented
20C2×#13110092C22112yescomplemented
21C2×#13110090C22112yescomplemented
22#15693028C2218yescomplemented
23C2×#13110089C2216yescomplemented
242−1+4×C103—21no—
252−1+4×C103C2214yescomplemented
26H(5,𝔽2)×C103—23no—
27H(5,𝔽2)×C103C2214yescomplemented
28C2×#41×C103—24no—
29C2×#41×C103C2211yescharacteristic
30C2×#41×C103C2216yescomplemented
31C23×D103—23no—
32#13110106—28no—
33#13110105—26no—
34#13110105C23112yes—
35C2×Dic206C2311yescharacteristic
36C2×D412—23no—
37C2×D412—26no—
38C2×D412C2313yes—
39C2×D412C23112yes—
40C2×C22⋊D103—23no—
41D4×D103—224no—
42C2×C4×D103—23no—
43#13110095C2318yescomplemented
44#13110093C2318yescomplemented
45#13110092C23124yescomplemented
46#13110090C23124yescomplemented
47#13110089C2314yes—
48#13110089C23112yescomplemented
49C2×C103⋊C8C23112yescomplemented
50#41×C103—24no—
51#41×C103—212no—
52#41×C103—26no—
53#41×C103—26no—
54#41×C103D414yes—
55#41×C103C2314yes—
56#41×C103C2314yes—
57#41×C103C23112yescomplemented
58C2×Q8×C103—24no—
59C2×Q8×C103D411yescharacteristic
60C2×Q8×C103C2316yescomplemented
61C2×D4×C103—26no—
62C2×D4×C103—212no—
63C2×D4×C103D413yes—
64C2×D4×C103C2316yescomplemented
65C22×C4×C103—26no—
66C22×C4×C103C2313yes—
67C23×C103—26no—
68C23×C103C2×D413yes—
69C22×D103—23no—
70C22×D103—212no—
71C22×D103—86no—
72Q8×C103—26no—
73Q8×C103C2×D414yes—
74Q8×C103C2414yes—
75D4×C103—24no—
76D4×C103—212no—
77D4×C103—26no—
78D4×C103C2×D4112yes—
79D4×C103C2414yes—
80C2×C4×C103—26no—
81C2×C4×C103—24no—
82C2×C4×C103—212no—
83C2×C4×C103C2×D416yes—
84C2×C4×C103C2×D413yes—
85C2×C4×C103C2411yescharacteristic
86C2×C4×C103C2416yes—
87C22⋊D103—212no—
88C2×Dic103—21no—
89D412—212no—
90D412—23no—
91D412C2416yes—
92C4×D103—212no—
93Dic206—21no—
94Dic206C2412yes—
95C103⋊C8C2418yes—
96C22×C103—24no—
97C22×C103—43no—
98C22×C103—86no—
99C22×C103C22×D411yescharacteristic
100C22×C103C22×D416yes—
101C2×D103—26no—
102C2×D103—46no—
103C2×D103—812no—
104C4×C103—24no—
105C4×C103C22×D411yescharacteristic
106C4×C103C22×D416yes—
107C4×C103C2511yescharacteristic
108Dic103—22no—
109C2×C103—21no—
110C2×C103—26no—
111C2×C103—44no—
112C2×C103C23×D411yescharacteristic
113D103—46no—
114#3585—1031nomaximal, Sylow
115C103#358511yescharacteristic, complemented, Sylow
116#585D10311yescharacteristic, complemented
117C2×H(5,𝔽2)—1031no—
118#577—1038no—
119#576—1038no—
120#575—1036no—
121C2×AGL(1,ℤ/8)—1036no—
122#568—1031no—
1232−1+4—21no—
1242−1+4C2×D10314yescomplemented
125H(5,𝔽2)—23no—
126H(5,𝔽2)—1038no—
127H(5,𝔽2)—2064no—
128H(5,𝔽2)C2×D10314yescomplemented
129C2×#41—24no—
130C2×#41—1033no—
131C2×#41—2061no—
132C2×#41C2×D10311yescharacteristic
133C2×#41C2×D10316yescomplemented
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#131—1038no—
145C2×M16—1036no—
146C24—2063no—
147#41—24no—
148#41—212no—
149#41—26no—
150#41—26no—
151#41—10312no—
152#41—2068no—
153#41—2066no—
154#41C22⋊D10314yes—
155#41C22×D10314yes—
156#41C22×D10314yes—
157#41C22×D103112yescomplemented
158C2×Q8—24no—
159C2×Q8—1031no—
160C2×Q8C22⋊D10311yescharacteristic
161C2×Q8C22×D10316yescomplemented
162C2×D4—26no—
163C2×D4—212no—
164C2×D4—10312no—
165C2×D4—1033no—
166C2×D4—2063no—
167C2×D4—20624no—
168C2×D4—2063no—
169C2×D4—2066no—
170C2×D4C22⋊D10313yes—
171C2×D4C22×D10316yescomplemented
172C22×C4—26no—
173C22×C4—2063no—
174C22×C4C22×D10313yes—
175Q16—1038no—
176SD16—10324no—
177SD16—1038no—
178D8—10324no—
179M16—1034no—
180M16—10312no—
181C2×C8—10312no—
182C23—26no—
183C23—8246no—
184C23—20612no—
185C23—2063no—
186C23C2×C22⋊D10313yes—
187Q8—26no—
188Q8—1032no—
189Q8—2061no—
190Q8C2×C22⋊D10314yes—
191Q8C23×D10314yes—
192D4—24no—
193D4—212no—
194D4—26no—
195D4—1036no—
196D4—20612no—
197D4—2063no—
198D4—20612no—
199D4C2×C22⋊D103112yes—
200D4C23×D10314yes—
201C2×C4—26no—
202C2×C4—24no—
203C2×C4—212no—
204C2×C4—2061no—
205C2×C4—20612no—
206C2×C4C2×C22⋊D10316yes—
207C2×C4C2×C22⋊D10313yes—
208C2×C4C23×D10316yes—
209C2×C4C23×D10311yescharacteristic
210C8—1038no—
211C22—24no—
212C22—43no—
213C22—86no—
214C22—82412no—
215C22—4126no—
216C22—2066no—
217C22C22×C22⋊D10316yes—
218C22C22×C22⋊D10311yescharacteristic
219C4—24no—
220C4—2062no—
221C4C22×C22⋊D10311yescharacteristic
222C4C22×C22⋊D10316yes—
223C4C24×D10311yescharacteristic
224C2—21no—
225C2—26no—
226C2—44no—
227C2—4126no—
228C2C23×C22⋊D10311yescharacteristic
subgroupG/H
centerC2C23×C22⋊D103
commutator, cosocleC4×C103C25
FrattiniC4C24×D103
Fitting#585×C103C2
socleC2×C103C23×D4
hypercenter#585D103
hypocenterC103#3585
LCSG ▷ C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C2 ◁ C2×#41 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#15693067C2semidirect
2. #20268599C2semidirect
3. #20268594C2semidirect
4. #20268589C2semidirect
5. C2×#15693046C2semidirect
6. #20268558C2semidirect
7. #585×C103C2semidirect
9. #15693067C22semidirect
11. C2×#13110105C22semidirect
12. C22×D412C22semidirect
14. #15693052C22semidirect
15. #15693051C22semidirect
16. #15693047C22semidirect
17. #15693046C22semidirect
18. #15693046C22semidirect
19. #15693041C22semidirect
20. C2×#13110092C22semidirect
21. C2×#13110090C22semidirect
22. #15693028C22semidirect
23. C2×#13110089C22semidirect
25. 2−1+4×C103C22semidirect
27. H(5,𝔽2)×C103C22semidirect
30. C2×#41×C103C22semidirect
43. #13110095C23semidirect
44. #13110093C23semidirect
45. #13110092C23semidirect
46. #13110090C23semidirect
48. #13110089C23semidirect
49. C2×C103⋊C8C23semidirect
57. #41×C103C23semidirect
60. C2×Q8×C103C23semidirect
64. C2×D4×C103C23semidirect
115. C103#3585semidirect
116. #585D103semidirect
124. 2−1+4C2×D103semidirect
128. H(5,𝔽2)C2×D103semidirect
133. C2×#41C2×D103semidirect
157. #41C22×D103semidirect
161. C2×Q8C22×D103semidirect
171. C2×D4C22×D103semidirect

Maximal subgroups

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

subgroupindex
C2×#156930672
#202685992
#202685942
#202685892
C2×#156930462
#202685582
#585×C1032
#3585103