Finite Group Databasebeta

← #19149376

Subgroups — #19149376

Order 5504.

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#14583735C211yesmaximal, characteristic, complemented
2C2×#13063371C211yesmaximal, characteristic, complemented
3#14583711C218yesmaximal, complemented
4#14583704C218yesmaximal, complemented
5#14583699C216yesmaximal, complemented
6C2×#13063353C216yesmaximal, complemented
7C43×#568C211yesmaximal, characteristic, complemented
8#13063373—21no—
9#13063372C2214yescomplemented
10#13063371—24no—
11#13063371C2218yescomplemented
12#13063371C2214yescomplemented
13#13063370—23no—
14C2×#10621902—21no—
15C2×#10621898C2213yescomplemented
16C2×#10621898C2216yescomplemented
17#41×D43—24no—
18C22×D172C2213yescomplemented
19C2×D4×D43—23no—
20#13063358C2214yescomplemented
21#13063358C22112yescomplemented
22#13063353C22112yescomplemented
23#13063353C22124yescomplemented
24#13063353C22112yescomplemented
25#13063347C22124yescomplemented
26C2×#10621873C22112yescomplemented
27C2×D344C22112yescomplemented
28C2×#41×C43C2211yescharacteristic
29#131×C43C2218yescomplemented
30C2×M16×C43C2216yescomplemented
31C23×D43—23no—
32#41×C43D414yes—
33#41×C43C2314yes—
34C2×Q8×C43D411yescharacteristic
35C2×D4×C43D413yes—
36C22×C4×C43C2313yes—
37C2×C22⋊D43—23no—
38C2×C22⋊D43—26no—
39#10621902—24no—
40#10621902—28no—
41Q8×D43—24no—
42#10621900—212no—
43D4×D43—224no—
44D4×D43—212no—
45#10621898—26no—
46#10621898—26no—
47#10621898—26no—
48#10621898C23112yes—
49#10621898C2314yes—
50#10621898C23112yescomplemented
51C2×D172—23no—
52C2×D172—26no—
53C2×D172C23112yes—
54C2×D172C2313yes—
55C2×D172C2316yescomplemented
56C2×C4×D43—23no—
57C2×C4×D43—26no—
58C2×Dic86C2311yescharacteristic
59C2×Dic86C2316yescomplemented
60M16×C43C2314yes—
61M16×C43C23112yescomplemented
62C2×C8×C43C23112yescomplemented
63Dic172C2318yescomplemented
64D344C23124yescomplemented
65#10621873C23124yescomplemented
66#10621873C2318yescomplemented
67C23×C43C2×D413yes—
68C22×D43—23no—
69C22×D43—26no—
70C22×D43—212no—
71C22×D43—86no—
72Q8×C43C2×D414yes—
73D4×C43C2×D4112yes—
74C2×C4×C43C2×D413yes—
75C2×C4×C43C2×D416yes—
76C2×C4×C43C2416yes—
77C2×C4×C43C2411yescharacteristic
78C22⋊D43—24no—
79C22⋊D43—212no—
80C22⋊D43—212no—
81C2×Dic43—21no—
82C2×Dic43—26no—
83D172—23no—
84D172—212no—
85D172—26no—
86D172C2416yes—
87D172C2414yes—
88C4×D43—24no—
89C4×D43—212no—
90C4×D43—212no—
91Dic86—21no—
92Dic86—26no—
93Dic86C2414yes—
94Dic86C2412yes—
95C8×C43C2418yes—
96C22×C43—43no—
97C22×C43C22×D411yescharacteristic
98C22×C43C22×D416yes—
99C2×D43—26no—
100C2×D43—24no—
101C2×D43—46no—
102C2×D43—86no—
103C2×D43—812no—
104C4×C43C22×D411yescharacteristic
105C4×C43C22×D416yes—
106C4×C43C2511yescharacteristic
107Dic43—24no—
108Dic43—22no—
109#3585—431nomaximal, Sylow
110C2×C43—21no—
111C2×C43—26no—
112C2×C43C23×D411yescharacteristic
113D43—46no—
114D43—44no—
115#585—431no—
116C2×H(5,𝔽2)—431no—
117#577—438no—
118#576—438no—
119#575—436no—
120C2×AGL(1,ℤ/8)—436no—
121#568D4311yescharacteristic, complemented
122C43#358511yescharacteristic, complemented, Sylow
1232−1+4—434no—
1242−1+4—861no—
125H(5,𝔽2)—434no—
126H(5,𝔽2)—438no—
127H(5,𝔽2)—864no—
128H(5,𝔽2)—863no—
129C2×#41—436no—
130C2×#41—433no—
131C2×#41—864no—
132C2×#41—861no—
133C2×#41C2×D4311yescharacteristic
134C22×D4—433no—
135C22×D4—863no—
136#137—434no—
137#137—4312no—
138AGL(1,ℤ/8)—4312no—
139AGL(1,ℤ/8)—4324no—
140AGL(1,ℤ/8)—4312no—
141#135—4324no—
142C2×SD16—4312no—
143C2×D8—4312no—
144#131C2×D4318yescomplemented
145C2×M16C2×D4316yescomplemented
146C24—863no—
147#41—4312no—
148#41—4312no—
149#41—434no—
150#41—868no—
151#41—864no—
152#41—8612no—
153#41—866no—
154#41—866no—
155#41—866no—
156#41D17214yes—
157#41C22×D4314yes—
158C2×Q8—436no—
159C2×Q8—431no—
160C2×Q8—864no—
161C2×Q8D17211yescharacteristic
162C2×D4—436no—
163C2×D4—4312no—
164C2×D4—433no—
165C2×D4—863no—
166C2×D4—866no—
167C2×D4—8624no—
168C2×D4—8612no—
169C2×D4—863no—
170C2×D4—866no—
171C2×D4D17213yes—
172C22×C4—863no—
173C22×C4—866no—
174C22×C4C22×D4313yes—
175Q16—438no—
176SD16—4324no—
177SD16—438no—
178D8—4324no—
179M16C22×D4314yes—
180M16C22×D43112yescomplemented
181C2×C8C22×D43112yescomplemented
182C23—866no—
183C23—8612no—
184C23—863no—
185C23—3446no—
186C23C2×D17213yes—
187Q8—434no—
188Q8—432no—
189Q8—866no—
190Q8—861no—
191Q8C2×D17214yes—
192D4—434no—
193D4—436no—
194D4—8612no—
195D4—864no—
196D4—8612no—
197D4—863no—
198D4—8612no—
199D4—866no—
200D4C2×D172112yes—
201C2×C4—861no—
202C2×C4—866no—
203C2×C4—864no—
204C2×C4—8612no—
205C2×C4—8612no—
206C2×C4C2×D17213yes—
207C2×C4C2×D17216yes—
208C2×C4C23×D4316yes—
209C2×C4C23×D4311yescharacteristic
210C8C23×D4318yes—
211C22—43no—
212C22—864no—
213C22—866no—
214C22—34412no—
215C22—3446no—
216C22—1726no—
217C22C22×D17211yescharacteristic
218C22C22×D17216yes—
219C4—862no—
220C4—864no—
221C4C22×D17211yescharacteristic
222C4C22×D17216yes—
223C4C24×D4311yescharacteristic
224C2—21no—
225C2—26no—
226C2—1726no—
227C2—1724no—
228C2C23×D17211yescharacteristic
subgroupG/H
centerC2C23×D172
commutatorC4×C43C25
FrattiniC4C24×D43
FittingC43×#568C2
socleC2×C43C23×D4
hypercenter#568D43
hypocenterC43#3585
LCSG ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2 ◁ C2×#41 ◁ #568

Complements

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

NH ≅ G/Ntype
1. #14583735C2semidirect
2. C2×#13063371C2semidirect
3. #14583711C2semidirect
4. #14583704C2semidirect
5. #14583699C2semidirect
6. C2×#13063353C2semidirect
7. C43×#568C2semidirect
9. #13063372C22semidirect
11. #13063371C22semidirect
12. #13063371C22semidirect
15. C2×#10621898C22semidirect
16. C2×#10621898C22semidirect
18. C22×D172C22semidirect
20. #13063358C22semidirect
21. #13063358C22semidirect
22. #13063353C22semidirect
23. #13063353C22semidirect
24. #13063353C22semidirect
25. #13063347C22semidirect
26. C2×#10621873C22semidirect
27. C2×D344C22semidirect
29. #131×C43C22semidirect
30. C2×M16×C43C22semidirect
50. #10621898C23semidirect
55. C2×D172C23semidirect
59. C2×Dic86C23semidirect
61. M16×C43C23semidirect
62. C2×C8×C43C23semidirect
63. Dic172C23semidirect
64. D344C23semidirect
65. #10621873C23semidirect
66. #10621873C23semidirect
121. #568D43semidirect
122. C43#3585semidirect
144. #131C2×D43semidirect
145. C2×M16C2×D43semidirect
180. M16C22×D43semidirect
181. C2×C8C22×D43semidirect