Finite Group Databasebeta

← #3292×C227 #44397248

Subgroups — #3292×C227 #44397248

Order 29056.

476 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 184 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#576×C227C214yesmaximal, complemented
2#493×C227C211yesmaximal, characteristic, complemented
3#472×C227C214yesmaximal, complemented
4#453×C227C214yesmaximal, complemented
5#443×C227C212yesmaximal, complemented
6H(5,𝔽2)×C227C2214yescomplemented
7AGL(1,ℤ/8)×C227—28no—
8AGL(1,ℤ/8)×C227C2218yescomplemented
9#135×C227—24no—
10#135×C227C2212yes—
11C2×D8×C227—24no—
12C2×D8×C227C2214yescomplemented
13#131×C227C2214yescomplemented
14#127×C227C2212yescomplemented
15#108×C227C2212yescomplemented
16#104×C227C2214yescomplemented
17#100×C227C2214yescomplemented
18C4×C8×C227C2211yescharacteristic, complemented
19#41×C227—28no—
20#41×C227C2314yes—
21C2×D4×C227—28no—
22C2×D4×C227—42no—
23C2×D4×C227—44no—
24C2×D4×C227C2314yes—
25Q16×C227D412yescomplemented
26SD16×C227—28no—
27SD16×C227D414yescomplemented
28D8×C227—28no—
29D8×C227—24no—
30D8×C227—24no—
31D8×C227D412yescomplemented
32M16×C227—24no—
33M16×C227C2314yes—
34C2×C8×C227—24no—
35C2×C8×C227C2312yes—
36C42×C227C2311yescharacteristic
37C23×C227—24no—
38C23×C227—44no—
39Q8×C227C2×D414yes—
40D4×C227—24no—
41D4×C227—28no—
42D4×C227—48no—
43D4×C227—84no—
44D4×C227C2×D414yes—
45C2×C4×C227—21no—
46C2×C4×C227—24no—
47C2×C4×C227—44no—
48C2×C4×C227C2411yescharacteristic
49C8×C227—24no—
50C8×C227C2×D412yes—
51C8×C227C2×D412yes—
52C22×C227—24no—
53C22×C227—44no—
54C22×C227—84no—
55C22×C227—164no—
56C22×C227H(5,𝔽2)11yescharacteristic
57C4×C227—22no—
58C4×C227—24no—
59C4×C227C22×D412yes—
60C2×C227—21no—
61C2×C227—44no—
62C2×C227—84no—
63C2×C227D4211yescharacteristic
64C227#329211yescharacteristic, direct product factor, Sylow
65#3292C22711yesmaximal, characteristic, direct product factor, Sylow
66#576C2×C22714yescomplemented
67#493C2×C22711yescharacteristic, complemented
68#472C2×C22714yescomplemented
69#453C2×C22714yescomplemented
70#443C2×C22712yescomplemented
71H(5,𝔽2)C22×C22714yescomplemented
72AGL(1,ℤ/8)—28no—
73AGL(1,ℤ/8)C22×C22718yescomplemented
74#135—24no—
75#135C22×C22712yes—
76C2×D8—24no—
77C2×D8C22×C22714yescomplemented
78#131C22×C22714yescomplemented
79#127C22×C22712yescomplemented
80#108C22×C22712yescomplemented
81#104C22×C22714yescomplemented
82#100C22×C22714yescomplemented
83C4×C8C22×C22711yescharacteristic, complemented
84#41—28no—
85#41C23×C22714yes—
86C2×D4—28no—
87C2×D4—42no—
88C2×D4—44no—
89C2×D4C23×C22714yes—
90Q16D4×C22712yescomplemented
91SD16—28no—
92SD16D4×C22714yescomplemented
93D8—28no—
94D8—24no—
95D8—24no—
96D8D4×C22712yescomplemented
97M16—24no—
98M16C23×C22714yes—
99C2×C8—24no—
100C2×C8C23×C22712yes—
101C42C23×C22711yescharacteristic
102C23—24no—
103C23—44no—
104Q8C2×D4×C22714yes—
105D4—24no—
106D4—28no—
107D4—48no—
108D4—84no—
109D4C2×D4×C22714yes—
110C2×C4—21no—
111C2×C4—24no—
112C2×C4—44no—
113C2×C4C24×C22711yescharacteristic
114C8—24no—
115C8C2×D4×C22712yes—
116C8C2×D4×C22712yes—
117C22—24no—
118C22—44no—
119C22—84no—
120C22—164no—
121C22H(5,𝔽2)×C22711yescharacteristic
122C4—22no—
123C4—24no—
124C4C22×D4×C22712yes—
125C2—21no—
126C2—44no—
127C2—84no—
128C2D42×C22711yescharacteristic
subgroupG/H
center, socleC2×C227D42
commutator, Frattini, cosocleC2×C4C24×C227
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C227 ◁ C2×C4×C227 ◁ G

Complements

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

NH ≅ G/Ntype
1. #576×C227C2semidirect
2. #493×C227C2semidirect
3. #472×C227C2semidirect
4. #453×C227C2semidirect
5. #443×C227C2semidirect
6. H(5,𝔽2)×C227C22semidirect
8. AGL(1,ℤ/8)×C227C22semidirect
12. C2×D8×C227C22semidirect
13. #131×C227C22semidirect
14. #127×C227C22semidirect
15. #108×C227C22semidirect
16. #104×C227C22semidirect
17. #100×C227C22semidirect
18. C4×C8×C227C22semidirect
25. Q16×C227D4semidirect
27. SD16×C227D4semidirect
31. D8×C227D4semidirect
64. C227#3292direct (N × H)
65. #3292C227direct (N × H)
66. #576C2×C227semidirect
67. #493C2×C227semidirect
68. #472C2×C227semidirect
69. #453C2×C227semidirect
70. #443C2×C227semidirect
71. H(5,𝔽2)C22×C227semidirect
73. AGL(1,ℤ/8)C22×C227semidirect
77. C2×D8C22×C227semidirect
78. #131C22×C227semidirect
79. #127C22×C227semidirect
80. #108C22×C227semidirect
81. #104C22×C227semidirect
82. #100C22×C227semidirect
83. C4×C8C22×C227semidirect
90. Q16D4×C227semidirect
92. SD16D4×C227semidirect
96. D8D4×C227semidirect

Maximal subgroups

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

subgroupindex
#576×C2272
#493×C2272
#472×C2272
#453×C2272
#443×C2272
#3292227