Finite Group Databasebeta

← C23×#3292 #15670911

Subgroups — C23×#3292 #15670911

Order 2944.

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
1C23×#576C214yesmaximal, complemented
2C23×#493C211yesmaximal, characteristic, complemented
3C23×#472C214yesmaximal, complemented
4C23×#453C214yesmaximal, complemented
5C23×#443C212yesmaximal, complemented
6C23×H(5,𝔽2)C2214yescomplemented
7C23×AGL(1,ℤ/8)—28no—
8C23×AGL(1,ℤ/8)C2218yescomplemented
9C23×#135—24no—
10C23×#135C2212yes—
11C2×D8×C23—24no—
12C2×D8×C23C2214yescomplemented
13C23×#131C2214yescomplemented
14C23×#127C2212yescomplemented
15C23×#108C2212yescomplemented
16C23×#104C2214yescomplemented
17C23×#100C2214yescomplemented
18C4×C8×C23C2211yescharacteristic, complemented
19#41×C23—28no—
20#41×C23C2314yes—
21C2×D4×C23—28no—
22C2×D4×C23—42no—
23C2×D4×C23—44no—
24C2×D4×C23C2314yes—
25Q16×C23D412yescomplemented
26SD16×C23—28no—
27SD16×C23D414yescomplemented
28D8×C23—28no—
29D8×C23—24no—
30D8×C23—24no—
31D8×C23D412yescomplemented
32M16×C23—24no—
33M16×C23C2314yes—
34C2×C8×C23—24no—
35C2×C8×C23C2312yes—
36C42×C23C2311yescharacteristic
37C23×C23—24no—
38C23×C23—44no—
39Q8×C23C2×D414yes—
40D4×C23—24no—
41D4×C23—28no—
42D4×C23—48no—
43D4×C23—84no—
44D4×C23C2×D414yes—
45C2×C4×C23—21no—
46C2×C4×C23—24no—
47C2×C4×C23—44no—
48C2×C4×C23C2411yescharacteristic
49C8×C23—24no—
50C8×C23C2×D412yes—
51C8×C23C2×D412yes—
52#3292C2311yesmaximal, characteristic, direct product factor, Sylow
53C22×C23—24no—
54C22×C23—44no—
55C22×C23—84no—
56C22×C23—164no—
57C22×C23H(5,𝔽2)11yescharacteristic
58C4×C23—22no—
59C4×C23—24no—
60C4×C23C22×D412yes—
61#576C2×C2314yescomplemented
62#493C2×C2311yescharacteristic, complemented
63#472C2×C2314yescomplemented
64#453C2×C2314yescomplemented
65#443C2×C2312yescomplemented
66C2×C23—21no—
67C2×C23—44no—
68C2×C23—84no—
69C2×C23D4211yescharacteristic
70H(5,𝔽2)C22×C2314yescomplemented
71AGL(1,ℤ/8)—28no—
72AGL(1,ℤ/8)C22×C2318yescomplemented
73#135—24no—
74#135C22×C2312yes—
75C2×D8—24no—
76C2×D8C22×C2314yescomplemented
77#131C22×C2314yescomplemented
78#127C22×C2312yescomplemented
79#108C22×C2312yescomplemented
80#104C22×C2314yescomplemented
81#100C22×C2314yescomplemented
82C4×C8C22×C2311yescharacteristic, complemented
83C23#329211yescharacteristic, direct product factor, Sylow
84#41—28no—
85#41C23×C2314yes—
86C2×D4—28no—
87C2×D4—42no—
88C2×D4—44no—
89C2×D4C23×C2314yes—
90Q16D4×C2312yescomplemented
91SD16—28no—
92SD16D4×C2314yescomplemented
93D8—28no—
94D8—24no—
95D8—24no—
96D8D4×C2312yescomplemented
97M16—24no—
98M16C23×C2314yes—
99C2×C8—24no—
100C2×C8C23×C2312yes—
101C42C23×C2311yescharacteristic
102C23—24no—
103C23—44no—
104Q8C2×D4×C2314yes—
105D4—24no—
106D4—28no—
107D4—48no—
108D4—84no—
109D4C2×D4×C2314yes—
110C2×C4—21no—
111C2×C4—24no—
112C2×C4—44no—
113C2×C4C24×C2311yescharacteristic
114C8—24no—
115C8C2×D4×C2312yes—
116C8C2×D4×C2312yes—
117C22—24no—
118C22—44no—
119C22—84no—
120C22—164no—
121C22C23×H(5,𝔽2)11yescharacteristic
122C4—22no—
123C4—24no—
124C4C22×D4×C2312yes—
125C2—21no—
126C2—44no—
127C2—84no—
128C2D42×C2311yescharacteristic
subgroupG/H
center, socleC2×C23D42
commutator, Frattini, cosocleC2×C4C24×C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ C2×C4×C23 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#576C2semidirect
2. C23×#493C2semidirect
3. C23×#472C2semidirect
4. C23×#453C2semidirect
5. C23×#443C2semidirect
6. C23×H(5,𝔽2)C22semidirect
8. C23×AGL(1,ℤ/8)C22semidirect
12. C2×D8×C23C22semidirect
13. C23×#131C22semidirect
14. C23×#127C22semidirect
15. C23×#108C22semidirect
16. C23×#104C22semidirect
17. C23×#100C22semidirect
18. C4×C8×C23C22semidirect
25. Q16×C23D4semidirect
27. SD16×C23D4semidirect
31. D8×C23D4semidirect
52. #3292C23direct (N × H)
61. #576C2×C23semidirect
62. #493C2×C23semidirect
63. #472C2×C23semidirect
64. #453C2×C23semidirect
65. #443C2×C23semidirect
70. H(5,𝔽2)C22×C23semidirect
72. AGL(1,ℤ/8)C22×C23semidirect
76. C2×D8C22×C23semidirect
77. #131C22×C23semidirect
78. #127C22×C23semidirect
79. #108C22×C23semidirect
80. #104C22×C23semidirect
81. #100C22×C23semidirect
82. C4×C8C22×C23semidirect
83. C23#3292direct (N × H)
90. Q16D4×C23semidirect
92. SD16D4×C23semidirect
96. D8D4×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#5762
C23×#4932
C23×#4722
C23×#4532
C23×#4432
#329223