Finite Group Databasebeta

← C23×#3415 #15671034

Subgroups — C23×#3415 #15671034

Order 2944.

370 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 180 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×#576C212yesmaximal, complemented
2C23×#509C216yesmaximal, complemented
3C23×#508C213yesmaximal, complemented
4C2×C23×D16C213yesmaximal, complemented
5C23×#504C211yesmaximal, characteristic, complemented
6C23×H(5,𝔽2)—22no—
7C23×AGL(1,ℤ/8)—26no—
8C23×#135C2216yescomplemented
9C2×D8×C23C2216yescomplemented
10C23×#131C2211yescharacteristic
11C23×Q32C2211yescharacteristic, complemented
12C23×SD32C2216yescomplemented
13C23×D16C2213yescomplemented
14C23×D16C2216yescomplemented
15C23×M32C2213yescomplemented
16C2×C16×C23C2213yescomplemented
17#41×C23—26no—
18#41×C23—42no—
19#41×C23D411yescharacteristic
20C2×D4×C23—26no—
21C2×D4×C23—46no—
22Q16×C23C2312yes—
23SD16×C23—26no—
24D8×C23—26no—
25D8×C23C2316yes—
26M16×C23D413yes—
27C2×C8×C23C2313yes—
28C16×C23C2311yescharacteristic
29C16×C23C2313yes—
30C23×C23—46no—
31Q8×C23—22no—
32Q8×C23D811yescharacteristic
33D4×C23—26no—
34D4×C23—46no—
35D4×C23—46no—
36D4×C23D813yes—
37C2×C4×C23—46no—
38C2×C4×C23C2×D413yes—
39C8×C23C2×D413yes—
40C8×C23C2411yescharacteristic
41#3415C2311yesmaximal, characteristic, direct product factor, Sylow
42C22×C23—46no—
43C22×C23—166no—
44C22×C23C2×D813yes—
45C4×C23—42no—
46C4×C23C2×D813yes—
47C4×C23C22×D411yescharacteristic
48#576C2×C2312yescomplemented
49#509C2×C2316yescomplemented
50#508C2×C2313yescomplemented
51C2×D16C2×C2313yescomplemented
52#504C2×C2311yescharacteristic, complemented
53C2×C23—23no—
54C2×C23—86no—
55C2×C23C22×D811yescharacteristic
56H(5,𝔽2)—22no—
57AGL(1,ℤ/8)—26no—
58#135C22×C2316yescomplemented
59C2×D8C22×C2316yescomplemented
60#131C22×C2311yescharacteristic
61Q32C22×C2311yescharacteristic, complemented
62SD32C22×C2316yescomplemented
63D16C22×C2313yescomplemented
64D16C22×C2316yescomplemented
65M32C22×C2313yescomplemented
66C2×C16C22×C2313yescomplemented
67C23#341511yescharacteristic, direct product factor, Sylow
68#41—26no—
69#41—42no—
70#41D4×C2311yescharacteristic
71C2×D4—26no—
72C2×D4—46no—
73Q16C23×C2312yes—
74SD16—26no—
75D8—26no—
76D8C23×C2316yes—
77M16D4×C2313yes—
78C2×C8C23×C2313yes—
79C16C23×C2311yescharacteristic
80C16C23×C2313yes—
81C23—46no—
82Q8—22no—
83Q8D8×C2311yescharacteristic
84D4—26no—
85D4—46no—
86D4—46no—
87D4D8×C2313yes—
88C2×C4—46no—
89C2×C4C2×D4×C2313yes—
90C8C2×D4×C2313yes—
91C8C24×C2311yescharacteristic
92C22—46no—
93C22—166no—
94C22C2×D8×C2313yes—
95C4—42no—
96C4C2×D8×C2313yes—
97C4C22×D4×C2311yescharacteristic
98C2—23no—
99C2—86no—
100C2C22×D8×C2311yescharacteristic
subgroupG/H
center, socleC2×C23C22×D8
commutator, Frattini, cosocleC8C24×C23
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C23 ◁ #41×C23 ◁ C23×#131 ◁ G

Complements

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

NH ≅ G/Ntype
1. C23×#576C2semidirect
2. C23×#509C2semidirect
3. C23×#508C2semidirect
4. C2×C23×D16C2semidirect
5. C23×#504C2semidirect
8. C23×#135C22semidirect
9. C2×D8×C23C22semidirect
11. C23×Q32C22semidirect
12. C23×SD32C22semidirect
13. C23×D16C22semidirect
14. C23×D16C22semidirect
15. C23×M32C22semidirect
16. C2×C16×C23C22semidirect
41. #3415C23direct (N × H)
48. #576C2×C23semidirect
49. #509C2×C23semidirect
50. #508C2×C23semidirect
51. C2×D16C2×C23semidirect
52. #504C2×C23semidirect
58. #135C22×C23semidirect
59. C2×D8C22×C23semidirect
61. Q32C22×C23semidirect
62. SD32C22×C23semidirect
63. D16C22×C23semidirect
64. D16C22×C23semidirect
65. M32C22×C23semidirect
66. C2×C16C22×C23semidirect
67. C23#3415direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#5762
C23×#5092
C23×#5082
C2×C23×D162
C23×#5042
#341523