Finite Group Databasebeta

← C23×C3×#1044 #72259790

Subgroups — C23×C3×#1044 #72259790

Order 2400.

1212 conjugacy classes of subgroups; only the 332 normal subgroups are shown.

normal subgroups only

This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 332 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×C3×C5⋊D5C211yesmaximal, characteristic
2C22×C3×#1044C2114yesmaximal, direct product factor
3C23×#1044C311yesmaximal, characteristic, direct product factor, Hall
4C23×C3×C52C411yescharacteristic, complemented
5C22×C3×C5⋊D5C417yescomplemented
6C22×C3×C5⋊D5C2217yes—
7C2×C3×#1044C22128yesdirect product factor
8C23×C3×AGL(1,𝔽5)—52nomaximal
9C23×C5⋊D5C2×C311yescharacteristic
10C22×#1044C2×C3114yesdirect product factor
11C22×C3×C52C2×C417yescomplemented
12C2×C3×C5⋊D5C2×C4121yescomplemented
13C2×C3×C5⋊D5C2317yes—
14C3×#1044C2318yesdirect product factor
15C23×C3×D5—52no—
16C23×C3×D5—102no—
17C22×C3×AGL(1,𝔽5)—528no—
18C23×C52C3×C411yescharacteristic, complemented
19C22×C5⋊D5C3×C417yescomplemented
20C22×C5⋊D5C22×C317yes—
21C2×#1044C22×C3128yesdirect product factor
22C23×AGL(1,𝔽5)—52no—
23C2×C3×C52C22×C417yescomplemented
24C3×C5⋊D5C22×C417yescomplemented
25C3×C5⋊D5C2411yescharacteristic
26C23×C3×C5—22no—
27C23×C3×C5AGL(1,𝔽5)12yescomplemented
28C22×C3×D5—514no—
29C22×C3×D5—514no—
30C22×C3×D5—1014no—
31C22×C3×D5—1014no—
32C2×C3×AGL(1,𝔽5)—556no—
33C22×C52C2×C3×C417yescomplemented
34C2×C5⋊D5C2×C3×C4121yescomplemented
35C2×C5⋊D5C23×C317yes—
36#1044C23×C318yesdirect product factor
37C23×C3×C4—251noHall
38C23×D5—52no—
39C23×D5—102no—
40C22×AGL(1,𝔽5)—528no—
41C3×C52C23×C411yescharacteristic, complemented, Hall
42C22×C3×C5—214no—
43C22×C3×C5C2×AGL(1,𝔽5)114yescomplemented
44C2×C3×D5—514no—
45C2×C3×D5—542no—
46C2×C3×D5—1042no—
47C2×C3×D5—1014no—
48C3×AGL(1,𝔽5)—516no—
49C2×C52C22×C3×C417yescomplemented
50C5⋊D5C22×C3×C417yescomplemented
51C5⋊D5C24×C311yescharacteristic
52C24×C3—251no—
53C22×C3×C4—2514no—
54C23×C5—22no—
55C23×C5C3×AGL(1,𝔽5)12yescomplemented
56C22×D5—514no—
57C22×D5—514no—
58C22×D5—1014no—
59C22×D5—1014no—
60C2×AGL(1,𝔽5)—556no—
61C23×C4—251noSylow
62C2×C3×C5—214no—
63C2×C3×C5C22×AGL(1,𝔽5)114yescomplemented
64C3×D5—514no—
65C3×D5—52no—
66C3×D5—1014no—
67C3×D5—102no—
68C52C23×C3×C411yescharacteristic, complemented, Sylow
69C23×C3—257no—
70C23×C3—257no—
71C23×C3#104411yescharacteristic, direct product factor
72C2×C3×C4—2528no—
73C22×C5—214no—
74C22×C5C2×C3×AGL(1,𝔽5)114yescomplemented
75C2×D5—542no—
76C2×D5—514no—
77C2×D5—1042no—
78C2×D5—1014no—
79AGL(1,𝔽5)—516no—
80C24—251no—
81C22×C4—2514no—
82C3×C5—22no—
83C3×C5C23×AGL(1,𝔽5)12yescomplemented
84C22×C3—257no—
85C22×C3—2521no—
86C22×C3C2×#104417yesdirect product factor
87C3×C4—258no—
88C2×C5—214no—
89C2×C5C22×C3×AGL(1,𝔽5)114yescomplemented
90D5—514no—
91D5—52no—
92D5—1014no—
93D5—102no—
94C23—257no—
95C23—257no—
96C23C3×#104411yescharacteristic, direct product factor
97C2×C4—2528no—
98C2×C3—251no—
99C2×C3—257no—
100C2×C3C22×#104417yesdirect product factor
101C5—22no—
102C5C23×C3×AGL(1,𝔽5)12yescomplemented
103C22—257no—
104C22—2521no—
105C22C2×C3×#104417yesdirect product factor
106C4—258no—
107C3C23×#104411yescharacteristic, direct product factor, Sylow
108C2—251no—
109C2—257no—
110C2C22×C3×#104417yesdirect product factor
subgroupG/H
center, hypercenterC23×C3#1044
commutator, hypocenterC52C23×C3×C4
FrattiniC1G
Fitting, socleC23×C3×C52C4
prime radicalC23×C3×C5⋊D5C2
cosocleC5⋊D5C24×C3
LCSG ▷ C52
UCSC1 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
2. C22×C3×#1044C2direct (N × H)
3. C23×#1044C3direct (N × H)
4. C23×C3×C52C4semidirect
5. C22×C3×C5⋊D5C4semidirect
7. C2×C3×#1044C22direct (N × H)
10. C22×#1044C2×C3direct (N × H)
11. C22×C3×C52C2×C4semidirect
12. C2×C3×C5⋊D5C2×C4semidirect
14. C3×#1044C23direct (N × H)
18. C23×C52C3×C4semidirect
19. C22×C5⋊D5C3×C4semidirect
21. C2×#1044C22×C3direct (N × H)
23. C2×C3×C52C22×C4semidirect
24. C3×C5⋊D5C22×C4semidirect
27. C23×C3×C5AGL(1,𝔽5)semidirect
33. C22×C52C2×C3×C4semidirect
34. C2×C5⋊D5C2×C3×C4semidirect
36. #1044C23×C3direct (N × H)
41. C3×C52C23×C4semidirect
43. C22×C3×C5C2×AGL(1,𝔽5)semidirect
49. C2×C52C22×C3×C4semidirect
50. C5⋊D5C22×C3×C4semidirect
55. C23×C5C3×AGL(1,𝔽5)semidirect
63. C2×C3×C5C22×AGL(1,𝔽5)semidirect
68. C52C23×C3×C4semidirect
71. C23×C3#1044direct (N × H)
74. C22×C5C2×C3×AGL(1,𝔽5)semidirect
83. C3×C5C23×AGL(1,𝔽5)semidirect
86. C22×C3C2×#1044direct (N × H)
89. C2×C5C22×C3×AGL(1,𝔽5)semidirect
96. C23C3×#1044direct (N × H)
100. C2×C3C22×#1044direct (N × H)
102. C5C23×C3×AGL(1,𝔽5)semidirect
105. C22C2×C3×#1044direct (N × H)
107. C3C23×#1044direct (N × H)
110. C2C22×C3×#1044direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C5⋊D52
C22×C3×#10442
C23×#10443
C23×C3×AGL(1,𝔽5)5