Finite Group Databasebeta

← C3×D4×AGL(1,𝔽5) #91978

Subgroups — C3×D4×AGL(1,𝔽5) #91978

Order 480.

188 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 76 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
1C22×C3×AGL(1,𝔽5)C212yesmaximal, complemented
2C3×D4×D5C211yesmaximal, characteristic
3C3×C22⋊AGL(1,𝔽5)C212yesmaximal, complemented
4C3×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
5C3×C4×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
6D4×AGL(1,𝔽5)C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×D5C2212yes—
8C2×C3×AGL(1,𝔽5)—24no—
9C2×C3×AGL(1,𝔽5)C2211yescharacteristic
10C2×C3×AGL(1,𝔽5)C2212yes—
11C2×C3×AGL(1,𝔽5)C2211yescharacteristic
12C3×C5×D4C411yescharacteristic, complemented
13C3×C22⋊D5C412yescomplemented
14C3×D20C411yescharacteristic, complemented
15C3×C4×D5C2211yescharacteristic
16C3×C4×D4—51nomaximal, Hall
17C22×AGL(1,𝔽5)C2×C312yescomplemented
18D4×D5C2×C311yescharacteristic
19C22⋊AGL(1,𝔽5)C2×C312yescomplemented
20C4⋊AGL(1,𝔽5)C2×C311yescharacteristic, complemented
21C4×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
22C22×C3×C5C2×C412yescomplemented
23C2×C3×D5—22no—
24C2×C3×D5—22no—
25C2×C3×D5C2×C412yescomplemented
26C2×C3×D5C2311yescharacteristic
27C3×AGL(1,𝔽5)—22no—
28C3×AGL(1,𝔽5)—21no—
29C3×AGL(1,𝔽5)D412yesdirect product factor
30C3×C4×C5C2×C411yescharacteristic, complemented
31C3×Dic5C2×C411yescharacteristic, complemented
32C2×C3×D4—51no—
33C22×C3×C4—52no—
34C3×C4⋊C4—51no—
35C3×C22⋊C4—52no—
36C3×C42—51no—
37C22×D5C22×C312yes—
38C2×AGL(1,𝔽5)—24no—
39C2×AGL(1,𝔽5)C22×C311yescharacteristic
40C2×AGL(1,𝔽5)C22×C312yes—
41C2×AGL(1,𝔽5)C22×C311yescharacteristic
42C5×D4C3×C411yescharacteristic, complemented
43C22⋊D5C3×C412yescomplemented
44D20C3×C411yescharacteristic, complemented
45C4×D5C22×C311yescharacteristic
46C4×D4—51noSylow
47C2×C3×C5—22no—
48C2×C3×C5C22×C411yescharacteristic
49C3×D5—22no—
50C3×D5C2×D411yescharacteristic
51C3×D5#4111yescharacteristic
52C23×C3—52no—
53C3×D4—52no—
54C3×D4—51no—
55C3×D4AGL(1,𝔽5)11yescharacteristic, direct product factor
56C2×C3×C4—51no—
57C2×C3×C4—51no—
58C2×C3×C4—51no—
59C2×C3×C4—52no—
60C2×C3×C4—104no—
61C22×C5C2×C3×C412yescomplemented
62C2×D5—22no—
63C2×D5—22no—
64C2×D5C2×C3×C412yescomplemented
65C2×D5C23×C311yescharacteristic
66AGL(1,𝔽5)—22no—
67AGL(1,𝔽5)—21no—
68AGL(1,𝔽5)C3×D412yesdirect product factor
69C4×C5C2×C3×C411yescharacteristic, complemented
70Dic5C2×C3×C411yescharacteristic, complemented
71C2×D4—51no—
72C22×C4—52no—
73C4⋊C4—51no—
74C22⋊C4—52no—
75C42—51no—
76C3×C5C4×D411yescharacteristic, complemented, Hall
77C22×C3—51no—
78C22×C3—52no—
79C22×C3—102no—
80C22×C3—102no—
81C22×C3C2×AGL(1,𝔽5)12yescomplemented
82C3×C4—52no—
83C3×C4—51no—
84C3×C4—101no—
85C3×C4—102no—
86C3×C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
87C2×C5—22no—
88C2×C5C22×C3×C411yescharacteristic
89D5—22no—
90D5C2×C3×D411yescharacteristic
91D5C3×#4111yescharacteristic
92C23—52no—
93D4—52no—
94D4—51no—
95D4C3×AGL(1,𝔽5)11yescharacteristic, direct product factor
96C2×C4—51no—
97C2×C4—51no—
98C2×C4—51no—
99C2×C4—52no—
100C2×C4—104no—
101C2×C3—22no—
102C2×C3—51no—
103C2×C3—51no—
104C2×C3—102no—
105C2×C3C22×AGL(1,𝔽5)11yescharacteristic
106C5C3×C4×D411yescharacteristic, complemented, Sylow
107C22—51no—
108C22—52no—
109C22—102no—
110C22—102no—
111C22C2×C3×AGL(1,𝔽5)12yescomplemented
112C4—52no—
113C4—51no—
114C4—102no—
115C4—101no—
116C4C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
117C3D4×AGL(1,𝔽5)11yescharacteristic, direct product factor, Sylow
118C2—22no—
119C2—51no—
120C2—51no—
121C2—102no—
122C2C22×C3×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C3C22×AGL(1,𝔽5)
commutatorC2×C5C22×C3×C4
FrattiniC2C22×C3×AGL(1,𝔽5)
FittingC3×C5×D4C4
socleC2×C3×C5C22×C4
prime radicalC3×D4×D5C2
cosocleC2×D5C23×C3
hypercenterC3×D4AGL(1,𝔽5)
hypocenterC5C3×C4×D4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C3×D4

Complements

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

NH ≅ G/Ntype
1. C22×C3×AGL(1,𝔽5)C2semidirect
3. C3×C22⋊AGL(1,𝔽5)C2semidirect
4. C3×C4⋊AGL(1,𝔽5)C2semidirect
5. C3×C4×AGL(1,𝔽5)C2semidirect
6. D4×AGL(1,𝔽5)C3direct (N × H)
12. C3×C5×D4C4semidirect
13. C3×C22⋊D5C4semidirect
14. C3×D20C4semidirect
17. C22×AGL(1,𝔽5)C2×C3semidirect
19. C22⋊AGL(1,𝔽5)C2×C3semidirect
20. C4⋊AGL(1,𝔽5)C2×C3semidirect
21. C4×AGL(1,𝔽5)C2×C3semidirect
22. C22×C3×C5C2×C4semidirect
25. C2×C3×D5C2×C4semidirect
29. C3×AGL(1,𝔽5)D4direct (N × H)
30. C3×C4×C5C2×C4semidirect
31. C3×Dic5C2×C4semidirect
42. C5×D4C3×C4semidirect
43. C22⋊D5C3×C4semidirect
44. D20C3×C4semidirect
55. C3×D4AGL(1,𝔽5)direct (N × H)
61. C22×C5C2×C3×C4semidirect
64. C2×D5C2×C3×C4semidirect
68. AGL(1,𝔽5)C3×D4direct (N × H)
69. C4×C5C2×C3×C4semidirect
70. Dic5C2×C3×C4semidirect
76. C3×C5C4×D4semidirect
81. C22×C3C2×AGL(1,𝔽5)semidirect
86. C3×C4C2×AGL(1,𝔽5)semidirect
95. D4C3×AGL(1,𝔽5)direct (N × H)
106. C5C3×C4×D4semidirect
111. C22C2×C3×AGL(1,𝔽5)semidirect
116. C4C2×C3×AGL(1,𝔽5)semidirect
117. C3D4×AGL(1,𝔽5)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×AGL(1,𝔽5)2
C3×D4×D52
C3×C22⋊AGL(1,𝔽5)2
C3×C4⋊AGL(1,𝔽5)2
C3×C4×AGL(1,𝔽5)2
D4×AGL(1,𝔽5)3
C3×C4×D45