Finite Group Databasebeta

← C2×C3×C5×ΓL(1,𝔽8)×D23 #44394523

Subgroups — C2×C3×C5×ΓL(1,𝔽8)×D23 #44394523

Order 28980.

240 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C2×C3×C5×ΓL(1,𝔽8)×C23C211yesmaximal, characteristic, complemented
2C3×C5×ΓL(1,𝔽8)×D23C212yesmaximal, direct product factor
3C2×C3×C5×C7×D23C311yesmaximal, characteristic, complemented
4C2×C5×ΓL(1,𝔽8)×D23C313yesmaximal, direct product factor
5C3×C5×ΓL(1,𝔽8)×C23C2211yescharacteristic, complemented, Hall
6C2×C3×ΓL(1,𝔽8)×D23C511yesmaximal, characteristic, direct product factor, Hall
7C5×ΓL(1,𝔽8)×D23C2×C316yesdirect product factor
8C3×C5×C7×D23C2×C312yescomplemented
9C2×C5×ΓL(1,𝔽8)×C23C2×C313yescomplemented
10C2×C3×C5×C7×C23C2×C311yescharacteristic, complemented
11C2×C32×C5×D23—71nomaximal, Hall
12C2×C5×C7×D23C3211yescharacteristic, complemented, Hall
13C2×C3×ΓL(1,𝔽8)×C23C2×C511yescharacteristic, complemented
14C3×ΓL(1,𝔽8)×D23C2×C512yesdirect product factor
15C5×ΓL(1,𝔽8)×C23C22×C313yescomplemented
16C3×C5×C7×C23C22×C311yescharacteristic, complemented
17C32×C5×D23—72no—
18C2×C32×C5×C23—71no—
19C2×C3×C7×D23C3×C511yescharacteristic, complemented
20C2×ΓL(1,𝔽8)×D23C3×C513yesdirect product factor
21C2×C5×C7×C23C2×C3211yescharacteristic, complemented
22C5×C7×D23C2×C3212yescomplemented
23C3×ΓL(1,𝔽8)×C23C22×C511yescharacteristic, complemented, Hall
24C2×C3×C5×D23—73no—
25C2×C3×C5×D23ΓL(1,𝔽8)11yescharacteristic, direct product factor
26C22×C3×C5×ΓL(1,𝔽8)—231nomaximal, Hall
27C32×C5×C23—71noHall
28C2×C3×C7×C23C2×C3×C511yescharacteristic, complemented
29C3×C7×D23C2×C3×C512yescomplemented
30C2×ΓL(1,𝔽8)×C23C2×C3×C513yescomplemented
31ΓL(1,𝔽8)×D23C2×C3×C516yesdirect product factor
32C2×C32×D23—71noHall
33C5×C7×C23C22×C3211yescharacteristic, complemented, Hall
34C2×C3×C5×C23—73no—
35C2×C3×C5×C23C2×ΓL(1,𝔽8)11yescharacteristic, complemented
36C3×C5×D23—76no—
37C3×C5×D23C2×ΓL(1,𝔽8)12yesdirect product factor
38C2×C7×D23C32×C511yescharacteristic, complemented, Hall
39C2×C3×C5×ΓL(1,𝔽8)—232no—
40C2×C3×C5×ΓL(1,𝔽8)D2311yescharacteristic, direct product factor
41C3×C7×C23C22×C3×C511yescharacteristic, complemented
42ΓL(1,𝔽8)×C23C22×C3×C513yescomplemented
43C2×C5×D23C3×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
44C22×C3×C5×C7—231no—
45C22×C5×ΓL(1,𝔽8)—233no—
46C2×C32×C23—71no—
47C32×D23—72no—
48C3×C5×C23—73no—
49C3×C5×C23C22×ΓL(1,𝔽8)11yescharacteristic, complemented
50C2×C7×C23C2×C32×C511yescharacteristic, complemented
51C7×D23C2×C32×C512yescomplemented
52C3×C5×ΓL(1,𝔽8)C2×D2311yescharacteristic, direct product factor, Hall
53C2×C3×D23—73no—
54C2×C3×D23C5×ΓL(1,𝔽8)11yescharacteristic, direct product factor
55C22×C3×ΓL(1,𝔽8)—231noHall
56C2×C5×C23C2×C3×ΓL(1,𝔽8)11yescharacteristic, complemented
57C5×D23C2×C3×ΓL(1,𝔽8)12yesdirect product factor
58C2×C3×C5×C7—232no—
59C2×C3×C5×C7C3×D2311yescharacteristic, complemented
60C2×C5×ΓL(1,𝔽8)—236no—
61C2×C5×ΓL(1,𝔽8)C3×D2313yesdirect product factor
62C32×C23—71noHall
63C22×C32×C5—1611noHall
64C7×C23C22×C32×C511yescharacteristic, complemented, Hall
65C22×C5×C7—231noHall
66C2×C3×C23—73no—
67C2×C3×C23C2×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
68C3×D23—76no—
69C3×D23C2×C5×ΓL(1,𝔽8)12yesdirect product factor
70C2×C3×ΓL(1,𝔽8)—232no—
71C2×C3×ΓL(1,𝔽8)C5×D2311yescharacteristic, direct product factor
72C5×C23C22×C3×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
73C3×C5×C7C2×C3×D2311yescharacteristic, complemented
74C5×ΓL(1,𝔽8)C2×C3×D2313yesdirect product factor
75C2×D23C3×C5×ΓL(1,𝔽8)11yescharacteristic, direct product factor, Hall
76C2×C32×C5—71no—
77C2×C32×C5—1612no—
78C22×C3×C7—231no—
79C22×ΓL(1,𝔽8)—233no—
80C2×C5×C7—232no—
81C2×C5×C7C32×D2311yescharacteristic, complemented
82C3×C23—73no—
83C3×C23C22×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
84C3×ΓL(1,𝔽8)C2×C5×D2311yescharacteristic, direct product factor, Hall
85C22×C3×C5—231no—
86C22×C3×C5—1613no—
87C2×C23C2×C3×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
88D23C2×C3×C5×ΓL(1,𝔽8)12yesdirect product factor
89C32×C5—71noHall
90C2×C3×C7—232no—
91C2×C3×C7C3×C5×D2311yescharacteristic, complemented
92C2×ΓL(1,𝔽8)—236no—
93C2×ΓL(1,𝔽8)C3×C5×D2313yesdirect product factor
94C22×C32—1611noHall
95C5×C7C2×C32×D2311yescharacteristic, complemented, Hall
96C2×C3×C5—73no—
97C2×C3×C5—232no—
98C2×C3×C5—1616no—
99C2×C3×C5ΓL(1,𝔽8)×D2311yescharacteristic, direct product factor
100C22×C7—231noHall
101C23C22×C3×C5×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
102C3×C7C2×C3×C5×D2311yescharacteristic, complemented
103ΓL(1,𝔽8)C2×C3×C5×D2313yesdirect product factor
104C22×C5—231noHall
105C2×C32—71no—
106C2×C32—1612no—
107C3×C5—73no—
108C3×C5C2×ΓL(1,𝔽8)×D2311yescharacteristic, direct product factor
109C2×C7—232no—
110C2×C7C32×C5×D2311yescharacteristic, complemented
111C22×C3—231no—
112C22×C3—1613no—
113C2×C5—232no—
114C2×C5C3×ΓL(1,𝔽8)×D2311yescharacteristic, direct product factor
115C32—71noSylow
116C7C2×C32×C5×D2311yescharacteristic, complemented, Sylow
117C2×C3—73no—
118C2×C3—232no—
119C2×C3—1616no—
120C2×C3C5×ΓL(1,𝔽8)×D2311yescharacteristic, direct product factor
121C5C2×C3×ΓL(1,𝔽8)×D2311yescharacteristic, direct product factor, Sylow
122C22—231noSylow
123C3—73no—
124C3C2×C5×ΓL(1,𝔽8)×D2311yescharacteristic, direct product factor
125C2—232no—
126C2C3×C5×ΓL(1,𝔽8)×D2311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C3×C5ΓL(1,𝔽8)×D23
commutator, cosocle, hypocenterC7×C23C22×C32×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C23C2×C3
LCSG ▷ C7×C23
UCSC1 ◁ C2×C3×C5

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×ΓL(1,𝔽8)×C23C2semidirect
2. C3×C5×ΓL(1,𝔽8)×D23C2direct (N × H)
3. C2×C3×C5×C7×D23C3semidirect
4. C2×C5×ΓL(1,𝔽8)×D23C3direct (N × H)
5. C3×C5×ΓL(1,𝔽8)×C23C22semidirect
6. C2×C3×ΓL(1,𝔽8)×D23C5direct (N × H)
7. C5×ΓL(1,𝔽8)×D23C2×C3direct (N × H)
8. C3×C5×C7×D23C2×C3semidirect
9. C2×C5×ΓL(1,𝔽8)×C23C2×C3semidirect
10. C2×C3×C5×C7×C23C2×C3semidirect
12. C2×C5×C7×D23C32semidirect
13. C2×C3×ΓL(1,𝔽8)×C23C2×C5semidirect
14. C3×ΓL(1,𝔽8)×D23C2×C5direct (N × H)
15. C5×ΓL(1,𝔽8)×C23C22×C3semidirect
16. C3×C5×C7×C23C22×C3semidirect
19. C2×C3×C7×D23C3×C5semidirect
20. C2×ΓL(1,𝔽8)×D23C3×C5direct (N × H)
21. C2×C5×C7×C23C2×C32semidirect
22. C5×C7×D23C2×C32semidirect
23. C3×ΓL(1,𝔽8)×C23C22×C5semidirect
25. C2×C3×C5×D23ΓL(1,𝔽8)direct (N × H)
28. C2×C3×C7×C23C2×C3×C5semidirect
29. C3×C7×D23C2×C3×C5semidirect
30. C2×ΓL(1,𝔽8)×C23C2×C3×C5semidirect
31. ΓL(1,𝔽8)×D23C2×C3×C5direct (N × H)
33. C5×C7×C23C22×C32semidirect
35. C2×C3×C5×C23C2×ΓL(1,𝔽8)semidirect
37. C3×C5×D23C2×ΓL(1,𝔽8)direct (N × H)
38. C2×C7×D23C32×C5semidirect
40. C2×C3×C5×ΓL(1,𝔽8)D23direct (N × H)
41. C3×C7×C23C22×C3×C5semidirect
42. ΓL(1,𝔽8)×C23C22×C3×C5semidirect
43. C2×C5×D23C3×ΓL(1,𝔽8)direct (N × H)
49. C3×C5×C23C22×ΓL(1,𝔽8)semidirect
50. C2×C7×C23C2×C32×C5semidirect
51. C7×D23C2×C32×C5semidirect
52. C3×C5×ΓL(1,𝔽8)C2×D23direct (N × H)
54. C2×C3×D23C5×ΓL(1,𝔽8)direct (N × H)
56. C2×C5×C23C2×C3×ΓL(1,𝔽8)semidirect
57. C5×D23C2×C3×ΓL(1,𝔽8)direct (N × H)
59. C2×C3×C5×C7C3×D23semidirect
61. C2×C5×ΓL(1,𝔽8)C3×D23direct (N × H)
64. C7×C23C22×C32×C5semidirect
67. C2×C3×C23C2×C5×ΓL(1,𝔽8)semidirect
69. C3×D23C2×C5×ΓL(1,𝔽8)direct (N × H)
71. C2×C3×ΓL(1,𝔽8)C5×D23direct (N × H)
72. C5×C23C22×C3×ΓL(1,𝔽8)semidirect
73. C3×C5×C7C2×C3×D23semidirect
74. C5×ΓL(1,𝔽8)C2×C3×D23direct (N × H)
75. C2×D23C3×C5×ΓL(1,𝔽8)direct (N × H)
81. C2×C5×C7C32×D23semidirect
83. C3×C23C22×C5×ΓL(1,𝔽8)semidirect
84. C3×ΓL(1,𝔽8)C2×C5×D23direct (N × H)
87. C2×C23C2×C3×C5×ΓL(1,𝔽8)semidirect
88. D23C2×C3×C5×ΓL(1,𝔽8)direct (N × H)
91. C2×C3×C7C3×C5×D23semidirect
93. C2×ΓL(1,𝔽8)C3×C5×D23direct (N × H)
95. C5×C7C2×C32×D23semidirect
99. C2×C3×C5ΓL(1,𝔽8)×D23direct (N × H)
101. C23C22×C3×C5×ΓL(1,𝔽8)semidirect
102. C3×C7C2×C3×C5×D23semidirect
103. ΓL(1,𝔽8)C2×C3×C5×D23direct (N × H)
108. C3×C5C2×ΓL(1,𝔽8)×D23direct (N × H)
110. C2×C7C32×C5×D23semidirect
114. C2×C5C3×ΓL(1,𝔽8)×D23direct (N × H)
116. C7C2×C32×C5×D23semidirect
120. C2×C3C5×ΓL(1,𝔽8)×D23direct (N × H)
121. C5C2×C3×ΓL(1,𝔽8)×D23direct (N × H)
124. C3C2×C5×ΓL(1,𝔽8)×D23direct (N × H)
126. C2C3×C5×ΓL(1,𝔽8)×D23direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×ΓL(1,𝔽8)×C232
C3×C5×ΓL(1,𝔽8)×D232
C2×C3×C5×C7×D233
C2×C5×ΓL(1,𝔽8)×D233
C2×C3×ΓL(1,𝔽8)×D235
C2×C32×C5×D237
C22×C3×C5×ΓL(1,𝔽8)23