Finite Group Databasebeta

← C5×C3⋊(C7⋊#63122) #41031998

Subgroups — C5×C3⋊(C7⋊#63122) #41031998

Order 27300.

104 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 42 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
1C3×C5×C7×D65C211yesmaximal, characteristic
2C5×C7⋊#63122—31nomaximal, Hall
3C3×C52×C7×C13C411yescharacteristic, complemented, Hall
4C3⋊(C7⋊#63122)C511yesmaximal, characteristic, direct product factor
5C5×C3⋊(C7⋊#261)—51nomaximal
6C5×C7×D65D311yescharacteristic
7C5×C3⋊#63122—71nomaximal, Hall
8C3×C7×D65C2×C511yescharacteristic
9C3×C5×C7×D13—51no—
10C52×C7×C13Dic311yescharacteristic, complemented, Hall
11C5×C3⋊(C7⋊AGL(1,𝔽5))—131nomaximal, Hall
12C3×C5×D65D711yescharacteristic
13C7⋊#63122—31no—
14C5×C7⋊#261—151no—
15C3×C5×C7×C13—41no—
16C3×C5×C7×C13C4×C511yescharacteristic, complemented
17C3×C5×C7×C13AGL(1,𝔽5)11yescharacteristic, complemented
18C5×#63122—211noHall
19C3⋊(C7⋊#261)—51noHall
20C3×C5×C7×D5—131no—
21C3×C52×C13Dic711yescharacteristic, complemented, Hall
22C7×D65C5×D311yescharacteristic
23C5×C7×D13—51no—
24C3⋊#63122—71no—
25C5×C3⋊#261—351no—
26C5×C7⋊AGL(1,𝔽5)—391noHall
27C5×D65D2111yescharacteristic
28C3×C7×D13—51no—
29C3×C52×C7#26111yescharacteristic, complemented, Hall
30C5×C7×C13—41no—
31C5×C7×C13C5×Dic311yescharacteristic, complemented
32C5×C7×C13ΓL(1,𝔽16)11yescharacteristic, complemented
33C3⋊(C7⋊AGL(1,𝔽5))—131no—
34C5×Dic21—651no—
35C3×D65C5×D711yescharacteristic
36C3×C5×D13—51no—
37C7⋊#261—151noHall
38C5×C7×D5—131no—
39C52×C13Dic2111yescharacteristic, complemented, Hall
40C5×ΓL(1,𝔽16)—911noHall
41C3×C7×C13C5×AGL(1,𝔽5)11yescharacteristic, complemented, Hall
42#63122—211no—
43C5×#261—1051no—
44C2×C3×C5×C7—651no—
45C3×C7×D5—131no—
46C3×C5×C13—41no—
47C3×C5×C13C5×Dic711yescharacteristic, complemented
48C3×C5×C13C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
49C7×D13—51no—
50C52×C7C3⋊#26111yescharacteristic, complemented, Hall
51C3⋊#261—351noHall
52C3×C5×D5—131no—
53C7⋊AGL(1,𝔽5)—391no—
54C5×Dic7—1951no—
55D65C5×D2111yescharacteristic
56C5×D13—51no—
57C3×C5×C7—41no—
58C3×C5×C7C5×#26111yescharacteristic, complemented
59C3×C5×C7#6312211yescharacteristic, complemented
60C5×AGL(1,𝔽5)—2731noHall
61C7×C13C5×ΓL(1,𝔽16)11yescharacteristic, complemented, Hall
62Dic21—651noHall
63C3×D13—51no—
64C3×C52C7⋊#26111yescharacteristic, complemented, Hall
65C2×C5×C7—651no—
66C7×D5—131no—
67C5×C13—41no—
68C5×C13C5×Dic2111yescharacteristic, complemented
69C5×C13C3⋊(C7⋊AGL(1,𝔽5))11yescharacteristic, complemented
70ΓL(1,𝔽16)—911no—
71C5×Dic3—4551no—
72#261—1051noHall
73C5×D5—131no—
74C2×C3×C7—651no—
75C3×C13C5×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented, Hall
76C5×C7—41no—
77C5×C7C5×C3⋊#26111yescharacteristic, complemented
78C5×C7C3⋊#6312211yescharacteristic, complemented
79C2×C3×C5—651no—
80C3×D5—131no—
81Dic7—1951noHall
82D13—51no—
83C52C3⋊(C7⋊#261)11yescharacteristic, complemented, Sylow
84C3×C7C5×#6312211yescharacteristic, complemented, Hall
85AGL(1,𝔽5)—2731no—
86C4×C5—13651no—
87C3×C5—41no—
88C3×C5C5×C7⋊#26111yescharacteristic, complemented
89C3×C5C7⋊#6312211yescharacteristic, complemented
90C2×C7—651no—
91C13C5×C3⋊(C7⋊AGL(1,𝔽5))11yescharacteristic, complemented, Sylow
92Dic3—4551noHall
93C2×C5—651no—
94D5—131no—
95C7C5×C3⋊#6312211yescharacteristic, complemented, Sylow
96C2×C3—651no—
97C5—41no—
98C5C5×C3⋊(C7⋊#261)11yescharacteristic, complemented
99C5C3⋊(C7⋊#63122)11yescharacteristic, direct product factor
100C4—13651noSylow
101C3C5×C7⋊#6312211yescharacteristic, complemented, Sylow
102C2—651no—
subgroupG/H
center, hypercenterC5C3⋊(C7⋊#63122)
commutator, hypocenterC3×C5×C7×C13C4×C5
FrattiniC1G
Fitting, socleC3×C52×C7×C13C4
prime radicalC3×C5×C7×D65C2
cosocleC3×C7×D65C2×C5
LCSG ▷ C3×C5×C7×C13
UCSC1 ◁ C5

Complements

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

NH ≅ G/Ntype
3. C3×C52×C7×C13C4semidirect
4. C3⋊(C7⋊#63122)C5direct (N × H)
10. C52×C7×C13Dic3semidirect
16. C3×C5×C7×C13C4×C5semidirect
17. C3×C5×C7×C13AGL(1,𝔽5)semidirect
21. C3×C52×C13Dic7semidirect
29. C3×C52×C7#261semidirect
31. C5×C7×C13C5×Dic3semidirect
32. C5×C7×C13ΓL(1,𝔽16)semidirect
39. C52×C13Dic21semidirect
41. C3×C7×C13C5×AGL(1,𝔽5)semidirect
47. C3×C5×C13C5×Dic7semidirect
48. C3×C5×C13C7⋊AGL(1,𝔽5)semidirect
50. C52×C7C3⋊#261semidirect
58. C3×C5×C7C5×#261semidirect
59. C3×C5×C7#63122semidirect
61. C7×C13C5×ΓL(1,𝔽16)semidirect
64. C3×C52C7⋊#261semidirect
68. C5×C13C5×Dic21semidirect
69. C5×C13C3⋊(C7⋊AGL(1,𝔽5))semidirect
75. C3×C13C5×C7⋊AGL(1,𝔽5)semidirect
77. C5×C7C5×C3⋊#261semidirect
78. C5×C7C3⋊#63122semidirect
83. C52C3⋊(C7⋊#261)semidirect
84. C3×C7C5×#63122semidirect
88. C3×C5C5×C7⋊#261semidirect
89. C3×C5C7⋊#63122semidirect
91. C13C5×C3⋊(C7⋊AGL(1,𝔽5))semidirect
95. C7C5×C3⋊#63122semidirect
98. C5C5×C3⋊(C7⋊#261)semidirect
99. C5C3⋊(C7⋊#63122)direct (N × H)
101. C3C5×C7⋊#63122semidirect

Maximal subgroups

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

subgroupindex
C3×C5×C7×D652
C5×C7⋊#631223
C3⋊(C7⋊#63122)5
C5×C3⋊(C7⋊#261)5
C5×C3⋊#631227
C5×C3⋊(C7⋊AGL(1,𝔽5))13