Finite Group Databasebeta

← C7×C3⋊(C7⋊#63120) #50304422

Subgroups — C7×C3⋊(C7⋊#63120) #50304422

Order 38220.

144 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 50 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×C72×D5×C13C211yesmaximal, characteristic
2C7×C7⋊#63120—31nomaximal, Hall
3C3×C5×C72×C13C411yescharacteristic, complemented, Hall
4C7×Dic273—51nomaximal, Hall
5C72×D5×C13D311yescharacteristic
6C3⋊(C7⋊#63120)C711yesmaximal, characteristic, direct product factor
7C7×C3⋊#63120—71nomaximal
8C2×C3×C72×C13—51no—
9C5×C72×C13Dic311yescharacteristic, complemented, Hall
10C7×C3⋊(C7⋊AGL(1,𝔽5))—131nomaximal, Hall
11C3×C7×D5×C13—23no—
12C3×C7×D5×C13D711yescharacteristic
13C3×C7×D5×C13C2×C711yescharacteristic
14C7×Dic91—151noHall
15C3×C72×C13AGL(1,𝔽5)11yescharacteristic, complemented, Hall
16C7⋊#63120—31no—
17C7×#63120—211no—
18C3×C72×D5D1311yescharacteristic
19C3×C5×C7×C13—23no—
20C3×C5×C7×C13Dic711yescharacteristic, complemented
21C3×C5×C7×C13C4×C711yescharacteristic, complemented
22C2×C72×C13—51no—
23Dic273—51no—
24C7×Dic39—351no—
25C7×C7⋊AGL(1,𝔽5)—391noHall
26C7×D5×C13—23no—
27C7×D5×C13D3×C711yescharacteristic
28C7×D5×C13D2111yescharacteristic
29C3⋊#63120—71noHall
30C3×C5×C72Dic1311yescharacteristic, complemented, Hall
31C72×C13ΓL(1,𝔽16)11yescharacteristic, complemented, Hall
32C7×Dic21—651noHall
33C2×C3×C7×C13—51no—
34C2×C3×C7×C13—51no—
35C2×C3×C7×C13—103no—
36C72×D5D3911yescharacteristic
37C5×C7×C13—23no—
38C5×C7×C13C7×Dic311yescharacteristic, complemented
39C5×C7×C13Dic2111yescharacteristic, complemented
40C3⋊(C7⋊AGL(1,𝔽5))—131no—
41C7×ΓL(1,𝔽16)—911no—
42C3×D5×C13C7×D711yescharacteristic
43Dic91—151no—
44C7×Dic13—1051no—
45C2×C3×C72—51no—
46C3×C7×C13—23no—
47C3×C7×C13C7×AGL(1,𝔽5)11yescharacteristic, complemented
48C3×C7×C13C7⋊AGL(1,𝔽5)11yescharacteristic, complemented
49#63120—211noHall
50C5×C72Dic3911yescharacteristic, complemented, Hall
51C3×C7×D5—23no—
52C3×C7×D5C7×D1311yescharacteristic
53C3×C7×D5D9111yescharacteristic
54C7×Dic7—1951noHall
55C3×C5×C13C7×Dic711yescharacteristic, complemented, Hall
56C2×C7×C13—51no—
57C2×C7×C13—51no—
58C2×C7×C13—103no—
59Dic39—351noHall
60C3×C72#6312011yescharacteristic, complemented, Hall
61C7⋊AGL(1,𝔽5)—391no—
62C7×AGL(1,𝔽5)—2731no—
63D5×C13C7×D2111yescharacteristic
64C3×C5×C7—23no—
65C3×C5×C7C7×Dic1311yescharacteristic, complemented
66C3×C5×C7Dic9111yescharacteristic, complemented
67C2×C72—51no—
68C7×C13—23no—
69C7×C13C7×ΓL(1,𝔽16)11yescharacteristic, complemented
70C7×C13C3⋊(C7⋊AGL(1,𝔽5))11yescharacteristic, complemented
71Dic21—651no—
72C7×Dic3—4551no—
73C2×C3×C13—51no—
74C7×D5—23no—
75C7×D5C7×D3911yescharacteristic
76C7×D5D27311yescharacteristic
77C5×C13C7×Dic2111yescharacteristic, complemented, Hall
78ΓL(1,𝔽16)—911noHall
79Dic13—1051noHall
80C72C3⋊#6312011yescharacteristic, complemented, Sylow
81C2×C3×C7—51no—
82C2×C3×C7—51no—
83C2×C3×C7—103no—
84C3×C13C7×C7⋊AGL(1,𝔽5)11yescharacteristic, complemented, Hall
85C5×C7—23no—
86C5×C7C7×Dic3911yescharacteristic, complemented
87C5×C7Dic27311yescharacteristic, complemented
88C3×D5C7×D9111yescharacteristic
89C4×C7—13651no—
90Dic7—1951no—
91C2×C13—51no—
92C3×C7—23no—
93C3×C7C7×#6312011yescharacteristic, complemented
94C3×C7C7⋊#6312011yescharacteristic, complemented
95AGL(1,𝔽5)—2731noHall
96C3×C5C7×Dic9111yescharacteristic, complemented, Hall
97C2×C7—51no—
98C2×C7—51no—
99C2×C7—103no—
100C13C7×C3⋊(C7⋊AGL(1,𝔽5))11yescharacteristic, complemented, Sylow
101Dic3—4551noHall
102D5C7×D27311yescharacteristic
103C7—23no—
104C7C7×C3⋊#6312011yescharacteristic, complemented
105C7C3⋊(C7⋊#63120)11yescharacteristic, direct product factor
106C2×C3—51no—
107C5C7×Dic27311yescharacteristic, complemented, Sylow
108C4—13651noSylow
109C3C7×C7⋊#6312011yescharacteristic, complemented, Sylow
110C2—51no—
subgroupG/H
center, hypercenterC7C3⋊(C7⋊#63120)
commutator, hypocenterC3×C5×C7×C13C4×C7
FrattiniC1G
Fitting, socleC3×C5×C72×C13C4
prime radicalC3×C72×D5×C13C2
cosocleC3×C7×D5×C13C2×C7
LCSG ▷ C3×C5×C7×C13
UCSC1 ◁ C7

Complements

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

NH ≅ G/Ntype
3. C3×C5×C72×C13C4semidirect
6. C3⋊(C7⋊#63120)C7direct (N × H)
9. C5×C72×C13Dic3semidirect
15. C3×C72×C13AGL(1,𝔽5)semidirect
20. C3×C5×C7×C13Dic7semidirect
21. C3×C5×C7×C13C4×C7semidirect
30. C3×C5×C72Dic13semidirect
31. C72×C13ΓL(1,𝔽16)semidirect
38. C5×C7×C13C7×Dic3semidirect
39. C5×C7×C13Dic21semidirect
47. C3×C7×C13C7×AGL(1,𝔽5)semidirect
48. C3×C7×C13C7⋊AGL(1,𝔽5)semidirect
50. C5×C72Dic39semidirect
55. C3×C5×C13C7×Dic7semidirect
60. C3×C72#63120semidirect
65. C3×C5×C7C7×Dic13semidirect
66. C3×C5×C7Dic91semidirect
69. C7×C13C7×ΓL(1,𝔽16)semidirect
70. C7×C13C3⋊(C7⋊AGL(1,𝔽5))semidirect
77. C5×C13C7×Dic21semidirect
80. C72C3⋊#63120semidirect
84. C3×C13C7×C7⋊AGL(1,𝔽5)semidirect
86. C5×C7C7×Dic39semidirect
87. C5×C7Dic273semidirect
93. C3×C7C7×#63120semidirect
94. C3×C7C7⋊#63120semidirect
96. C3×C5C7×Dic91semidirect
100. C13C7×C3⋊(C7⋊AGL(1,𝔽5))semidirect
104. C7C7×C3⋊#63120semidirect
105. C7C3⋊(C7⋊#63120)direct (N × H)
107. C5C7×Dic273semidirect
109. C3C7×C7⋊#63120semidirect

Maximal subgroups

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

subgroupindex
C3×C72×D5×C132
C7×C7⋊#631203
C7×Dic2735
C3⋊(C7⋊#63120)7
C7×C3⋊#631207
C7×C3⋊(C7⋊AGL(1,𝔽5))13