Finite Group Databasebeta

← C2×C7×D5×#63270 #50304279

Subgroups — C2×C7×D5×#63270 #50304279

Order 38220.

200 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 70 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×C5×C7×#63270C211yesmaximal, characteristic, complemented
2C7×D5×#63270C212yesmaximal, direct product factor
3C2×C72×D5×C13C311yesmaximal, characteristic, complemented, Hall
4C5×C7×#63270C2211yescharacteristic, complemented, Hall
5C22×C7×#63270—51nomaximal, Hall
6C72×D5×C13C2×C312yescomplemented
7C2×C5×C72×C13C2×C311yescharacteristic, complemented
8C2×D5×#63270C711yesmaximal, characteristic, direct product factor
9C2×C7×D5×C13⋊C3—71nomaximal
10C2×C7×#63270—52no—
11C2×C7×#63270D511yescharacteristic, direct product factor
12C5×C72×C13C22×C311yescharacteristic, complemented, Hall
13C2×C7×D5×ΓL(1,𝔽8)—131nomaximal, Hall
14D5×#63270C2×C712yesdirect product factor
15C7×D5×C13⋊C3—72no—
16C2×C5×#63270C2×C711yescharacteristic, complemented
17C2×C5×C7×C13⋊C3—71no—
18C22×C72×C13—51noHall
19C7×#63270C2×D511yescharacteristic, direct product factor, Hall
20C2×C7×D5×C13—32no—
21C2×C7×D5×C13ΓL(1,𝔽8)11yescharacteristic, complemented
22C2×C7×D5×C13C3×C711yescharacteristic, complemented
23C2×C5×C7×ΓL(1,𝔽8)—131no—
24C7×D5×ΓL(1,𝔽8)—132no—
25C5×#63270C22×C711yescharacteristic, complemented
26C5×C7×C13⋊C3—71no—
27C2×C72×C13—52no—
28C2×C72×C13C3×D511yescharacteristic, complemented
29C22×#63270—51no—
30C22×C7×C13⋊C3—351no—
31C2×C72×D5C13⋊C311yescharacteristic, complemented, Hall
32C2×C5×C7×C13—32no—
33C2×C5×C7×C13C2×ΓL(1,𝔽8)11yescharacteristic, complemented
34C2×C5×C7×C13C2×C3×C711yescharacteristic, complemented
35C7×D5×C13—34no—
36C7×D5×C13C2×ΓL(1,𝔽8)12yescomplemented
37C7×D5×C13C2×C3×C712yescomplemented
38C2×D5×C13⋊C3—71noHall
39C5×C7×ΓL(1,𝔽8)—131noHall
40C72×C13C2×C3×D511yescharacteristic, complemented, Hall
41C22×C7×ΓL(1,𝔽8)—651noHall
42C2×#63270—52no—
43C2×#63270C7×D511yescharacteristic, direct product factor
44C2×C7×C13⋊C3—71no—
45C2×C7×C13⋊C3—352no—
46C2×C5×C72C2×C13⋊C311yescharacteristic, complemented
47C72×D5C2×C13⋊C312yescomplemented
48C5×C7×C13—32no—
49C5×C7×C13C22×ΓL(1,𝔽8)11yescharacteristic, complemented
50C5×C7×C13C22×C3×C711yescharacteristic, complemented
51C2×C3×C7×D5—911no—
52C2×D5×ΓL(1,𝔽8)—131no—
53C2×C5×C13⋊C3—71no—
54D5×C13⋊C3—72no—
55C22×C7×C13—51no—
56C22×C7×C13—51no—
57C22×C7×C13—152no—
58C2×C7×ΓL(1,𝔽8)—131no—
59C2×C7×ΓL(1,𝔽8)—652no—
60#63270C2×C7×D511yescharacteristic, direct product factor
61C7×C13⋊C3—71no—
62C2×D5×C13C7×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
63C5×C72C22×C13⋊C311yescharacteristic, complemented, Hall
64C2×C3×C5×C7—911no—
65C3×C7×D5—912no—
66C2×C5×ΓL(1,𝔽8)—131no—
67D5×ΓL(1,𝔽8)—132no—
68C22×C72—51noHall
69C5×C13⋊C3—71noHall
70C2×C7×C13—32no—
71C2×C7×C13—52no—
72C2×C7×C13—52no—
73C2×C7×C13—154no—
74C2×C7×C13D5×ΓL(1,𝔽8)11yescharacteristic, complemented
75C2×C7×C13C3×C7×D511yescharacteristic, complemented
76C22×C13⋊C3—351noHall
77C7×ΓL(1,𝔽8)—131noHall
78C2×C7×D5—32no—
79C2×C7×D5C7×C13⋊C311yescharacteristic, complemented
80C2×C7×D5#6327011yescharacteristic, direct product factor
81C2×C5×C13C2×C7×ΓL(1,𝔽8)11yescharacteristic, complemented
82D5×C13C2×C7×ΓL(1,𝔽8)12yescomplemented
83C3×C5×C7—911no—
84C5×ΓL(1,𝔽8)—131no—
85C2×C72—52no—
86C2×C72D5×C13⋊C311yescharacteristic, complemented
87C7×C13—32no—
88C7×C13C2×D5×ΓL(1,𝔽8)11yescharacteristic, complemented
89C7×C13C2×C3×C7×D511yescharacteristic, complemented
90C22×C3×C7—4551no—
91C22×ΓL(1,𝔽8)—651no—
92C2×C13⋊C3—71no—
93C2×C13⋊C3—352no—
94C2×C5×C7—32no—
95C2×C5×C7C2×C7×C13⋊C311yescharacteristic, complemented
96C2×C5×C7C2×#6327011yescharacteristic, complemented
97C7×D5—34no—
98C7×D5C2×C7×C13⋊C312yescomplemented
99C7×D5C2×#6327012yesdirect product factor
100C5×C13C22×C7×ΓL(1,𝔽8)11yescharacteristic, complemented, Hall
101C2×C3×D5—911noHall
102C22×C13—51noHall
103C72C2×D5×C13⋊C311yescharacteristic, complemented, Sylow
104C2×C3×C7—911no—
105C2×C3×C7—4552no—
106C2×ΓL(1,𝔽8)—131no—
107C2×ΓL(1,𝔽8)—652no—
108C13⋊C3—71noHall
109C5×C7—32no—
110C5×C7C22×C7×C13⋊C311yescharacteristic, complemented
111C5×C7C22×#6327011yescharacteristic, complemented
112C2×C3×C5—911no—
113C3×D5—912no—
114C22×C7—51no—
115C22×C7—51no—
116C22×C7—152no—
117C2×C13—52no—
118C2×C13C7×D5×ΓL(1,𝔽8)11yescharacteristic, complemented
119C3×C7—911no—
120ΓL(1,𝔽8)—131no—
121C2×D5C7×#6327011yescharacteristic, direct product factor, Hall
122C3×C5—911noHall
123C2×C7—32no—
124C2×C7—52no—
125C2×C7—52no—
126C2×C7—154no—
127C2×C7C7×D5×C13⋊C311yescharacteristic, complemented
128C2×C7D5×#6327011yescharacteristic, direct product factor
129C13C2×C7×D5×ΓL(1,𝔽8)11yescharacteristic, complemented, Sylow
130C22×C3—4551noHall
131C2×C5C2×C7×#6327011yescharacteristic, complemented
132D5C2×C7×#6327012yesdirect product factor
133C7—32no—
134C7C2×C7×D5×C13⋊C311yescharacteristic, complemented
135C7C2×D5×#6327011yescharacteristic, direct product factor
136C2×C3—911no—
137C2×C3—4552no—
138C5C22×C7×#6327011yescharacteristic, complemented, Sylow
139C22—51noSylow
140C3—911noSylow
141C2—52no—
142C2C7×D5×#6327011yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C7D5×#63270
commutator, cosocle, hypocenterC5×C7×C13C22×C3×C7
FrattiniC1G
Fitting, socleC2×C5×C72×C13C2×C3
LCSG ▷ C5×C7×C13
UCSC1 ◁ C2×C7

Complements

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

NH ≅ G/Ntype
1. C2×C5×C7×#63270C2semidirect
2. C7×D5×#63270C2direct (N × H)
3. C2×C72×D5×C13C3semidirect
4. C5×C7×#63270C22semidirect
6. C72×D5×C13C2×C3semidirect
7. C2×C5×C72×C13C2×C3semidirect
8. C2×D5×#63270C7direct (N × H)
11. C2×C7×#63270D5direct (N × H)
12. C5×C72×C13C22×C3semidirect
14. D5×#63270C2×C7direct (N × H)
16. C2×C5×#63270C2×C7semidirect
19. C7×#63270C2×D5direct (N × H)
21. C2×C7×D5×C13ΓL(1,𝔽8)semidirect
22. C2×C7×D5×C13C3×C7semidirect
25. C5×#63270C22×C7semidirect
28. C2×C72×C13C3×D5semidirect
31. C2×C72×D5C13⋊C3semidirect
33. C2×C5×C7×C13C2×ΓL(1,𝔽8)semidirect
34. C2×C5×C7×C13C2×C3×C7semidirect
36. C7×D5×C13C2×ΓL(1,𝔽8)semidirect
37. C7×D5×C13C2×C3×C7semidirect
40. C72×C13C2×C3×D5semidirect
43. C2×#63270C7×D5direct (N × H)
46. C2×C5×C72C2×C13⋊C3semidirect
47. C72×D5C2×C13⋊C3semidirect
49. C5×C7×C13C22×ΓL(1,𝔽8)semidirect
50. C5×C7×C13C22×C3×C7semidirect
60. #63270C2×C7×D5direct (N × H)
62. C2×D5×C13C7×ΓL(1,𝔽8)semidirect
63. C5×C72C22×C13⋊C3semidirect
74. C2×C7×C13D5×ΓL(1,𝔽8)semidirect
75. C2×C7×C13C3×C7×D5semidirect
79. C2×C7×D5C7×C13⋊C3semidirect
80. C2×C7×D5#63270direct (N × H)
81. C2×C5×C13C2×C7×ΓL(1,𝔽8)semidirect
82. D5×C13C2×C7×ΓL(1,𝔽8)semidirect
86. C2×C72D5×C13⋊C3semidirect
88. C7×C13C2×D5×ΓL(1,𝔽8)semidirect
89. C7×C13C2×C3×C7×D5semidirect
95. C2×C5×C7C2×C7×C13⋊C3semidirect
96. C2×C5×C7C2×#63270semidirect
98. C7×D5C2×C7×C13⋊C3semidirect
99. C7×D5C2×#63270direct (N × H)
100. C5×C13C22×C7×ΓL(1,𝔽8)semidirect
103. C72C2×D5×C13⋊C3semidirect
110. C5×C7C22×C7×C13⋊C3semidirect
111. C5×C7C22×#63270semidirect
118. C2×C13C7×D5×ΓL(1,𝔽8)semidirect
121. C2×D5C7×#63270direct (N × H)
127. C2×C7C7×D5×C13⋊C3semidirect
128. C2×C7D5×#63270direct (N × H)
129. C13C2×C7×D5×ΓL(1,𝔽8)semidirect
131. C2×C5C2×C7×#63270semidirect
132. D5C2×C7×#63270direct (N × H)
134. C7C2×C7×D5×C13⋊C3semidirect
135. C7C2×D5×#63270direct (N × H)
138. C5C22×C7×#63270semidirect
142. C2C7×D5×#63270direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×C7×#632702
C7×D5×#632702
C2×C72×D5×C133
C22×C7×#632705
C2×D5×#632707
C2×C7×D5×C13⋊C37
C2×C7×D5×ΓL(1,𝔽8)13