Finite Group Databasebeta

← C2×C5×C11×#64423 #44568772

Subgroups — C2×C5×C11×#64423 #44568772

Order 32340.

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

normal subgroups only

This group has 208 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
1C2×C5×C11×#3861C211yesmaximal, characteristic, complemented
2C5×C11×#64423C212yesmaximal, direct product factor
3C2×C5×C7×C11×D7C311yesmaximal, characteristic, complemented, Hall
4C5×C11×#3861C2211yescharacteristic, complemented, Hall
5C2×C11×#64423C511yesmaximal, characteristic, direct product factor, Hall
6C5×C7×C11×D7C2×C312yescomplemented
7C2×C5×C72×C11C2×C311yescharacteristic, complemented
8C22×C5×C11×ΓL(1,𝔽8)—71nomaximal
9C2×C5×C11×AGL(1,𝔽7)—71nomaximal
10C2×C11×#3861C2×C511yescharacteristic, complemented
11C11×#64423C2×C512yesdirect product factor
12C2×C5×#64423C1111yesmaximal, characteristic, direct product factor, Hall
13C5×C72×C11C22×C311yescharacteristic, complemented, Hall
14C5×C11×AGL(1,𝔽7)—72no—
15C2×C5×C11×ΓL(1,𝔽8)—72no—
16C2×C5×C11×ΓL(1,𝔽8)—71no—
17C2×C5×C11×ΓL(1,𝔽8)—71no—
18C2×C5×C11×ΓL(1,𝔽8)—143no—
19C2×C7×C11×D7C3×C511yescharacteristic, complemented, Hall
20C11×#3861C22×C511yescharacteristic, complemented, Hall
21C22×C5×C7×C11—71no—
22C2×C5×C11×D7ΓL(1,𝔽8)11yescharacteristic, complemented
23C2×C5×#3861C2×C1111yescharacteristic, complemented
24C5×#64423C2×C1112yesdirect product factor
25C5×C11×ΓL(1,𝔽8)—71no—
26C5×C11×ΓL(1,𝔽8)—71no—
27C5×C11×ΓL(1,𝔽8)—143no—
28C2×C72×C11C2×C3×C511yescharacteristic, complemented
29C7×C11×D7C2×C3×C512yescomplemented
30C2×C5×C7×D7C3×C1111yescharacteristic, complemented, Hall
31C22×C11×ΓL(1,𝔽8)—71no—
32C2×C11×AGL(1,𝔽7)—71no—
33C2×C5×C7×C11—23no—
34C2×C5×C7×C11—72no—
35C2×C5×C7×C11AGL(1,𝔽7)11yescharacteristic, complemented
36C2×C5×C7×C11C2×ΓL(1,𝔽8)11yescharacteristic, complemented
37C5×C11×D7C2×ΓL(1,𝔽8)12yescomplemented
38C5×#3861C22×C1111yescharacteristic, complemented, Hall
39C22×C3×C5×C11—491noHall
40C2×#64423C5×C1111yescharacteristic, direct product factor, Hall
41C72×C11C22×C3×C511yescharacteristic, complemented, Hall
42C2×C5×C72C2×C3×C1111yescharacteristic, complemented
43C5×C7×D7C2×C3×C1112yescomplemented
44C2×C11×ΓL(1,𝔽8)—72no—
45C2×C11×ΓL(1,𝔽8)—71no—
46C2×C11×ΓL(1,𝔽8)—71no—
47C2×C11×ΓL(1,𝔽8)—143no—
48C11×AGL(1,𝔽7)—72no—
49C22×C5×ΓL(1,𝔽8)—71no—
50C2×C5×AGL(1,𝔽7)—71no—
51C5×C7×C11—23no—
52C5×C7×C11C2×AGL(1,𝔽7)11yescharacteristic, complemented
53C5×C7×C11C22×ΓL(1,𝔽8)11yescharacteristic, complemented
54C2×C3×C5×C11—492no—
55C2×C3×C5×C11—491no—
56C22×C7×C11—71no—
57C2×C11×D7C5×ΓL(1,𝔽8)11yescharacteristic, complemented
58C2×#3861C2×C5×C1111yescharacteristic, complemented
59#64423C2×C5×C1112yesdirect product factor
60C5×C72C22×C3×C1111yescharacteristic, complemented, Hall
61C11×ΓL(1,𝔽8)—71no—
62C11×ΓL(1,𝔽8)—71no—
63C11×ΓL(1,𝔽8)—143no—
64C22×C5×C11—71noHall
65C2×C5×ΓL(1,𝔽8)—72no—
66C2×C5×ΓL(1,𝔽8)—71no—
67C2×C5×ΓL(1,𝔽8)—71no—
68C2×C5×ΓL(1,𝔽8)—143no—
69C5×AGL(1,𝔽7)—72no—
70C2×C7×D7C3×C5×C1111yescharacteristic, complemented, Hall
71C3×C5×C11—491noHall
72C2×C7×C11—23no—
73C2×C7×C11—72no—
74C2×C7×C11C5×AGL(1,𝔽7)11yescharacteristic, complemented
75C2×C7×C11C2×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
76C11×D7C2×C5×ΓL(1,𝔽8)12yescomplemented
77#3861C22×C5×C1111yescharacteristic, complemented, Hall
78C22×C5×C7—71no—
79C2×C5×D7C11×ΓL(1,𝔽8)11yescharacteristic, complemented
80C22×C3×C11—491noHall
81C2×C5×C11—72no—
82C2×C5×C11#6442311yescharacteristic, direct product factor
83C5×ΓL(1,𝔽8)—71no—
84C5×ΓL(1,𝔽8)—71no—
85C5×ΓL(1,𝔽8)—143no—
86C2×C72C2×C3×C5×C1111yescharacteristic, complemented
87C7×D7C2×C3×C5×C1112yescomplemented
88C22×ΓL(1,𝔽8)—71no—
89C2×AGL(1,𝔽7)—71no—
90C7×C11—23no—
91C7×C11C2×C5×AGL(1,𝔽7)11yescharacteristic, complemented
92C7×C11C22×C5×ΓL(1,𝔽8)11yescharacteristic, complemented
93C2×C5×C7—23no—
94C2×C5×C7—72no—
95C2×C5×C7C11×AGL(1,𝔽7)11yescharacteristic, complemented
96C2×C5×C7C2×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
97C5×D7C2×C11×ΓL(1,𝔽8)12yescomplemented
98C2×C3×C11—492no—
99C2×C3×C11—491no—
100C22×C3×C5—491noHall
101C5×C11C2×#6442311yescharacteristic, direct product factor, Hall
102C72C22×C3×C5×C1111yescharacteristic, complemented, Sylow
103C22×C11—71noHall
104C2×ΓL(1,𝔽8)—72no—
105C2×ΓL(1,𝔽8)—71no—
106C2×ΓL(1,𝔽8)—71no—
107C2×ΓL(1,𝔽8)—143no—
108AGL(1,𝔽7)—72no—
109C5×C7—23no—
110C5×C7C2×C11×AGL(1,𝔽7)11yescharacteristic, complemented
111C5×C7C22×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
112C3×C11—491noHall
113C2×C3×C5—492no—
114C2×C3×C5—491no—
115C22×C7—71no—
116C2×D7C5×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
117C2×C11—72no—
118C2×C11C5×#6442311yescharacteristic, direct product factor
119ΓL(1,𝔽8)—71no—
120ΓL(1,𝔽8)—71no—
121ΓL(1,𝔽8)—143no—
122C22×C5—71noHall
123C3×C5—491noHall
124C2×C7—23no—
125C2×C7—72no—
126C2×C7C2×C5×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
127C2×C7C5×C11×AGL(1,𝔽7)11yescharacteristic, complemented
128D7C2×C5×C11×ΓL(1,𝔽8)12yescomplemented
129C22×C3—491noHall
130C11C2×C5×#6442311yescharacteristic, direct product factor, Sylow
131C2×C5—72no—
132C2×C5C11×#6442311yescharacteristic, direct product factor
133C7—23no—
134C7C2×C5×C11×AGL(1,𝔽7)11yescharacteristic, complemented
135C7C22×C5×C11×ΓL(1,𝔽8)11yescharacteristic, complemented
136C2×C3—492no—
137C2×C3—491no—
138C5C2×C11×#6442311yescharacteristic, direct product factor, Sylow
139C22—71noSylow
140C3—491noSylow
141C2—72no—
142C2C5×C11×#6442311yescharacteristic, direct product factor
subgroupG/H
center, hypercenterC2×C5×C11#64423
commutator, cosocle, hypocenterC72C22×C3×C5×C11
FrattiniC1G
Fitting, socleC2×C5×C72×C11C2×C3
LCSG ▷ C72
UCSC1 ◁ C2×C5×C11

Complements

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

NH ≅ G/Ntype
1. C2×C5×C11×#3861C2semidirect
2. C5×C11×#64423C2direct (N × H)
3. C2×C5×C7×C11×D7C3semidirect
4. C5×C11×#3861C22semidirect
5. C2×C11×#64423C5direct (N × H)
6. C5×C7×C11×D7C2×C3semidirect
7. C2×C5×C72×C11C2×C3semidirect
10. C2×C11×#3861C2×C5semidirect
11. C11×#64423C2×C5direct (N × H)
12. C2×C5×#64423C11direct (N × H)
13. C5×C72×C11C22×C3semidirect
19. C2×C7×C11×D7C3×C5semidirect
20. C11×#3861C22×C5semidirect
22. C2×C5×C11×D7ΓL(1,𝔽8)semidirect
23. C2×C5×#3861C2×C11semidirect
24. C5×#64423C2×C11direct (N × H)
28. C2×C72×C11C2×C3×C5semidirect
29. C7×C11×D7C2×C3×C5semidirect
30. C2×C5×C7×D7C3×C11semidirect
35. C2×C5×C7×C11AGL(1,𝔽7)semidirect
36. C2×C5×C7×C11C2×ΓL(1,𝔽8)semidirect
37. C5×C11×D7C2×ΓL(1,𝔽8)semidirect
38. C5×#3861C22×C11semidirect
40. C2×#64423C5×C11direct (N × H)
41. C72×C11C22×C3×C5semidirect
42. C2×C5×C72C2×C3×C11semidirect
43. C5×C7×D7C2×C3×C11semidirect
52. C5×C7×C11C2×AGL(1,𝔽7)semidirect
53. C5×C7×C11C22×ΓL(1,𝔽8)semidirect
57. C2×C11×D7C5×ΓL(1,𝔽8)semidirect
58. C2×#3861C2×C5×C11semidirect
59. #64423C2×C5×C11direct (N × H)
60. C5×C72C22×C3×C11semidirect
70. C2×C7×D7C3×C5×C11semidirect
74. C2×C7×C11C5×AGL(1,𝔽7)semidirect
75. C2×C7×C11C2×C5×ΓL(1,𝔽8)semidirect
76. C11×D7C2×C5×ΓL(1,𝔽8)semidirect
77. #3861C22×C5×C11semidirect
79. C2×C5×D7C11×ΓL(1,𝔽8)semidirect
82. C2×C5×C11#64423direct (N × H)
86. C2×C72C2×C3×C5×C11semidirect
87. C7×D7C2×C3×C5×C11semidirect
91. C7×C11C2×C5×AGL(1,𝔽7)semidirect
92. C7×C11C22×C5×ΓL(1,𝔽8)semidirect
95. C2×C5×C7C11×AGL(1,𝔽7)semidirect
96. C2×C5×C7C2×C11×ΓL(1,𝔽8)semidirect
97. C5×D7C2×C11×ΓL(1,𝔽8)semidirect
101. C5×C11C2×#64423direct (N × H)
102. C72C22×C3×C5×C11semidirect
110. C5×C7C2×C11×AGL(1,𝔽7)semidirect
111. C5×C7C22×C11×ΓL(1,𝔽8)semidirect
116. C2×D7C5×C11×ΓL(1,𝔽8)semidirect
118. C2×C11C5×#64423direct (N × H)
126. C2×C7C2×C5×C11×ΓL(1,𝔽8)semidirect
127. C2×C7C5×C11×AGL(1,𝔽7)semidirect
128. D7C2×C5×C11×ΓL(1,𝔽8)semidirect
130. C11C2×C5×#64423direct (N × H)
132. C2×C5C11×#64423direct (N × H)
134. C7C2×C5×C11×AGL(1,𝔽7)semidirect
135. C7C22×C5×C11×ΓL(1,𝔽8)semidirect
138. C5C2×C11×#64423direct (N × H)
142. C2C5×C11×#64423direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×C11×#38612
C5×C11×#644232
C2×C5×C7×C11×D73
C2×C11×#644235
C22×C5×C11×ΓL(1,𝔽8)7
C2×C5×C11×AGL(1,𝔽7)7
C2×C5×#6442311