Finite Group Databasebeta

← C2×#63307⋊C3 #13132433

Subgroups — C2×#63307⋊C3 #13132433

Order 1680.

216 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 67 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
1C22×C5⋊AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
2C22×D5×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
3C2×C5×ΓL(1,𝔽8)⋊C4C211yesmaximal, characteristic, complemented
4#63307⋊C3C214yesmaximal, direct product factor
5C2×#63307C311yesmaximal, characteristic, complemented, Hall
6C22×C5×ΓL(1,𝔽8)C2211yescharacteristic
7C2×C5⋊AGL(1,𝔽7)—22no—
8C2×C5⋊AGL(1,𝔽7)C2212yescomplemented
9C2×D5×ΓL(1,𝔽8)—22no—
10C2×D5×ΓL(1,𝔽8)C2212yescomplemented
11C5×ΓL(1,𝔽8)⋊C4C2212yescomplemented
12C2×(C22⋊D7)⋊C3—51nomaximal, Hall
13C22×D35C2×C311yescharacteristic, complemented
14C22×C7×D5C2×C311yescharacteristic, complemented
15C2×C5×Dic7C2×C311yescharacteristic, complemented
16#63307C2×C314yescomplemented
17C2×C3×D20—71nomaximal, Hall
18C2×C5×ΓL(1,𝔽8)D412yescomplemented
19C2×C5×ΓL(1,𝔽8)C2311yescharacteristic
20C5⋊AGL(1,𝔽7)—22no—
21D5×ΓL(1,𝔽8)—22no—
22C23×ΓL(1,𝔽8)—51no—
23C22×AGL(1,𝔽7)—51no—
24(C22⋊D7)⋊C3—54no—
25C2×ΓL(1,𝔽8)⋊C4D511yescharacteristic, complemented
26C22×C5×C7C22×C311yescharacteristic
27C2×D35—22no—
28C2×D35C22×C312yescomplemented
29C2×C7×D5—22no—
30C2×C7×D5C22×C312yescomplemented
31C5×Dic7C22×C312yescomplemented
32C22×C3×D5—71no—
33C22×C3×D5—71no—
34C2×C3×C4×C5—71no—
35C3×D20—74no—
36C2×C22⋊D7—51noHall
37C5×ΓL(1,𝔽8)C2×D411yescharacteristic, complemented, Hall
38C22×ΓL(1,𝔽8)—52no—
39C22×ΓL(1,𝔽8)—102no—
40C22×ΓL(1,𝔽8)C2×D511yescharacteristic
41C2×AGL(1,𝔽7)—52no—
42C2×AGL(1,𝔽7)—102no—
43ΓL(1,𝔽8)⋊C4C2×D512yescomplemented
44C2×D20—71noHall
45C2×C5×C7C3×D412yescomplemented
46C2×C5×C7C23×C311yescharacteristic
47D35—22no—
48C7×D5—22no—
49C22×C3×C5—71no—
50C2×C3×D5—72no—
51C2×C3×D5—72no—
52C2×C3×D5—142no—
53C2×C3×D5—142no—
54C3×C4×C5—72no—
55C23×C7—51no—
56C22×D7—51no—
57C22⋊D7—54no—
58C2×Dic7C3×D511yescharacteristic, complemented
59C2×C3×D4—351noHall
60C2×ΓL(1,𝔽8)—102no—
61C2×ΓL(1,𝔽8)D2012yescomplemented
62C2×ΓL(1,𝔽8)C22×D511yescharacteristic
63AGL(1,𝔽7)—102no—
64C22×D5—71no—
65C22×D5AGL(1,𝔽7)11yescharacteristic, complemented
66C2×C4×C5—71no—
67D20—74no—
68C5×C7C2×C3×D411yescharacteristic, complemented, Hall
69C2×C3×C5—72no—
70C2×C3×C5—71no—
71C3×D5—142no—
72C3×D5—142no—
73C22×C7—52no—
74C22×C7—102no—
75C22×C7C2×C3×D511yescharacteristic
76C2×D7—52no—
77C2×D7—102no—
78Dic7C2×C3×D512yescomplemented
79C23×C3—351no—
80C23×C3—351no—
81C3×D4—354no—
82C2×C3×C4—71no—
83ΓL(1,𝔽8)C2×D2011yescharacteristic, complemented, Hall
84C22×C5C2×AGL(1,𝔽7)11yescharacteristic
85C2×D5—22no—
86C2×D5—72no—
87C2×D5—142no—
88C2×D5C2×AGL(1,𝔽7)12yescomplemented
89C4×C5—72no—
90C2×D4—351noSylow
91C3×C5—71noHall
92C2×C7—102no—
93C2×C7C3×D2012yescomplemented
94C2×C7C22×C3×D511yescharacteristic
95D7—102no—
96C22×C3—71no—
97C22×C3—352no—
98C22×C3—352no—
99C22×C3—702no—
100C22×C3—702no—
101C3×C4—72no—
102C2×C5(C22⋊D7)⋊C312yescomplemented
103C2×C5C22×AGL(1,𝔽7)11yescharacteristic
104D5—22no—
105D5—142no—
106C23—51no—
107C23—351no—
108D4—354no—
109C2×C4—71no—
110C7C2×C3×D2011yescharacteristic, complemented, Sylow
111C2×C3—72no—
112C2×C3—71no—
113C2×C3—702no—
114C2×C3—702no—
115C5C2×(C22⋊D7)⋊C311yescharacteristic, complemented, Sylow
116C22—52no—
117C22—102no—
118C22—352no—
119C22—702no—
120C22D5×AGL(1,𝔽7)11yescharacteristic
121C4—72no—
122C3—71noSylow
123C2—102no—
124C2—702no—
125C2#63307⋊C312yesdirect product factor
126C2C2×D5×AGL(1,𝔽7)11yescharacteristic
subgroupG/H
center, hypercenterC22D5×AGL(1,𝔽7)
commutator, cosocleC2×C5×C7C23×C3
FrattiniC2C2×D5×AGL(1,𝔽7)
Fitting, socleC22×C5×C7C22×C3
hypocenterC5×C7C2×C3×D4
LCSG ▷ C2×C5×C7 ▷ C5×C7
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C22×C5⋊AGL(1,𝔽7)C2semidirect
2. C22×D5×ΓL(1,𝔽8)C2semidirect
3. C2×C5×ΓL(1,𝔽8)⋊C4C2semidirect
4. #63307⋊C3C2direct (N × H)
5. C2×#63307C3semidirect
8. C2×C5⋊AGL(1,𝔽7)C22semidirect
10. C2×D5×ΓL(1,𝔽8)C22semidirect
11. C5×ΓL(1,𝔽8)⋊C4C22semidirect
13. C22×D35C2×C3semidirect
14. C22×C7×D5C2×C3semidirect
15. C2×C5×Dic7C2×C3semidirect
16. #63307C2×C3semidirect
18. C2×C5×ΓL(1,𝔽8)D4semidirect
25. C2×ΓL(1,𝔽8)⋊C4D5semidirect
28. C2×D35C22×C3semidirect
30. C2×C7×D5C22×C3semidirect
31. C5×Dic7C22×C3semidirect
37. C5×ΓL(1,𝔽8)C2×D4semidirect
43. ΓL(1,𝔽8)⋊C4C2×D5semidirect
45. C2×C5×C7C3×D4semidirect
58. C2×Dic7C3×D5semidirect
61. C2×ΓL(1,𝔽8)D20semidirect
65. C22×D5AGL(1,𝔽7)semidirect
68. C5×C7C2×C3×D4semidirect
78. Dic7C2×C3×D5semidirect
83. ΓL(1,𝔽8)C2×D20semidirect
88. C2×D5C2×AGL(1,𝔽7)semidirect
93. C2×C7C3×D20semidirect
102. C2×C5(C22⋊D7)⋊C3semidirect
110. C7C2×C3×D20semidirect
115. C5C2×(C22⋊D7)⋊C3semidirect
125. C2#63307⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C5⋊AGL(1,𝔽7)2
C22×D5×ΓL(1,𝔽8)2
C2×C5×ΓL(1,𝔽8)⋊C42
#63307⋊C32
C2×#633073
C2×(C22⋊D7)⋊C35
C2×C3×D207