Finite Group Databasebeta

← C2×#63306⋊C3 #13132432

Subgroups — C2×#63306⋊C3 #13132432

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×C5×AGL(1,𝔽7)C211yesmaximal, characteristic, complemented
3C2×Dic5×ΓL(1,𝔽8)C211yesmaximal, characteristic, complemented
4#63306⋊C3C214yesmaximal, direct product factor
5C2×#63306C311yesmaximal, characteristic, complemented, Hall
6C22×C5×ΓL(1,𝔽8)C2211yescharacteristic
7C2×C5⋊AGL(1,𝔽7)—22no—
8C2×C5⋊AGL(1,𝔽7)C2212yescomplemented
9C2×C5×AGL(1,𝔽7)—22no—
10C2×C5×AGL(1,𝔽7)C2212yescomplemented
11Dic5×ΓL(1,𝔽8)C2212yescomplemented
12C2×C4⋊AGL(1,𝔽7)—51nomaximal, Hall
13C22×D35C2×C311yescharacteristic, complemented
14C22×C5×D7C2×C311yescharacteristic, complemented
15C2×C7×Dic5C2×C311yescharacteristic, complemented
16#63306C2×C314yescomplemented
17C2×C3×C22⋊D5—71nomaximal, Hall
18C2×C5×ΓL(1,𝔽8)D412yescomplemented
19C2×C5×ΓL(1,𝔽8)C2311yescharacteristic
20C5⋊AGL(1,𝔽7)—22no—
21C5×AGL(1,𝔽7)—22no—
22C22×AGL(1,𝔽7)—51no—
23C22×AGL(1,𝔽7)D511yescharacteristic, complemented
24C2×C4×ΓL(1,𝔽8)—51no—
25C4⋊AGL(1,𝔽7)—54no—
26C22×C5×C7C22×C311yescharacteristic
27C2×D35—22no—
28C2×D35C22×C312yescomplemented
29C2×C5×D7—22no—
30C2×C5×D7C22×C312yescomplemented
31C7×Dic5C22×C312yescomplemented
32C23×C3×C5—71no—
33C22×C3×D5—71no—
34C3×C22⋊D5—74no—
35C2×C3×Dic5—71no—
36C2×D28—51noHall
37C5×ΓL(1,𝔽8)C2×D411yescharacteristic, complemented, Hall
38C22×ΓL(1,𝔽8)C2×D511yescharacteristic
39C2×AGL(1,𝔽7)—22no—
40C2×AGL(1,𝔽7)—52no—
41C2×AGL(1,𝔽7)—102no—
42C2×AGL(1,𝔽7)C2×D512yescomplemented
43C4×ΓL(1,𝔽8)—52no—
44C2×C22⋊D5—71noHall
45C2×C5×C7C3×D412yescomplemented
46C2×C5×C7C23×C311yescharacteristic
47D35—22no—
48C5×D7—22no—
49C22×C3×C5—72no—
50C22×C3×C5—71no—
51C22×C3×C5—142no—
52C2×C3×D5—72no—
53C2×C3×D5—142no—
54C3×Dic5—72no—
55C22×D7—51no—
56C22×D7C3×D511yescharacteristic, complemented
57C2×C4×C7—51no—
58D28—54no—
59C2×C3×D4—351noHall
60C2×ΓL(1,𝔽8)C22⋊D512yescomplemented
61C2×ΓL(1,𝔽8)C22×D511yescharacteristic
62AGL(1,𝔽7)—22no—
63AGL(1,𝔽7)—102no—
64C23×C5—71no—
65C22×D5—71no—
66C22⋊D5—74no—
67C2×Dic5AGL(1,𝔽7)11yescharacteristic, complemented
68C5×C7C2×C3×D411yescharacteristic, complemented, Hall
69C2×C3×C5—72no—
70C2×C3×C5—71no—
71C2×C3×C5—142no—
72C3×D5—142no—
73C22×C7C2×C3×D511yescharacteristic
74C2×D7—22no—
75C2×D7—52no—
76C2×D7—102no—
77C2×D7C2×C3×D512yescomplemented
78C4×C7—52no—
79C23×C3—71no—
80C23×C3—351no—
81C3×D4—354no—
82C2×C3×C4—351no—
83ΓL(1,𝔽8)C2×C22⋊D511yescharacteristic, complemented, Hall
84C22×C5—72no—
85C22×C5—142no—
86C22×C5C2×AGL(1,𝔽7)11yescharacteristic
87C2×D5—72no—
88C2×D5—142no—
89Dic5C2×AGL(1,𝔽7)12yescomplemented
90C2×D4—351noSylow
91C3×C5—71noHall
92C2×C7C3×C22⋊D512yescomplemented
93C2×C7C22×C3×D511yescharacteristic
94D7—22no—
95D7—102no—
96C22×C3—72no—
97C22×C3—71no—
98C22×C3—142no—
99C22×C3—352no—
100C22×C3—702no—
101C3×C4—352no—
102C2×C5—142no—
103C2×C5C4⋊AGL(1,𝔽7)12yescomplemented
104C2×C5C22×AGL(1,𝔽7)11yescharacteristic
105D5—142no—
106C23—71no—
107C23—351no—
108D4—354no—
109C2×C4—51no—
110C7C2×C3×C22⋊D511yescharacteristic, complemented, Sylow
111C2×C3—72no—
112C2×C3—71no—
113C2×C3—142no—
114C2×C3—702no—
115C5C2×C4⋊AGL(1,𝔽7)11yescharacteristic, complemented, Sylow
116C22—72no—
117C22—142no—
118C22—352no—
119C22—702no—
120C22D5×AGL(1,𝔽7)11yescharacteristic
121C4—52no—
122C3—71noSylow
123C2—142no—
124C2—702no—
125C2#63306⋊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×C5×AGL(1,𝔽7)C2semidirect
3. C2×Dic5×ΓL(1,𝔽8)C2semidirect
4. #63306⋊C3C2direct (N × H)
5. C2×#63306C3semidirect
8. C2×C5⋊AGL(1,𝔽7)C22semidirect
10. C2×C5×AGL(1,𝔽7)C22semidirect
11. Dic5×ΓL(1,𝔽8)C22semidirect
13. C22×D35C2×C3semidirect
14. C22×C5×D7C2×C3semidirect
15. C2×C7×Dic5C2×C3semidirect
16. #63306C2×C3semidirect
18. C2×C5×ΓL(1,𝔽8)D4semidirect
23. C22×AGL(1,𝔽7)D5semidirect
28. C2×D35C22×C3semidirect
30. C2×C5×D7C22×C3semidirect
31. C7×Dic5C22×C3semidirect
37. C5×ΓL(1,𝔽8)C2×D4semidirect
42. C2×AGL(1,𝔽7)C2×D5semidirect
45. C2×C5×C7C3×D4semidirect
56. C22×D7C3×D5semidirect
60. C2×ΓL(1,𝔽8)C22⋊D5semidirect
67. C2×Dic5AGL(1,𝔽7)semidirect
68. C5×C7C2×C3×D4semidirect
77. C2×D7C2×C3×D5semidirect
83. ΓL(1,𝔽8)C2×C22⋊D5semidirect
89. Dic5C2×AGL(1,𝔽7)semidirect
92. C2×C7C3×C22⋊D5semidirect
103. C2×C5C4⋊AGL(1,𝔽7)semidirect
110. C7C2×C3×C22⋊D5semidirect
115. C5C2×C4⋊AGL(1,𝔽7)semidirect
125. C2#63306⋊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×C5×AGL(1,𝔽7)2
C2×Dic5×ΓL(1,𝔽8)2
#63306⋊C32
C2×#633063
C2×C4⋊AGL(1,𝔽7)5
C2×C3×C22⋊D57