Finite Group Databasebeta

← C7×#440 #90086

Subgroups — C7×#440 #90086

Order 448.

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

normal subgroups only

This group has 120 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×C7×SD16C211yesmaximal, characteristic
2C4×C7×Q8C211yesmaximal, characteristic, complemented
3C4×C7×D4C211yesmaximal, characteristic
4C7×#107C211yesmaximal, characteristic, complemented
5C7×#103C211yesmaximal, characteristic, complemented
6C7×#102C211yesmaximal, characteristic
7C7×#97C211yesmaximal, characteristic, complemented
8C2×C7×Q8C2211yescharacteristic
9C2×C7×D4C2211yescharacteristic
10C22×C4×C7—21no—
11C7×SD16C414yescomplemented
12C2×C7×C8C2211yescharacteristic
13C2×C7×C8C2211yescharacteristic
14C7×C4⋊C4—21no—
15C7×C4⋊C4C2211yescharacteristic
16C7×C4⋊C4C2211yescharacteristic
17C7×C22⋊C4—21no—
18C42×C7—21no—
19C42×C7C2211yescharacteristic
20#440C711yesmaximal, characteristic, direct product factor, Sylow
21C23×C7—21no—
22C7×Q8—21no—
23C7×Q8C2×C412yescomplemented
24C7×D4—21no—
25C7×D4C2×C412yes—
26C2×C4×C7—21no—
27C2×C4×C7—21no—
28C2×C4×C7—21no—
29C2×C4×C7—42no—
30C2×C4×C7D411yescharacteristic
31C2×C4×C7D411yescharacteristic
32C2×C4×C7C2311yescharacteristic
33C7×C8—21no—
34C7×C8C2×C412yescomplemented
35C2×SD16C2×C711yescharacteristic
36C4×Q8C2×C711yescharacteristic, complemented
37C4×D4C2×C711yescharacteristic
38#107C2×C711yescharacteristic, complemented
39#103C2×C711yescharacteristic, complemented
40#102C2×C711yescharacteristic
41#97C2×C711yescharacteristic, complemented
42C22×C7—22no—
43C22×C7—41no—
44C22×C7—41no—
45C22×C7C2×D411yescharacteristic
46C4×C7—22no—
47C4×C7—21no—
48C4×C7—21no—
49C4×C7—41no—
50C4×C7—41no—
51C4×C7C22×C411yescharacteristic
52C4×C7#4111yescharacteristic
53C2×Q8C22×C711yescharacteristic
54C2×D4C22×C711yescharacteristic
55C22×C4—21no—
56SD16C4×C714yescomplemented
57C2×C8C22×C711yescharacteristic
58C2×C8C22×C711yescharacteristic
59C4⋊C4—21no—
60C4⋊C4C22×C711yescharacteristic
61C4⋊C4C22×C711yescharacteristic
62C22⋊C4—21no—
63C42—21no—
64C42C22×C711yescharacteristic
65C2×C7—42no—
66C2×C7C4×D411yescharacteristic
67C2×C7AGL(1,ℤ/8)11yescharacteristic
68C2×C7#13711yescharacteristic
69C23—21no—
70Q8—21no—
71Q8C2×C4×C712yescomplemented
72D4—21no—
73D4C2×C4×C712yes—
74C2×C4—21no—
75C2×C4—21no—
76C2×C4—21no—
77C2×C4—42no—
78C2×C4C7×D411yescharacteristic
79C2×C4C7×D411yescharacteristic
80C2×C4C23×C711yescharacteristic
81C8—21no—
82C8C2×C4×C712yescomplemented
83C7#44011yescharacteristic, direct product factor, Sylow
84C22—22no—
85C22—41no—
86C22—41no—
87C22C2×C7×D411yescharacteristic
88C4—22no—
89C4—21no—
90C4—21no—
91C4—41no—
92C4—41no—
93C4C22×C4×C711yescharacteristic
94C4C7×#4111yescharacteristic
95C2—42no—
96C2C4×C7×D411yescharacteristic
97C2C7×AGL(1,ℤ/8)11yescharacteristic
98C2C7×#13711yescharacteristic
subgroupG/H
center, socleC22×C7C2×D4
commutatorC4C22×C4×C7
Frattini, cosocleC2×C4C23×C7
prime radicalC2×C7×D4C22
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C7 ◁ C42×C7 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×C7×Q8C2semidirect
4. C7×#107C2semidirect
5. C7×#103C2semidirect
7. C7×#97C2semidirect
11. C7×SD16C4semidirect
20. #440C7direct (N × H)
23. C7×Q8C2×C4semidirect
34. C7×C8C2×C4semidirect
36. C4×Q8C2×C7semidirect
38. #107C2×C7semidirect
39. #103C2×C7semidirect
41. #97C2×C7semidirect
56. SD16C4×C7semidirect
71. Q8C2×C4×C7semidirect
82. C8C2×C4×C7semidirect
83. C7#440direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×SD162
C4×C7×Q82
C4×C7×D42
C7×#1072
C7×#1032
C7×#1022
C7×#972
#4407