Finite Group Databasebeta

← C7×#575 #90596

Subgroups — C7×#575 #90596

Order 448.

262 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 158 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×#41C212yesmaximal, complemented
2C7×#137C214yesmaximal, complemented
3C7×AGL(1,ℤ/8)C214yesmaximal, complemented
4C7×#135C214yesmaximal, complemented
5C2×C7×M16C211yesmaximal, characteristic, complemented
6C7×#41—22no—
7C7×#41—22no—
8C7×#41C2214yes—
9C7×#41C2214yescomplemented
10C2×C7×Q8C2212yescomplemented
11C2×C7×D4—22no—
12C2×C7×D4C2212yescomplemented
13C22×C4×C7—22no—
14C22×C4×C7C2211yescharacteristic
15C7×Q16C2214yescomplemented
16C7×SD16C2218yescomplemented
17C7×D8C2214yescomplemented
18C7×M16C2214yescomplemented
19C2×C7×C8C2212yescomplemented
20#575C711yesmaximal, characteristic, direct product factor, Sylow
21C23×C7—22no—
22C23×C7D411yescharacteristic
23C7×Q8—22no—
24C7×Q8C2314yes—
25C7×D4—24no—
26C7×D4—24no—
27C7×D4—22no—
28C7×D4C2314yes—
29C2×C4×C7—22no—
30C2×C4×C7—24no—
31C2×C4×C7—24no—
32C2×C4×C7D411yescharacteristic
33C2×C4×C7D412yes—
34C2×C4×C7C2312yes—
35C2×C4×C7C2311yescharacteristic
36C7×C8C2314yes—
37C2×#41C2×C712yescomplemented
38#137C2×C714yescomplemented
39AGL(1,ℤ/8)C2×C714yescomplemented
40#135C2×C714yescomplemented
41C2×M16C2×C711yescharacteristic, complemented
42C22×C7—24no—
43C22×C7—41no—
44C22×C7—44no—
45C22×C7C2×D412yes—
46C22×C7C2×D411yescharacteristic
47C4×C7—24no—
48C4×C7C2×D412yes—
49C4×C7C2×D411yescharacteristic
50C4×C7C2411yescharacteristic
51#41—22no—
52#41—22no—
53#41C22×C714yes—
54#41C22×C714yescomplemented
55C2×Q8C22×C712yescomplemented
56C2×D4—22no—
57C2×D4C22×C712yescomplemented
58C22×C4—22no—
59C22×C4C22×C711yescharacteristic
60Q16C22×C714yescomplemented
61SD16C22×C718yescomplemented
62D8C22×C714yescomplemented
63M16C22×C714yescomplemented
64C2×C8C22×C712yescomplemented
65C2×C7—22no—
66C2×C7—21no—
67C2×C7—44no—
68C2×C7C22×D411yescharacteristic
69C23—22no—
70C23C7×D411yescharacteristic
71Q8—22no—
72Q8C23×C714yes—
73D4—24no—
74D4—24no—
75D4—22no—
76D4C23×C714yes—
77C2×C4—22no—
78C2×C4—24no—
79C2×C4—24no—
80C2×C4C7×D412yes—
81C2×C4C7×D411yescharacteristic
82C2×C4C23×C711yescharacteristic
83C2×C4C23×C712yes—
84C8C23×C714yes—
85C7#57511yescharacteristic, direct product factor, Sylow
86C22—24no—
87C22—41no—
88C22—44no—
89C22C2×C7×D412yes—
90C22C2×C7×D411yescharacteristic
91C4—24no—
92C4C2×C7×D411yescharacteristic
93C4C2×C7×D412yes—
94C4C24×C711yescharacteristic
95C2—22no—
96C2—21no—
97C2—44no—
98C2C22×C7×D411yescharacteristic
subgroupG/H
centerC4×C7C2×D4
commutator, Frattini, cosocleC4C24×C7
socleC2×C7C22×D4
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C7 ◁ C22×C4×C7 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C7×#41C2semidirect
2. C7×#137C2semidirect
3. C7×AGL(1,ℤ/8)C2semidirect
4. C7×#135C2semidirect
5. C2×C7×M16C2semidirect
9. C7×#41C22semidirect
10. C2×C7×Q8C22semidirect
12. C2×C7×D4C22semidirect
15. C7×Q16C22semidirect
16. C7×SD16C22semidirect
17. C7×D8C22semidirect
18. C7×M16C22semidirect
19. C2×C7×C8C22semidirect
20. #575C7direct (N × H)
37. C2×#41C2×C7semidirect
38. #137C2×C7semidirect
39. AGL(1,ℤ/8)C2×C7semidirect
40. #135C2×C7semidirect
41. C2×M16C2×C7semidirect
54. #41C22×C7semidirect
55. C2×Q8C22×C7semidirect
57. C2×D4C22×C7semidirect
60. Q16C22×C7semidirect
61. SD16C22×C7semidirect
62. D8C22×C7semidirect
63. M16C22×C7semidirect
64. C2×C8C22×C7semidirect
85. C7#575direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#412
C7×#1372
C7×AGL(1,ℤ/8)2
C7×#1352
C2×C7×M162
#5757