Finite Group Databasebeta

← C7×#577 #90598

Subgroups — C7×#577 #90598

Order 448.

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

normal subgroups only

This group has 258 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
1C7×2−1+4C211yesmaximal, characteristic, complemented
2C7×H(5,𝔽2)C211yesmaximal, characteristic, complemented
3C7×#137C213yesmaximal, complemented
4C7×AGL(1,ℤ/8)C213yesmaximal, complemented
5C7×#135C213yesmaximal, complemented
6C2×C7×SD16C213yesmaximal, complemented
7C7×#131C211yesmaximal, characteristic, complemented
8C7×#41—21no—
9C7×#41—23no—
10C7×#41C2213yes—
11C7×#41C2211yescharacteristic
12C7×#41C2213yescomplemented
13C2×C7×Q8—21no—
14C2×C7×Q8C2213yescomplemented
15C2×C7×D4—23no—
16C2×C7×D4C2213yes—
17C7×Q16C2213yescomplemented
18C7×SD16C2213yescomplemented
19C7×SD16C2216yescomplemented
20C7×SD16C2211yescharacteristic, complemented
21C7×D8C2213yescomplemented
22C7×M16C2213yescomplemented
23C2×C7×C8C2213yescomplemented
24#577C711yesmaximal, characteristic, direct product factor, Sylow
25C23×C7—23no—
26C7×Q8—23no—
27C7×Q8D411yescharacteristic
28C7×Q8C2313yes—
29C7×Q8C2311yescharacteristic
30C7×D4—23no—
31C7×D4—23no—
32C7×D4—23no—
33C7×D4D413yes—
34C7×D4C2311yescharacteristic
35C7×D4C2313yes—
36C2×C4×C7—23no—
37C2×C4×C7—23no—
38C2×C4×C7—23no—
39C2×C4×C7C2313yes—
40C7×C8C2311yescharacteristic
41C7×C8C2313yes—
422−1+4C2×C711yescharacteristic, complemented
43H(5,𝔽2)C2×C711yescharacteristic, complemented
44#137C2×C713yescomplemented
45AGL(1,ℤ/8)C2×C713yescomplemented
46#135C2×C713yescomplemented
47C2×SD16C2×C713yescomplemented
48#131C2×C711yescharacteristic, complemented
49C22×C7—21no—
50C22×C7—23no—
51C22×C7—83no—
52C22×C7C2×D413yes—
53C4×C7—21no—
54C4×C7—23no—
55C4×C7C2×D413yes—
56C4×C7C2411yescharacteristic
57#41—21no—
58#41—23no—
59#41C22×C713yes—
60#41C22×C711yescharacteristic
61#41C22×C713yescomplemented
62C2×Q8—21no—
63C2×Q8C22×C713yescomplemented
64C2×D4—23no—
65C2×D4C22×C713yes—
66Q16C22×C713yescomplemented
67SD16C22×C713yescomplemented
68SD16C22×C716yescomplemented
69SD16C22×C711yescharacteristic, complemented
70D8C22×C713yescomplemented
71M16C22×C713yescomplemented
72C2×C8C22×C713yescomplemented
73C2×C7—23no—
74C2×C7—41no—
75C2×C7—43no—
76C2×C7C22×D411yescharacteristic
77C23—23no—
78Q8—23no—
79Q8C7×D411yescharacteristic
80Q8C23×C713yes—
81Q8C23×C711yescharacteristic
82D4—23no—
83D4—23no—
84D4—23no—
85D4C7×D413yes—
86D4C23×C711yescharacteristic
87D4C23×C713yes—
88C2×C4—23no—
89C2×C4—23no—
90C2×C4—23no—
91C2×C4C23×C713yes—
92C8C23×C711yescharacteristic
93C8C23×C713yes—
94C7#57711yescharacteristic, direct product factor, Sylow
95C22—21no—
96C22—23no—
97C22—83no—
98C22C2×C7×D413yes—
99C4—21no—
100C4—23no—
101C4C2×C7×D413yes—
102C4C24×C711yescharacteristic
103C2—23no—
104C2—43no—
105C2—41no—
106C2C22×C7×D411yescharacteristic
subgroupG/H
center, socleC2×C7C22×D4
commutator, Frattini, cosocleC4C24×C7
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C7×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C7×2−1+4C2semidirect
2. C7×H(5,𝔽2)C2semidirect
3. C7×#137C2semidirect
4. C7×AGL(1,ℤ/8)C2semidirect
5. C7×#135C2semidirect
6. C2×C7×SD16C2semidirect
7. C7×#131C2semidirect
12. C7×#41C22semidirect
14. C2×C7×Q8C22semidirect
17. C7×Q16C22semidirect
18. C7×SD16C22semidirect
19. C7×SD16C22semidirect
20. C7×SD16C22semidirect
21. C7×D8C22semidirect
22. C7×M16C22semidirect
23. C2×C7×C8C22semidirect
24. #577C7direct (N × H)
42. 2−1+4C2×C7semidirect
43. H(5,𝔽2)C2×C7semidirect
44. #137C2×C7semidirect
45. AGL(1,ℤ/8)C2×C7semidirect
46. #135C2×C7semidirect
47. C2×SD16C2×C7semidirect
48. #131C2×C7semidirect
61. #41C22×C7semidirect
63. C2×Q8C22×C7semidirect
66. Q16C22×C7semidirect
67. SD16C22×C7semidirect
68. SD16C22×C7semidirect
69. SD16C22×C7semidirect
70. D8C22×C7semidirect
71. M16C22×C7semidirect
72. C2×C8C22×C7semidirect
94. C7#577direct (N × H)

Maximal subgroups

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

subgroupindex
C7×2−1+42
C7×H(5,𝔽2)2
C7×#1372
C7×AGL(1,ℤ/8)2
C7×#1352
C2×C7×SD162
C7×#1312
#5777