Finite Group Databasebeta

← #577×C71 #17985708

Subgroups — #577×C71 #17985708

Order 4544.

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
12−1+4×C71C211yesmaximal, characteristic, complemented
2H(5,𝔽2)×C71C211yesmaximal, characteristic, complemented
3#137×C71C213yesmaximal, complemented
4AGL(1,ℤ/8)×C71C213yesmaximal, complemented
5#135×C71C213yesmaximal, complemented
6C2×SD16×C71C213yesmaximal, complemented
7#131×C71C211yesmaximal, characteristic, complemented
8#41×C71—21no—
9#41×C71—23no—
10#41×C71C2213yes—
11#41×C71C2211yescharacteristic
12#41×C71C2213yescomplemented
13C2×Q8×C71—21no—
14C2×Q8×C71C2213yescomplemented
15C2×D4×C71—23no—
16C2×D4×C71C2213yes—
17Q16×C71C2213yescomplemented
18SD16×C71C2213yescomplemented
19SD16×C71C2216yescomplemented
20SD16×C71C2211yescharacteristic, complemented
21D8×C71C2213yescomplemented
22M16×C71C2213yescomplemented
23C2×C8×C71C2213yescomplemented
24C23×C71—23no—
25Q8×C71—23no—
26Q8×C71D411yescharacteristic
27Q8×C71C2313yes—
28Q8×C71C2311yescharacteristic
29D4×C71—23no—
30D4×C71—23no—
31D4×C71—23no—
32D4×C71D413yes—
33D4×C71C2311yescharacteristic
34D4×C71C2313yes—
35C2×C4×C71—23no—
36C2×C4×C71—23no—
37C2×C4×C71—23no—
38C2×C4×C71C2313yes—
39C8×C71C2311yescharacteristic
40C8×C71C2313yes—
41C22×C71—21no—
42C22×C71—23no—
43C22×C71—83no—
44C22×C71C2×D413yes—
45C4×C71—21no—
46C4×C71—23no—
47C4×C71C2×D413yes—
48C4×C71C2411yescharacteristic
49C2×C71—23no—
50C2×C71—41no—
51C2×C71—43no—
52C2×C71C22×D411yescharacteristic
53C71#57711yescharacteristic, direct product factor, Sylow
54#577C7111yesmaximal, characteristic, direct product factor, Sylow
552−1+4C2×C7111yescharacteristic, complemented
56H(5,𝔽2)C2×C7111yescharacteristic, complemented
57#137C2×C7113yescomplemented
58AGL(1,ℤ/8)C2×C7113yescomplemented
59#135C2×C7113yescomplemented
60C2×SD16C2×C7113yescomplemented
61#131C2×C7111yescharacteristic, complemented
62#41—21no—
63#41—23no—
64#41C22×C7113yes—
65#41C22×C7111yescharacteristic
66#41C22×C7113yescomplemented
67C2×Q8—21no—
68C2×Q8C22×C7113yescomplemented
69C2×D4—23no—
70C2×D4C22×C7113yes—
71Q16C22×C7113yescomplemented
72SD16C22×C7113yescomplemented
73SD16C22×C7116yescomplemented
74SD16C22×C7111yescharacteristic, complemented
75D8C22×C7113yescomplemented
76M16C22×C7113yescomplemented
77C2×C8C22×C7113yescomplemented
78C23—23no—
79Q8—23no—
80Q8D4×C7111yescharacteristic
81Q8C23×C7113yes—
82Q8C23×C7111yescharacteristic
83D4—23no—
84D4—23no—
85D4—23no—
86D4D4×C7113yes—
87D4C23×C7111yescharacteristic
88D4C23×C7113yes—
89C2×C4—23no—
90C2×C4—23no—
91C2×C4—23no—
92C2×C4C23×C7113yes—
93C8C23×C7111yescharacteristic
94C8C23×C7113yes—
95C22—21no—
96C22—23no—
97C22—83no—
98C22C2×D4×C7113yes—
99C4—21no—
100C4—23no—
101C4C2×D4×C7113yes—
102C4C24×C7111yescharacteristic
103C2—23no—
104C2—43no—
105C2—41no—
106C2C22×D4×C7111yescharacteristic
subgroupG/H
center, socleC2×C71C22×D4
commutator, Frattini, cosocleC4C24×C71
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C71 ◁ #41×C71 ◁ G

Complements

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

NH ≅ G/Ntype
1. 2−1+4×C71C2semidirect
2. H(5,𝔽2)×C71C2semidirect
3. #137×C71C2semidirect
4. AGL(1,ℤ/8)×C71C2semidirect
5. #135×C71C2semidirect
6. C2×SD16×C71C2semidirect
7. #131×C71C2semidirect
12. #41×C71C22semidirect
14. C2×Q8×C71C22semidirect
17. Q16×C71C22semidirect
18. SD16×C71C22semidirect
19. SD16×C71C22semidirect
20. SD16×C71C22semidirect
21. D8×C71C22semidirect
22. M16×C71C22semidirect
23. C2×C8×C71C22semidirect
53. C71#577direct (N × H)
54. #577C71direct (N × H)
55. 2−1+4C2×C71semidirect
56. H(5,𝔽2)C2×C71semidirect
57. #137C2×C71semidirect
58. AGL(1,ℤ/8)C2×C71semidirect
59. #135C2×C71semidirect
60. C2×SD16C2×C71semidirect
61. #131C2×C71semidirect
66. #41C22×C71semidirect
68. C2×Q8C22×C71semidirect
71. Q16C22×C71semidirect
72. SD16C22×C71semidirect
73. SD16C22×C71semidirect
74. SD16C22×C71semidirect
75. D8C22×C71semidirect
76. M16C22×C71semidirect
77. C2×C8C22×C71semidirect

Maximal subgroups

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

subgroupindex
2−1+4×C712
H(5,𝔽2)×C712
#137×C712
AGL(1,ℤ/8)×C712
#135×C712
C2×SD16×C712
#131×C712
#57771