Finite Group Databasebeta

← C5×#6553 #11765432

Subgroups — C5×#6553 #11765432

Order 1120.

332 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C5×C7×#41C211yesmaximal, characteristic, complemented
2C5×#1161C212yesmaximal, complemented
3C5×D4×D7C216yesmaximal, complemented
4C5×#1157C213yesmaximal, complemented
5C2×C5×D28C213yesmaximal, complemented
6C22×C5×D7C2216yescomplemented
7C5×C7×Q8C2211yescharacteristic, complemented
8C5×C7×D4C2213yescomplemented
9C2×C4×C5×C7C2213yescomplemented
10C5×C22⋊D7C2216yescomplemented
11C5×D28C2213yescomplemented
12C5×D28C2216yescomplemented
13C4×C5×D7C2216yescomplemented
14C5×Dic14C2211yescharacteristic, complemented
15#6553C511yesmaximal, characteristic, direct product factor, Hall
16C5×H(5,𝔽2)—71nomaximal, Hall
17C22×C5×C7C2313yes—
18C2×C5×D7—46no—
19C2×C5×D7C2316yes—
20C4×C5×C7C2311yescharacteristic
21C4×C5×C7C2313yes—
22C5×Dic7C2312yes—
23C7×#41C2×C511yescharacteristic, complemented
24#1161C2×C512yescomplemented
25D4×D7C2×C516yescomplemented
26#1157C2×C513yescomplemented
27C2×D28C2×C513yescomplemented
28C5×#41—72no—
29C5×#41—73no—
30C5×#41D711yescharacteristic, complemented
31C2×C5×D4—76no—
32C2×C5×D4—73no—
33C2×C5×C7—23no—
34C2×C5×C7C2411yescharacteristic
35C5×D7—26no—
36C22×D7C22×C516yescomplemented
37C7×Q8C22×C511yescharacteristic, complemented
38C7×D4C22×C513yescomplemented
39C2×C4×C7C22×C513yescomplemented
40C22⋊D7C22×C516yescomplemented
41D28C22×C513yescomplemented
42D28C22×C516yescomplemented
43C4×D7C22×C516yescomplemented
44Dic14C22×C511yescharacteristic, complemented
45C23×C5—76no—
46C5×Q8—71no—
47C5×Q8C2×D711yescharacteristic, complemented
48C5×D4—76no—
49C5×D4—76no—
50C5×D4—73no—
51C5×D4C2×D713yescomplemented
52C2×C4×C5—76no—
53C2×C4×C5C2×D713yescomplemented
54C5×C7H(5,𝔽2)11yescharacteristic, complemented, Hall
55H(5,𝔽2)—71noSylow
56C22×C7C23×C513yes—
57C2×D7—46no—
58C2×D7C23×C516yes—
59C4×C7C23×C511yescharacteristic
60C4×C7C23×C513yes—
61Dic7C23×C512yes—
62C22×C5—76no—
63C22×C5—286no—
64C22×C5C22×D713yes—
65C4×C5—72no—
66C4×C5C22×D713yes—
67C4×C5C22×D711yescharacteristic
68#41—72no—
69#41—73no—
70#41C5×D711yescharacteristic, complemented
71C2×D4—76no—
72C2×D4—73no—
73C2×C7—23no—
74C2×C7C24×C511yescharacteristic
75D7—26no—
76C2×C5—23no—
77C2×C5—146no—
78C2×C5C23×D711yescharacteristic
79C23—76no—
80Q8—71no—
81Q8C2×C5×D711yescharacteristic, complemented
82D4—76no—
83D4—76no—
84D4—73no—
85D4C2×C5×D713yescomplemented
86C2×C4—76no—
87C2×C4C2×C5×D713yescomplemented
88C7C5×H(5,𝔽2)11yescharacteristic, complemented, Sylow
89C5#655311yescharacteristic, direct product factor, Sylow
90C22—76no—
91C22—286no—
92C22C22×C5×D713yes—
93C4—72no—
94C4C22×C5×D713yes—
95C4C22×C5×D711yescharacteristic
96C2—23no—
97C2—146no—
98C2C23×C5×D711yescharacteristic
subgroupG/H
centerC2×C5C23×D7
commutator, cosocleC2×C7C24×C5
FrattiniC2C23×C5×D7
FittingC5×C7×#41C2
socleC2×C5×C7C24
hypercenterC5×#41D7
hypocenterC7C5×H(5,𝔽2)
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C5 ◁ C5×#41

Complements

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

NH ≅ G/Ntype
1. C5×C7×#41C2semidirect
2. C5×#1161C2semidirect
3. C5×D4×D7C2semidirect
4. C5×#1157C2semidirect
5. C2×C5×D28C2semidirect
6. C22×C5×D7C22semidirect
7. C5×C7×Q8C22semidirect
8. C5×C7×D4C22semidirect
9. C2×C4×C5×C7C22semidirect
10. C5×C22⋊D7C22semidirect
11. C5×D28C22semidirect
12. C5×D28C22semidirect
13. C4×C5×D7C22semidirect
14. C5×Dic14C22semidirect
15. #6553C5direct (N × H)
23. C7×#41C2×C5semidirect
24. #1161C2×C5semidirect
25. D4×D7C2×C5semidirect
26. #1157C2×C5semidirect
27. C2×D28C2×C5semidirect
30. C5×#41D7semidirect
36. C22×D7C22×C5semidirect
37. C7×Q8C22×C5semidirect
38. C7×D4C22×C5semidirect
39. C2×C4×C7C22×C5semidirect
40. C22⋊D7C22×C5semidirect
41. D28C22×C5semidirect
42. D28C22×C5semidirect
43. C4×D7C22×C5semidirect
44. Dic14C22×C5semidirect
47. C5×Q8C2×D7semidirect
51. C5×D4C2×D7semidirect
53. C2×C4×C5C2×D7semidirect
54. C5×C7H(5,𝔽2)semidirect
70. #41C5×D7semidirect
81. Q8C2×C5×D7semidirect
85. D4C2×C5×D7semidirect
87. C2×C4C2×C5×D7semidirect
88. C7C5×H(5,𝔽2)semidirect
89. C5#6553direct (N × H)

Maximal subgroups

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

subgroupindex
C5×C7×#412
C5×#11612
C5×D4×D72
C5×#11572
C2×C5×D282
#65535
C5×H(5,𝔽2)7