Finite Group Databasebeta

← C5×#551 #66215

Subgroups — C5×#551 #66215

Order 320.

248 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 150 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×#126C212yesmaximal, complemented
2C5×#124C211yesmaximal, characteristic, complemented
3C5×#123C211yesmaximal, characteristic, complemented
4C5×#123C212yesmaximal, complemented
5C5×C22⋊Q8C212yesmaximal, complemented
6C5×#120C211yesmaximal, characteristic
7C4×C5×D4C212yesmaximal, complemented
8C5×#117C212yesmaximal, complemented
9C2×C5×C22⋊C4C212yesmaximal, complemented
10C24×C5C2211yescharacteristic
11C2×C5×Q8C2211yescharacteristic, complemented
12C2×C5×D4C2212yes—
13C2×C5×D4C2211yescharacteristic
14C22×C4×C5C2212yes—
15C22×C4×C5C2212yes—
16C22×C4×C5C2211yescharacteristic, complemented
17C5×C4⋊C4C2212yescomplemented
18C5×C4⋊C4C2212yescomplemented
19C5×C4⋊C4C2212yescomplemented
20C5×C4⋊C4C2212yescomplemented
21C5×C22⋊C4C2212yes—
22C5×C22⋊C4C2212yes—
23C5×C22⋊C4C2212yes—
24C5×C22⋊C4C2212yes—
25C5×C22⋊C4C2212yes—
26C5×C22⋊C4C2211yescharacteristic
27C5×C22⋊C4C2212yescomplemented
28C5×C22⋊C4C2211yescharacteristic, complemented
29C42×C5C2211yescharacteristic, complemented
30C42×C5C2212yescomplemented
31#551C511yesmaximal, characteristic, direct product factor, Sylow
32C23×C5—24no—
33C23×C5—41no—
34C23×C5C2311yescharacteristic
35C23×C5C2311yescharacteristic
36C23×C5C2312yes—
37C5×Q8—41no—
38C5×D4—24no—
39C5×D4—41no—
40C2×C4×C5—24no—
41C2×C4×C5—42no—
42C2×C4×C5—41no—
43C2×C4×C5C2312yes—
44C2×C4×C5C2312yes—
45C2×C4×C5C2312yes—
46C2×C4×C5C2311yescharacteristic
47C2×C4×C5C2312yes—
48C2×C4×C5C2312yes—
49#126C2×C512yescomplemented
50#124C2×C511yescharacteristic, complemented
51#123C2×C511yescharacteristic, complemented
52#123C2×C512yescomplemented
53C22⋊Q8C2×C512yescomplemented
54#120C2×C511yescharacteristic
55C4×D4C2×C512yescomplemented
56#117C2×C512yescomplemented
57C2×C22⋊C4C2×C512yescomplemented
58C22×C5—22no—
59C22×C5—22no—
60C22×C5—22no—
61C22×C5—22no—
62C22×C5—21no—
63C22×C5—21no—
64C22×C5—44no—
65C22×C5#4114yes—
66C22×C5C2411yescharacteristic
67C4×C5—22no—
68C4×C5—22no—
69C4×C5—22no—
70C4×C5—22no—
71C4×C5—22no—
72C4×C5—21no—
73C24C22×C511yescharacteristic
74C2×Q8C22×C511yescharacteristic, complemented
75C2×D4C22×C512yes—
76C2×D4C22×C511yescharacteristic
77C22×C4C22×C512yes—
78C22×C4C22×C512yes—
79C22×C4C22×C511yescharacteristic, complemented
80C4⋊C4C22×C512yescomplemented
81C4⋊C4C22×C512yescomplemented
82C4⋊C4C22×C512yescomplemented
83C4⋊C4C22×C512yescomplemented
84C22⋊C4C22×C512yes—
85C22⋊C4C22×C512yes—
86C22⋊C4C22×C512yes—
87C22⋊C4C22×C512yes—
88C22⋊C4C22×C512yes—
89C22⋊C4C22×C511yescharacteristic
90C22⋊C4C22×C512yescomplemented
91C22⋊C4C22×C511yescharacteristic, complemented
92C42C22×C511yescharacteristic, complemented
93C42C22×C512yescomplemented
94C2×C5—24no—
95C2×C5—41no—
96C2×C5—41no—
97C2×C5C2×#4112yes—
98C2×C5H(5,𝔽2)11yescharacteristic
99C23—24no—
100C23—41no—
101C23C23×C511yescharacteristic
102C23C23×C511yescharacteristic
103C23C23×C512yes—
104Q8—41no—
105D4—24no—
106D4—41no—
107C2×C4—24no—
108C2×C4—42no—
109C2×C4—41no—
110C2×C4C23×C512yes—
111C2×C4C23×C512yes—
112C2×C4C23×C512yes—
113C2×C4C23×C511yescharacteristic
114C2×C4C23×C512yes—
115C2×C4C23×C512yes—
116C5#55111yescharacteristic, direct product factor, Sylow
117C22—22no—
118C22—22no—
119C22—22no—
120C22—22no—
121C22—21no—
122C22—21no—
123C22—44no—
124C22C5×#4114yes—
125C22C24×C511yescharacteristic
126C4—22no—
127C4—22no—
128C4—22no—
129C4—22no—
130C4—22no—
131C4—21no—
132C2—24no—
133C2—41no—
134C2—41no—
135C2C2×C5×#4112yes—
136C2C5×H(5,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C5C24
commutator, Frattini, cosocleC22C24×C5
prime radicalC5×#120C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#126C2semidirect
2. C5×#124C2semidirect
3. C5×#123C2semidirect
4. C5×#123C2semidirect
5. C5×C22⋊Q8C2semidirect
7. C4×C5×D4C2semidirect
8. C5×#117C2semidirect
9. C2×C5×C22⋊C4C2semidirect
11. C2×C5×Q8C22semidirect
16. C22×C4×C5C22semidirect
17. C5×C4⋊C4C22semidirect
18. C5×C4⋊C4C22semidirect
19. C5×C4⋊C4C22semidirect
20. C5×C4⋊C4C22semidirect
27. C5×C22⋊C4C22semidirect
28. C5×C22⋊C4C22semidirect
29. C42×C5C22semidirect
30. C42×C5C22semidirect
31. #551C5direct (N × H)
49. #126C2×C5semidirect
50. #124C2×C5semidirect
51. #123C2×C5semidirect
52. #123C2×C5semidirect
53. C22⋊Q8C2×C5semidirect
55. C4×D4C2×C5semidirect
56. #117C2×C5semidirect
57. C2×C22⋊C4C2×C5semidirect
74. C2×Q8C22×C5semidirect
79. C22×C4C22×C5semidirect
80. C4⋊C4C22×C5semidirect
81. C4⋊C4C22×C5semidirect
82. C4⋊C4C22×C5semidirect
83. C4⋊C4C22×C5semidirect
90. C22⋊C4C22×C5semidirect
91. C22⋊C4C22×C5semidirect
92. C42C22×C5semidirect
93. C42C22×C5semidirect
116. C5#551direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#1262
C5×#1242
C5×#1232
C5×C22⋊Q82
C5×#1202
C4×C5×D42
C5×#1172
C2×C5×C22⋊C42
#5515