Finite Group Databasebeta

← #65321

Subgroups — #65321

Order 320.

198 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 39 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
1#4144C211yesmaximal, characteristic, complemented
2C5×#120C211yesmaximal, characteristic, complemented
3#4099C212yesmaximal, complemented
4#3966C211yesmaximal, characteristic, complemented
5#3938C212yesmaximal, complemented
6C24×C5C2211yescharacteristic, complemented
7C2×C5×D4—21no—
8C2×C5×D4C2211yescharacteristic
9C2×C22⋊D5—21no—
10C2×C22⋊D5—22no—
11C2×C22⋊D5C2212yes—
12#711—22no—
13D4×D5—22no—
14#709—21no—
15C5×C22⋊C4—21no—
16C5×C22⋊C4C2211yescharacteristic, complemented
17C22⋊Dic5—22no—
18C22⋊Dic5C2212yescomplemented
19B(4,𝔽2)—51nomaximal, Sylow
20C23×C5—22no—
21C23×C5—21no—
22C23×C5—42no—
23C23×C5D411yescharacteristic, complemented
24C23×C5C2311yescharacteristic
25C22×D5—21no—
26C22×D5D412yescomplemented
27C5×D4—22no—
28C5×D4—42no—
29C2×C4×C5—21no—
30C2×C4×C5D411yescharacteristic, complemented
31C22⋊D5—22no—
32C22⋊D5—22no—
33C22⋊D5—22no—
34C22⋊D5—44no—
35C2×Dic5—21no—
36C2×Dic5—22no—
37C2×Dic5D412yescomplemented
38D20—21no—
39C4×D5—22no—
40Dic10—21no—
41H(5,𝔽2)—51no—
42#120—52no—
43#120D511yescharacteristic, complemented
44#99—51no—
45#99—52no—
46C22×C5—24no—
47C22×C5—22no—
48C22×C5—21no—
49C22×C5—41no—
50C22×C5—41no—
51C22×C5—42no—
52C22×C5—41no—
53C22×C5—41no—
54C22×C5C2×D412yes—
55C22×C5C2×D411yescharacteristic
56C2×D5—22no—
57C2×D5—42no—
58C2×D5—81no—
59C4×C5—21no—
60C4×C5—41no—
61Dic5—22no—
62Dic5—42no—
63C24C2×D511yescharacteristic, complemented
64#41—102no—
65#41—101no—
66C2×D4—21no—
67C2×D4—52no—
68C2×D4—102no—
69C2×D4—101no—
70C2×D4—102no—
71C2×D4C2×D511yescharacteristic
72C22⋊C4—21no—
73C22⋊C4—52no—
74C22⋊C4—102no—
75C22⋊C4C2×D511yescharacteristic, complemented
76C2×C5—22no—
77C2×C5—21no—
78C2×C5—41no—
79C2×C5—42no—
80C2×C5#12011yescharacteristic
81D5—42no—
82C23—22no—
83C23—21no—
84C23—42no—
85C23—52no—
86C23—101no—
87C23C22⋊D511yescharacteristic, complemented
88C23C22×D511yescharacteristic
89Q8—101no—
90D4—22no—
91D4—42no—
92D4—101no—
93D4—102no—
94D4—102no—
95D4—102no—
96D4—204no—
97C2×C4—21no—
98C2×C4—52no—
99C2×C4—102no—
100C2×C4—101no—
101C2×C4—102no—
102C2×C4C22⋊D511yescharacteristic, complemented
103C5B(4,𝔽2)11yescharacteristic, complemented, Sylow
104C22—22no—
105C22—24no—
106C22—21no—
107C22—41no—
108C22—41no—
109C22—41no—
110C22—42no—
111C22—41no—
112C22—102no—
113C22—202no—
114C22—401no—
115C22D4×D512yes—
116C22C2×C22⋊D511yescharacteristic
117C4—21no—
118C4—41no—
119C4—102no—
120C4—202no—
121C2—21no—
122C2—22no—
123C2—42no—
124C2—41no—
125C2—202no—
126C2#408311yescharacteristic
subgroupG/H
centerC2#4083
commutator, cosocleC23×C5C23
FrattiniC23C22×D5
FittingC5×#120C2
socleC2×C5#120
hypercenter#120D5
hypocenterC5B(4,𝔽2)
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C23 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #4144C2semidirect
2. C5×#120C2semidirect
3. #4099C2semidirect
4. #3966C2semidirect
5. #3938C2semidirect
6. C24×C5C22semidirect
16. C5×C22⋊C4C22semidirect
18. C22⋊Dic5C22semidirect
23. C23×C5D4semidirect
26. C22×D5D4semidirect
30. C2×C4×C5D4semidirect
37. C2×Dic5D4semidirect
43. #120D5semidirect
63. C24C2×D5semidirect
75. C22⋊C4C2×D5semidirect
87. C23C22⋊D5semidirect
102. C2×C4C22⋊D5semidirect
103. C5B(4,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
#41442
C5×#1202
#40992
#39662
#39382
B(4,𝔽2)5