Finite Group Databasebeta

← #65030

Subgroups — #65030

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
2#4083C211yesmaximal, characteristic, complemented
3#4028C212yesmaximal, complemented
4C5×#99C211yesmaximal, characteristic, complemented
5#3938C212yesmaximal, complemented
6C23×D5C2211yescharacteristic, complemented
7C2×C5×D4C2211yescharacteristic
8C2×C22⋊D5—21no—
9C2×C22⋊D5—21no—
10C2×C22⋊D5C2212yes—
11#711—22no—
12D4×D5—22no—
13#709—21no—
14C2×D20—22no—
15C5×C22⋊C4C2212yescomplemented
16C22⋊Dic5C2211yescharacteristic, complemented
17#685—21no—
18#685—22no—
19B(4,𝔽2)—51nomaximal, Sylow
20C23×C5D411yescharacteristic, complemented
21C23×C5C2311yescharacteristic
22C22×D5—22no—
23C22×D5—21no—
24C22×D5—21no—
25C22×D5—42no—
26C22×D5D412yescomplemented
27C5×D4—22no—
28C2×C4×C5—22no—
29C2×C4×C5D411yescharacteristic, complemented
30C22⋊D5—22no—
31C22⋊D5—22no—
32C22⋊D5—22no—
33C22⋊D5—42no—
34C2×Dic5—21no—
35C2×Dic5—21no—
36C2×Dic5D412yescomplemented
37D20—21no—
38D20—44no—
39C4×D5—22no—
40Dic10—21no—
41H(5,𝔽2)—51no—
42#120—51no—
43#120—52no—
44#99—52no—
45#99D511yescharacteristic, complemented
46C22×C5—21no—
47C22×C5—41no—
48C22×C5—41no—
49C22×C5C2×D411yescharacteristic
50C22×C5C2×D412yes—
51C2×D5—24no—
52C2×D5—22no—
53C2×D5—22no—
54C2×D5—41no—
55C2×D5—42no—
56C2×D5—41no—
57C2×D5—42no—
58C2×D5—81no—
59C4×C5—21no—
60C4×C5—42no—
61Dic5—22no—
62Dic5—41no—
63C24—51no—
64#41—102no—
65#41—101no—
66C2×D4—52no—
67C2×D4—102no—
68C2×D4—102no—
69C2×D4—101no—
70C2×D4—101no—
71C2×D4C2×D511yescharacteristic
72C22⋊C4—51no—
73C22⋊C4—102no—
74C22⋊C4—101no—
75C22⋊C4C2×D512yescomplemented
76C2×C5—21no—
77C2×C5—22no—
78C2×C5—41no—
79C2×C5#12011yescharacteristic
80D5—42no—
81D5—42no—
82C23—52no—
83C23—101no—
84C23—101no—
85C23—102no—
86C23—202no—
87C23D2011yescharacteristic, complemented
88C23C22×D511yescharacteristic
89Q8—101no—
90D4—22no—
91D4—101no—
92D4—102no—
93D4—102no—
94D4—102no—
95D4—204no—
96D4—202no—
97C2×C4—22no—
98C2×C4—52no—
99C2×C4—102no—
100C2×C4—101no—
101C2×C4—101no—
102C2×C4D2011yescharacteristic, complemented
103C5B(4,𝔽2)11yescharacteristic, complemented, Sylow
104C22—21no—
105C22—41no—
106C22—41no—
107C22—104no—
108C22—102no—
109C22—102no—
110C22—202no—
111C22—202no—
112C22—201no—
113C22—201no—
114C22—401no—
115C22C2×D2011yescharacteristic
116C22D4×D512yes—
117C4—21no—
118C4—42no—
119C4—102no—
120C4—201no—
121C2—22no—
122C2—21no—
123C2—41no—
124C2—202no—
125C2—202no—
126C2#402811yescharacteristic
subgroupG/H
centerC2#4028
commutator, cosocleC23×C5C23
FrattiniC23C22×D5
FittingC5×#99C2
socleC2×C5#120
hypercenter#99D5
hypocenterC5B(4,𝔽2)
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
1. #4144C2semidirect
2. #4083C2semidirect
3. #4028C2semidirect
4. C5×#99C2semidirect
5. #3938C2semidirect
6. C23×D5C22semidirect
15. C5×C22⋊C4C22semidirect
16. C22⋊Dic5C22semidirect
20. C23×C5D4semidirect
26. C22×D5D4semidirect
29. C2×C4×C5D4semidirect
36. C2×Dic5D4semidirect
45. #99D5semidirect
75. C22⋊C4C2×D5semidirect
87. C23D20semidirect
102. C2×C4D20semidirect
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
#40832
#40282
C5×#992
#39382
B(4,𝔽2)5