Finite Group Databasebeta

← #65487

Subgroups — #65487

Order 320.

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

normal subgroups only

This group has 168 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#4149C211yesmaximal, characteristic, complemented
2C5×AGL(1,ℤ/8)C211yesmaximal, characteristic, complemented
3#4086C211yesmaximal, characteristic, complemented
4#4078C211yesmaximal, characteristic, complemented
5#3969C211yesmaximal, characteristic, complemented
6#3957C211yesmaximal, characteristic, complemented
7#3955C211yesmaximal, characteristic, complemented
8C5×#41C2211yescharacteristic
9C2×C5×D4C2211yescharacteristic, complemented
10C2×C22⋊D5—22no—
11#713—21no—
12D4×D5—22no—
13D4×D5—21no—
14#709—21no—
15#709C2211yescharacteristic
16C2×D20—21no—
17C2×D20C2211yescharacteristic
18C5×SD16—21no—
19C5×D8—21no—
20C5×M16C2211yescharacteristic, complemented
21#687—21no—
22#686—21no—
23C4×Dic5C2211yescharacteristic, complemented
24#681C2211yescharacteristic, complemented
25#453—51nomaximal, Sylow
26C23×C5—21no—
27C22×D5—22no—
28C22×D5D412yes—
29C5×Q8D411yescharacteristic, complemented
30C5×D4—21no—
31C5×D4—21no—
32C5×D4—21no—
33C5×D4D411yescharacteristic, complemented
34C2×C4×C5—21no—
35C2×C4×C5C2311yescharacteristic
36C22⋊D5—22no—
37C22⋊D5—21no—
38C22⋊D5—44no—
39C2×Dic5—21no—
40D20—22no—
41D20—21no—
42D20—21no—
43D20D411yescharacteristic, complemented
44C4×D5—22no—
45C4×D5—21no—
46Dic10D411yescharacteristic, complemented
47C5×C8—21no—
48#168—21no—
49H(5,𝔽2)—51no—
50AGL(1,ℤ/8)—51no—
51AGL(1,ℤ/8)D511yescharacteristic, complemented
52#127—51no—
53#104—51no—
54#104—51no—
55#100—51no—
56C22×C5—21no—
57C22×C5—42no—
58C22×C5—41no—
59C22×C5C2×D411yescharacteristic
60C2×D5—22no—
61C2×D5—21no—
62C2×D5—42no—
63C2×D5—82no—
64C4×C5—21no—
65C4×C5C2×D411yescharacteristic
66C4×C5C2×D411yescharacteristic
67Dic5—22no—
68Dic5—21no—
69#41—51no—
70#41—101no—
71#41—101no—
72#41C2×D511yescharacteristic
73C2×D4—51no—
74C2×D4—101no—
75C2×D4—102no—
76C2×D4—101no—
77C2×D4—102no—
78C2×D4C2×D511yescharacteristic, complemented
79SD16—21no—
80SD16—101no—
81D8—21no—
82D8—101no—
83M16—51no—
84M16C2×D511yescharacteristic, complemented
85C42—51no—
86C2×C5—21no—
87C2×C5—41no—
88C2×C5—81no—
89C2×C5#12011yescharacteristic
90D5—42no—
91D5—41no—
92C23—21no—
93C23—52no—
94C23—102no—
95Q8—51no—
96Q8C22⋊D511yescharacteristic, complemented
97D4—21no—
98D4—21no—
99D4—21no—
100D4—51no—
101D4—102no—
102D4—102no—
103D4—101no—
104D4—101no—
105D4—101no—
106D4—204no—
107D4C22⋊D511yescharacteristic, complemented
108C2×C4—21no—
109C2×C4—102no—
110C2×C4—101no—
111C2×C4—101no—
112C2×C4C22×D511yescharacteristic
113C8—21no—
114C8—101no—
115C5#45311yescharacteristic, complemented, Sylow
116C22—21no—
117C22—42no—
118C22—41no—
119C22—101no—
120C22—102no—
121C22—202no—
122C22—402no—
123C22D4×D511yescharacteristic
124C4—21no—
125C4—102no—
126C4—101no—
127C4D4×D511yescharacteristic
128C4C2×C22⋊D511yescharacteristic
129C2—21no—
130C2—41no—
131C2—81no—
132C2—202no—
133C2—201no—
134C2#408311yescharacteristic
subgroupG/H
centerC2#4083
commutator, cosocleC2×C4×C5C23
FrattiniC2×C4C22×D5
FittingC5×AGL(1,ℤ/8)C2
socleC2×C5#120
hypercenterAGL(1,ℤ/8)D5
hypocenterC5#453
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #4149C2semidirect
2. C5×AGL(1,ℤ/8)C2semidirect
3. #4086C2semidirect
4. #4078C2semidirect
5. #3969C2semidirect
6. #3957C2semidirect
7. #3955C2semidirect
9. C2×C5×D4C22semidirect
20. C5×M16C22semidirect
23. C4×Dic5C22semidirect
24. #681C22semidirect
29. C5×Q8D4semidirect
33. C5×D4D4semidirect
43. D20D4semidirect
46. Dic10D4semidirect
51. AGL(1,ℤ/8)D5semidirect
78. C2×D4C2×D5semidirect
84. M16C2×D5semidirect
96. Q8C22⋊D5semidirect
107. D4C22⋊D5semidirect
115. C5#453semidirect

Maximal subgroups

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

subgroupindex
#41492
C5×AGL(1,ℤ/8)2
#40862
#40782
#39692
#39572
#39552
#4535