Finite Group Databasebeta

← #10624052

Subgroups — #10624052

Order 720.

176 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 46 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#67157C211yesmaximal, characteristic, complemented
2#67151C212yesmaximal, complemented
3C5×#627C211yesmaximal, characteristic, complemented
4#67137C211yesmaximal, characteristic, complemented
5C3×#1200C212yesmaximal, complemented
6#6773—32nomaximal
7C2×C3⋊D15C2211yescharacteristic
8C2×C3×D15C2212yes—
9C5×C3⋊Dic3C2211yescharacteristic
10C3×C5×Dic3C2212yes—
11C32×Dic5C2211yescharacteristic
12#3798—51nomaximal, Hall
13D60—32no—
14C5×Dic6—32no—
15C3×C4×D5—32no—
16#1202—32no—
17#1202—32no—
18#1200—32no—
19#1200—61no—
20#1200D312yescomplemented
21C2×C32×C5C2311yescharacteristic
22C3⋊D15—21no—
23C3×D15—22no—
24#713—91noHall
25C4×C3⋊D3—51no—
26C3×C4×D3—52no—
27#627D511yescharacteristic, complemented
28#626—52no—
29#625—51no—
30C2×D15—32no—
31C2×D15—61no—
32C2×D15C2×D312yes—
33C2×C3×D5—32no—
34C3×C4×C5—32no—
35C3×Dic5—21no—
36C3×Dic5C2×D312yes—
37C5×Dic3—32no—
38C5×Dic3—61no—
39C5×Dic3C2×D312yes—
40#235—152no—
41C32×C5#4111yescharacteristic, complemented, Hall
42C5×Q8—91no—
43D20—92no—
44D20—91no—
45C4×D5—32no—
46C4×D5—91no—
47C2×C3⋊D3—51no—
48C2×C3×D3—52no—
49C32×C4—51no—
50C3⋊Dic3C2×D511yescharacteristic
51C3×Dic3C2×D512yes—
52C2×C3×C5—21no—
53C2×C3×C5C22×D312yes—
54D15—22no—
55D15—62no—
56D15—62no—
57C3×D5—62no—
58C2×C3×C4—152no—
59#67—152no—
60#67—152no—
61D12—152no—
62C4×D3—52no—
63C4×D3—152no—
64C4×D3—301no—
65Dic6—32no—
66C2×D5—32no—
67C2×D5—91no—
68C4×C5—32no—
69C4×C5—91no—
70Dic5D3211yescharacteristic
71C2×C32C22×D511yescharacteristic
72C3⋊D3—101no—
73C3×D3—102no—
74#41—451noSylow
75C3×C5—21no—
76C3×C5#23512yescomplemented
77C22×C3—152no—
78C2×D3—52no—
79C2×D3—152no—
80C2×D3—301no—
81C3×C4—32no—
82C3×C4—52no—
83C3×C4—101no—
84Dic3—32no—
85Dic3—61no—
86Dic3D3×D512yes—
87C2×C5C2×D3211yescharacteristic
88D5—62no—
89D5—181no—
90C32#71311yescharacteristic, complemented, Sylow
91Q8—91no—
92D4—452no—
93D4—451no—
94C2×C4—152no—
95C2×C4—451no—
96C2×C3—21no—
97C2×C3—302no—
98C2×C3C2×D3×D512yes—
99D3—102no—
100D3—302no—
101D3—302no—
102C5#379811yescharacteristic, complemented, Sylow
103C22—152no—
104C22—451no—
105C4—32no—
106C4—51no—
107C4—91no—
108C3—21no—
109C3#677312yescomplemented
110C2—302no—
111C2—901no—
112C2D32×D511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×D5
commutator, Fitting, socle, cosocleC2×C32×C5C23
hypocenterC32×C5#41
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
1. #67157C2semidirect
2. #67151C2semidirect
3. C5×#627C2semidirect
4. #67137C2semidirect
5. C3×#1200C2semidirect
20. #1200D3semidirect
27. #627D5semidirect
41. C32×C5#41semidirect
76. C3×C5#235semidirect
90. C32#713semidirect
102. C5#3798semidirect
109. C3#6773semidirect

Maximal subgroups

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

subgroupindex
#671572
#671512
C5×#6272
#671372
C3×#12002
#67733
#37985