Finite Group Databasebeta

← #10624046

Subgroups — #10624046

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#67158C211yesmaximal, characteristic, complemented
2D3×Dic15C212yesmaximal, complemented
3C5×#625C211yesmaximal, characteristic, complemented
4D5×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×#1201C212yesmaximal, complemented
6#6780—32nomaximal
7C2×C3×C5×D3C2212yes—
8C2×C32×D5C2211yescharacteristic
9C3⋊Dic15C2211yescharacteristic
10C5×C3⋊Dic3C2211yescharacteristic
11C3×Dic15C2212yes—
12#3805—51nomaximal, Hall
13C2×Dic15—32no—
14C5×#67—32no—
15C3×C22⋊D5—32no—
16#1204—32no—
17#1201D312yescomplemented
18D3×Dic5—32no—
19D5×Dic3—32no—
20D5×Dic3—61no—
21C2×C32×C5C2311yescharacteristic
22C3×C5×D3—22no—
23C32×D5—21no—
24#711—91noHall
25C2×C3⋊Dic3—51no—
26C3×#67—52no—
27#627—51no—
28#625D511yescharacteristic, complemented
29D3×Dic3—52no—
30C22×C3×C5—32no—
31C2×C5×D3C2×D312yes—
32C2×C3×D5—21no—
33C2×C3×D5C2×D312yes—
34Dic15—32no—
35Dic15—61no—
36Dic15C2×D312yes—
37C3×Dic5—32no—
38C5×Dic3—32no—
39C5×Dic3—61no—
40#237—152no—
41C32×C5#4111yescharacteristic, complemented, Hall
42C5×D4—91no—
43C22⋊D5—32no—
44C2×Dic5—92no—
45C4×D5—91no—
46Dic10—91no—
47C22×C32—51no—
48C2×C3×D3C2×D512yes—
49C3⋊Dic3—51no—
50C3⋊Dic3C2×D511yescharacteristic
51C3×Dic3—52no—
52C2×C3×C5—21no—
53C2×C3×C5—62no—
54C2×C3×C5C22×D312yes—
55C3×D5—22no—
56C3×D5—22no—
57C5×D3—22no—
58C3×D4—152no—
59#67—32no—
60#67—52no—
61C2×Dic3—152no—
62C2×Dic3—152no—
63C2×Dic3—301no—
64C4×D3—152no—
65Dic6—152no—
66C22×C5—32no—
67C2×D5D3211yescharacteristic
68C4×C5—91no—
69Dic5—32no—
70Dic5—91no—
71C2×C32—101no—
72C2×C32C22×D511yescharacteristic
73C3×D3—22no—
74#41—451noSylow
75C3×C5—21no—
76C3×C5#23712yescomplemented
77C22×C3—32no—
78C22×C3—52no—
79C22×C3—101no—
80C2×D3D3×D512yes—
81C3×C4—152no—
82Dic3—32no—
83Dic3—52no—
84Dic3—61no—
85Dic3—152no—
86Dic3—301no—
87C2×C5—62no—
88C2×C5C2×D3211yescharacteristic
89D5—21no—
90C32#71111yescharacteristic, complemented, Sylow
91Q8—451no—
92D4—91no—
93D4—152no—
94C2×C4—451no—
95C2×C4—452no—
96C2×C3—21no—
97C2×C3—62no—
98C2×C3—102no—
99C2×C3—102no—
100C2×C3C2×D3×D512yes—
101D3—22no—
102C5#380511yescharacteristic, complemented, Sylow
103C22—32no—
104C22—51no—
105C4—91no—
106C4—152no—
107C4—451no—
108C3—21no—
109C3#678012yescomplemented
110C2—62no—
111C2—101no—
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. #67158C2semidirect
2. D3×Dic15C2semidirect
3. C5×#625C2semidirect
4. D5×C3⋊Dic3C2semidirect
5. C3×#1201C2semidirect
17. #1201D3semidirect
28. #625D5semidirect
41. C32×C5#41semidirect
76. C3×C5#237semidirect
90. C32#711semidirect
102. C5#3805semidirect
109. C3#6780semidirect

Maximal subgroups

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

subgroupindex
#671582
D3×Dic152
C5×#6252
D5×C3⋊Dic32
C3×#12012
#67803
#38055