Finite Group Databasebeta

← #13086156

Subgroups — #13086156

Order 1440.

254 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 74 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#10623832C211yesmaximal, characteristic
2#10623814C212yesmaximal
3C5×#3724C211yesmaximal, characteristic
4Dic5×C3⋊Dic3C211yesmaximal, characteristic
5C3×Dic3⋊Dic5C212yesmaximal
6#91346—32nomaximal
7C2×C3⋊Dic15C2211yescharacteristic
8C2×C5×C3⋊Dic3C2211yescharacteristic
9C2×C3×Dic15C2212yes—
10C2×C3×C5×Dic3C2212yes—
11C2×C32×Dic5C2211yescharacteristic
12#63870—51nomaximal, Hall
13#6712—32no—
14C5×Dic3⋊C4—32no—
15C3×C4⋊Dic5—32no—
16Dic3⋊Dic5—32no—
17Dic3⋊Dic5D312yes—
18#6668—32no—
19Dic3×Dic5—32no—
20Dic3×Dic5—61no—
21C22×C32×C5C2311yescharacteristic
22C3⋊Dic15—21no—
23C5×C3⋊Dic3Q812yes—
24C3×Dic15—22no—
25C3×C5×Dic3—22no—
26C32×Dic5—21no—
27#4036—91noHall
28C4×C3⋊Dic3—51no—
29C3×Dic3⋊C4—52no—
30#3724—51no—
31#3724D511yescharacteristic
32Dic3⋊Dic3—52no—
33C2×C3×C4×C5—32no—
34C2×Dic15—32no—
35C2×Dic15—61no—
36C2×Dic15C2×D312yes—
37C2×C5×Dic3—32no—
38C2×C5×Dic3—61no—
39C2×C5×Dic3C2×D312yes—
40C2×C3×Dic5—21no—
41C2×C3×Dic5—32no—
42C2×C3×Dic5C2×D312yes—
43#889—152no—
44C2×C32×C5C2×Q811yescharacteristic
45C2×C32×C5#4111yescharacteristic
46C2×C32×C5#4111yescharacteristic
47C5×C4⋊C4—91no—
48C4⋊Dic5—32no—
49C4⋊Dic5—91no—
50#683—92no—
51C4×Dic5—91no—
52C2×C32×C4—51no—
53C2×C3⋊Dic3—51no—
54C2×C3⋊Dic3C2×D511yescharacteristic
55C2×C3×Dic3—52no—
56C2×C3×Dic3C2×D512yes—
57C22×C3×C5—21no—
58C22×C3×C5C22×D312yes—
59C3×C4×C5—62no—
60Dic15—22no—
61Dic15—62no—
62Dic15—62no—
63C3×Dic5—22no—
64C3×Dic5—22no—
65C3×Dic5—62no—
66C5×Dic3—22no—
67C5×Dic3—34no—
68C5×Dic3—62no—
69C3×C4⋊C4—152no—
70C4⋊Dic3—152no—
71Dic3⋊C4—32no—
72Dic3⋊C4—52no—
73Dic3⋊C4—152no—
74Dic3⋊C4—152no—
75C4×Dic3—152no—
76C4×Dic3—301no—
77C32×C5#12511yescharacteristic, complemented, Hall
78C2×C4×C5—32no—
79C2×C4×C5—91no—
80C2×Dic5—32no—
81C2×Dic5—91no—
82C2×Dic5D3211yescharacteristic
83C22×C32C22×D511yescharacteristic
84C32×C4—101no—
85C3⋊Dic3—101no—
86C3⋊Dic3Dic1012yes—
87C3×Dic3—22no—
88C3×Dic3—102no—
89#125—451noSylow
90C2×C3×C5—21no—
91C2×C3×C5—21no—
92C2×C3×C5—21no—
93C2×C3×C5#23512yes—
94C2×C3×C5#23712yes—
95C2×C3×C5D3×Q812yes—
96C2×C3×C4—32no—
97C2×C3×C4—52no—
98C2×C3×C4—101no—
99C2×C3×C4—152no—
100C2×Dic3—32no—
101C2×Dic3—52no—
102C2×Dic3—61no—
103C2×Dic3—152no—
104C2×Dic3—301no—
105C2×Dic3D3×D512yes—
106C22×C5C2×D3211yescharacteristic
107C4×C5—62no—
108C4×C5—92no—
109Dic5—21no—
110Dic5—62no—
111Dic5—181no—
112C2×C32C2×Dic1011yescharacteristic
113C2×C32#71111yescharacteristic
114C2×C32#71311yescharacteristic
115C4⋊C4—91no—
116C4⋊C4—152no—
117C4⋊C4—452no—
118C4⋊C4—451no—
119C42—451no—
120C3×C5—21no—
121C3×C5#88912yescomplemented
122C22×C3—21no—
123C22×C3C2×D3×D512yes—
124C3×C4—62no—
125C3×C4—102no—
126C3×C4—102no—
127C3×C4—302no—
128Dic3—22no—
129Dic3—34no—
130Dic3—62no—
131Dic3—102no—
132Dic3—302no—
133Dic3—302no—
134C2×C5#379711yescharacteristic
135C2×C5#379811yescharacteristic
136C2×C5#380511yescharacteristic
137C32#403611yescharacteristic, complemented, Sylow
138C2×C4—32no—
139C2×C4—51no—
140C2×C4—91no—
141C2×C4—152no—
142C2×C4—451no—
143C2×C3—21no—
144C2×C3—21no—
145C2×C3—21no—
146C2×C3#676612yes—
147C2×C3D3×Dic1012yes—
148C2×C3#678212yes—
149C5#6387011yescharacteristic, complemented, Sylow
150C22D32×D511yescharacteristic
151C4—62no—
152C4—92no—
153C4—101no—
154C4—302no—
155C4—901no—
156C3—21no—
157C3#9134612yescomplemented
158C2#1062403911yescharacteristic
159C2#1062404711yescharacteristic
160C2#1062404811yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutator, Fitting, socle, prime radical, cosocleC22×C32×C5C23
hypocenterC32×C5#125
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
77. C32×C5#125semidirect
121. C3×C5#889semidirect
137. C32#4036semidirect
149. C5#63870semidirect
157. C3#91346semidirect

Maximal subgroups

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

subgroupindex
#106238322
#106238142
C5×#37242
Dic5×C3⋊Dic32
C3×Dic3⋊Dic52
#913463
#638705