Finite Group Databasebeta

← #13085664

Subgroups — #13085664

Order 1440.

270 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 70 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#10624223C211yesmaximal, characteristic
2#10623795C212yesmaximal, complemented
3C5×#3712C211yesmaximal, characteristic, complemented
4C3⋊D3×#168C211yesmaximal, characteristic, complemented
5C3×#6648C212yesmaximal, complemented
6#91266—32nomaximal
7C4×C5×C3⋊D3C2211yescharacteristic
8C3×C4×D15C2212yes—
9#67158C411yescharacteristic
10#67157C411yescharacteristic
11#67156C412yes—
12C3⋊(C3⋊#168)C2211yescharacteristic
13C3×C5×C3⋊C8C2212yes—
14C32×#168C2211yescharacteristic
15#63803—51nomaximal, Hall
16#6771—32no—
17#6705—32no—
18C5×ΓL(1,𝔽25)—32no—
19C3×C8×D5—32no—
20#6649—32no—
21#6648D312yescomplemented
22D3×#168—32no—
23D3×#168—61no—
24C2×C5×C3⋊D3C2×C411yescharacteristic
25C2×C3×D15C2×C412yes—
26C32×C4×C5C2311yescharacteristic
27C5×C3⋊Dic3C2×C411yescharacteristic
28C3×Dic15C2×C412yes—
29#4053—91noHall
30#3798—51no—
31C8×C3⋊D3—51no—
32C3×D3×C8—52no—
33#3712D511yescharacteristic, complemented
34#3711—52no—
35C4×D15C2×D312yes—
36C4×C5×D3—32no—
37C4×C5×D3—61no—
38C3×C4×D5—32no—
39#1204—32no—
40#1203—32no—
41#1202—32no—
42#1201—32no—
43C3×C5×C8—32no—
44C3⋊#168—32no—
45C3⋊#168—61no—
46C3×#168—21no—
47C3×#168C2×D312yes—
48C5×C3⋊C8C2×D312yes—
49#901—152no—
50C2×C32×C5C22×C411yescharacteristic
51C5×C3⋊D3—21no—
52C3×D15—22no—
53#709—91no—
54C5×M16—91no—
55C2×#168—91no—
56#676—92no—
57C8×D5—32no—
58C4×C3⋊D3C2×D511yescharacteristic
59C3×C4×D3—52no—
60#627—51no—
61#626—52no—
62#625—51no—
63C32×C8—51no—
64C3⋊(C3⋊C8)—51no—
65C3×C3⋊C8C2×D512yes—
66C2×D15C4×D312yes—
67C2×C5×D3—32no—
68C2×C5×D3—61no—
69C2×C3×D5—32no—
70C3×C4×C5—21no—
71C3×C4×C5C22×D312yes—
72Dic15C4×D312yes—
73C3×Dic5—32no—
74C5×Dic3—32no—
75C5×Dic3—61no—
76#235—152no—
77C2×C3×C8—152no—
78#208—152no—
79ΓL(1,𝔽25)—32no—
80ΓL(1,𝔽25)—152no—
81D3×C8—52no—
82D3×C8—152no—
83D3×C8—301no—
84C32×C5#13111yescharacteristic, complemented, Hall
85C2×C4×C5—91no—
86C22⋊D5—92no—
87D20—91no—
88C4×D5—32no—
89Dic10—91no—
90C5×C8—32no—
91#168—91no—
92#168D3211yescharacteristic
93C2×C3⋊D3C4×D511yescharacteristic
94C2×C3×D3—52no—
95C32×C4C22×D511yescharacteristic
96C3⋊Dic3C4×D511yescharacteristic
97C3×Dic3—52no—
98#131—451noSylow
99C2×C3×C5—21no—
100C2×C3×C5C2×C4×D312yes—
101D15—22no—
102C3×D5—62no—
103C5×D3—62no—
104C5×D3—62no—
105C2×C3×C4—152no—
106#67—152no—
107#67—152no—
108D12—152no—
109C4×D3—32no—
110C4×D3—52no—
111C4×D3—61no—
112Dic6—152no—
113C3×C8—32no—
114C3×C8—52no—
115C3×C8—101no—
116C3⋊C8—152no—
117C3⋊C8—301no—
118C3⋊C8D3×D512yes—
119C22×C5—91no—
120C2×D5—32no—
121C4×C5—91no—
122C4×C5C2×D3211yescharacteristic
123Dic5—32no—
124C2×C32C2×C4×D511yescharacteristic
125C3⋊D3—21no—
126C3×D3—102no—
127#41—451no—
128M16—91no—
129M16—452no—
130C2×C8—152no—
131C2×C8—451no—
132C3×C5—21no—
133C3×C5#90112yescomplemented
134C22×C3—152no—
135C2×D3—32no—
136C2×D3—52no—
137C2×D3—61no—
138C3×C4—21no—
139C3×C4—152no—
140C3×C4C2×D3×D512yes—
141Dic3—32no—
142Dic3—52no—
143Dic3—61no—
144C2×C5—181no—
145C2×C5C4×D3211yescharacteristic
146D5—62no—
147C32#405311yescharacteristic, complemented, Sylow
148Q8—451no—
149D4—451no—
150D4—452no—
151C2×C4—91no—
152C2×C4—152no—
153C8—32no—
154C8—51no—
155C8—451no—
156C2×C3—21no—
157C2×C3—302no—
158C2×C3C4×D3×D512yes—
159D3—62no—
160D3—62no—
161D3—102no—
162C5#6380311yescharacteristic, complemented, Sylow
163C22—91no—
164C22—152no—
165C4—91no—
166C4—152no—
167C4D32×D511yescharacteristic
168C3—21no—
169C3#9126612yescomplemented
170C2—181no—
171C2—302no—
172C2#1062402211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC4D32×D5
commutator, socleC2×C32×C5C22×C4
Fitting, cosocleC32×C4×C5C23
prime radical#10624223C2
hypocenterC32×C5#131
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C4

Complements

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

NH ≅ G/Ntype
2. #10623795C2semidirect
3. C5×#3712C2semidirect
4. C3⋊D3×#168C2semidirect
5. C3×#6648C2semidirect
21. #6648D3semidirect
33. #3712D5semidirect
84. C32×C5#131semidirect
133. C3×C5#901semidirect
147. C32#4053semidirect
162. C5#63803semidirect
169. C3#91266semidirect

Maximal subgroups

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

subgroupindex
#106242232
#106237952
C5×#37122
C3⋊D3×#1682
C3×#66482
#912663
#638035