Finite Group Databasebeta

← #13088213

Subgroups — #13088213

Order 1440.

360 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×C32×C5×Q8C211yesmaximal, characteristic, complemented
2#10624283C211yesmaximal, characteristic
3#10623912C212yesmaximal, complemented
4#10623911C212yesmaximal, complemented
5#10623904C211yesmaximal, characteristic, complemented
6#91831—34nomaximal
7C32×C5×Q8—21no—
8C32×C5×Q8C2212yes—
9C2×C32×C4×C5—21no—
10C2×C32×C4×C5C2211yescharacteristic
11#67184—21no—
12C4⋊(C3⋊D15)C2211yescharacteristic
13C4×C3⋊D15—21no—
14#67180C2211yescharacteristic
15C3⋊(C3⋊#168)C2212yes—
16#64151—51nomaximal, Hall
17C2×C3×C5×Q8D314yescomplemented
18#6817—34no—
19#6718—38no—
20#6717—38no—
21#6710—34no—
22C22×C32×C5D411yescharacteristic
23C2×C3⋊D15—21no—
24C32×C4×C5—22no—
25C32×C4×C5D411yescharacteristic
26C32×C4×C5C2311yescharacteristic
27C3⋊Dic15—21no—
28#4088—91noHall
29C2×C32×Q8D511yescharacteristic, complemented
30#3828—51no—
31#3756—52no—
32#3755—52no—
33#3748—51no—
34C3×C5×Q8—24no—
35C3×C5×Q8C2×D318yes—
36C2×C3×C4×C5—24no—
37C2×C3×C4×C5C2×D314yes—
38#1220—64no—
39D60—34no—
40C4×D15—64no—
41Dic30—34no—
42C3⋊#168—38no—
43#942—154no—
44C2×C32×C5—21no—
45C2×C32×C5C2×D411yescharacteristic
46C3⋊D15—41no—
47C2×C5×Q8C3⋊D311yescharacteristic, complemented
48#709—91no—
49#689—92no—
50#688—92no—
51#681—91no—
52C32×Q8—21no—
53C32×Q8C2×D512yes—
54C2×C32×C4—21no—
55C2×C32×C4C2×D511yescharacteristic
56#638—101no—
57C4⋊(C3⋊D3)—51no—
58C4×C3⋊D3—101no—
59#634—51no—
60C3⋊(C3⋊C8)—52no—
61C22×C3×C5#6714yes—
62C2×D15—64no—
63C3×C4×C5—28no—
64C3×C4×C5#6714yes—
65C3×C4×C5C22×D314yes—
66Dic15—64no—
67C2×C3×Q8D1514yescomplemented
68#235—154no—
69#216—158no—
70#215—158no—
71#208—154no—
72C32×C5#13711yescharacteristic, complemented, Hall
73C5×Q8—21no—
74C5×Q8C2×C3⋊D312yes—
75C2×C4×C5—21no—
76C2×C4×C5C2×C3⋊D311yescharacteristic
77C22⋊D5—181no—
78D20—91no—
79C4×D5—181no—
80Dic10—91no—
81#168—92no—
82C22×C32C22⋊D511yescharacteristic
83C2×C3⋊D3—101no—
84C32×C4—22no—
85C32×C4C22⋊D511yescharacteristic
86C32×C4C22×D511yescharacteristic
87C3⋊Dic3—101no—
88#137—451noSylow
89C2×C3×C5—24no—
90C2×C3×C5C2×#6714yes—
91D15—124no—
92C3×Q8—24no—
93C3×Q8C2×D1518yes—
94C2×C3×C4—24no—
95C2×C3×C4C2×D1514yes—
96#67—304no—
97D12—154no—
98C4×D3—304no—
99Dic6—154no—
100C3⋊C8—158no—
101C22×C5#63811yescharacteristic
102C2×D5—181no—
103C4×C5—22no—
104C4×C5#63811yescharacteristic
105C4×C5C22×C3⋊D311yescharacteristic
106Dic5—181no—
107C2×C32—21no—
108C2×C32C2×C22⋊D511yescharacteristic
109C3⋊D3—201no—
110#41—451no—
111C2×Q8C3⋊D1511yescharacteristic, complemented
112Q16—452no—
113SD16—452no—
114M16—451no—
115C3×C5#94214yescomplemented
116C22×C3#122014yes—
117C2×D3—304no—
118C3×C4—28no—
119C3×C4#122014yes—
120C3×C4C22×D1514yes—
121Dic3—304no—
122C2×C5—21no—
123C2×C5C2×#63811yescharacteristic
124D5—361no—
125C32#408811yescharacteristic, complemented, Sylow
126Q8—21no—
127Q8—451no—
128Q8C2×C3⋊D1512yes—
129D4—451no—
130D4—901no—
131C2×C4—21no—
132C2×C4—901no—
133C2×C4C2×C3⋊D1511yescharacteristic
134C8—452no—
135C2×C3—24no—
136C2×C3C2×#122014yes—
137D3—604no—
138C5#6415111yescharacteristic, complemented, Sylow
139C22—901no—
140C22#6718411yescharacteristic
141C4—22no—
142C4—901no—
143C4#6718411yescharacteristic
144C4C22×C3⋊D1511yescharacteristic
145C3#9183114yescomplemented
146C2—21no—
147C2—1801no—
148C2C2×#6718411yescharacteristic
subgroupG/H
centerC2C2×#67184
commutator, cosocleC32×C4×C5C23
FrattiniC4C22×C3⋊D15
FittingC2×C32×C5×Q8C2
socleC2×C32×C5C2×D4
prime radical#10624283C2
hypercenterC2×Q8C3⋊D15
hypocenterC32×C5#137
LCSG ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C5×Q8C2semidirect
3. #10623912C2semidirect
4. #10623911C2semidirect
5. #10623904C2semidirect
17. C2×C3×C5×Q8D3semidirect
29. C2×C32×Q8D5semidirect
47. C2×C5×Q8C3⋊D3semidirect
67. C2×C3×Q8D15semidirect
72. C32×C5#137semidirect
111. C2×Q8C3⋊D15semidirect
115. C3×C5#942semidirect
125. C32#4088semidirect
138. C5#64151semidirect
145. C3#91831semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C5×Q82
#106242832
#106239122
#106239112
#106239042
#918313
#641515