Finite Group Databasebeta

← #10624205

Subgroups — #10624205

Order 720.

232 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 52 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×D3×D15C212yesmaximal, complemented
2C4×C3⋊D15C211yesmaximal, characteristic, complemented
3C3×D60C211yesmaximal, characteristic, complemented
4C3×C5×D12C211yesmaximal, characteristic, complemented
5#67150C212yesmaximal, complemented
6D4×D15—31nomaximal
7#6777—31nomaximal
8C2×C3⋊D15C2211yescharacteristic
9C2×C3×D15C2212yescomplemented
10C2×C3×C5×D3C2212yescomplemented
11D3×D15—24no—
12C32×C4×C5C2211yescharacteristic, complemented
13C3⋊Dic15C2211yescharacteristic, complemented
14#3802—51nomaximal, Hall
15C22×D15—32no—
16C2×D3×D5—32no—
17C3×C5×D4—31no—
18#1220—32no—
19D60D311yescharacteristic, complemented
20C4×D15—31no—
21C4×D15—31no—
22C4×D15—61no—
23C5×D12D311yescharacteristic, complemented
24C3×D20—31no—
25#1201—32no—
26C2×C32×C5C2311yescharacteristic
27C3⋊D15D412yescomplemented
28C3×D15—22no—
29C3×C5×D3—22no—
30D4×D5—91noHall
31C2×D32—52no—
32C4×C3⋊D3—51no—
33C3×D12—51no—
34C3×D12D511yescharacteristic, complemented
35#625—52no—
36C22×C3×C5—32no—
37C2×D15—31no—
38C2×D15—31no—
39C2×D15—61no—
40C2×D15—64no—
41C2×D15C2×D312yescomplemented
42C2×C5×D3C2×D312yescomplemented
43C2×C3×D5—32no—
44D3×D5—64no—
45C3×C4×C5—21no—
46C3×C4×C5C2×D311yescharacteristic, complemented
47C3×C4×C5C2×D311yescharacteristic, complemented
48Dic15—31no—
49Dic15—31no—
50Dic15—61no—
51D3×D4—151no—
52D3×D4—151no—
53C32×C5C2×D411yescharacteristic, complemented, Hall
54C22×D5—92no—
55C5×D4—31no—
56C22⋊D5—92no—
57D20—31no—
58C4×D5—91no—
59C2×C3⋊D3—51no—
60C2×C3×D3—52no—
61C2×C3×D3C2×D512yescomplemented
62D32—104no—
63C32×C4C2×D511yescharacteristic, complemented
64C3⋊Dic3—51no—
65C2×C3×C5—21no—
66C2×C3×C5—62no—
67C2×C3×C5C22×D311yescharacteristic
68C2×C3×C5C22×D311yescharacteristic
69D15—22no—
70D15—32no—
71D15—32no—
72D15—62no—
73C3×D5—62no—
74C5×D3—22no—
75C22×D3—152no—
76C22×D3—152no—
77C3×D4—31no—
78C3×D4—151no—
79#67—152no—
80#67—152no—
81D12—51no—
82D12D1511yescharacteristic, complemented
83C4×D3—151no—
84C4×D3—151no—
85C4×D3—301no—
86C22×C5—32no—
87C2×D5—32no—
88C2×D5—91no—
89C2×D5—184no—
90C4×C5D3211yescharacteristic, complemented
91Dic5—91no—
92C2×C32C22×D511yescharacteristic
93C3⋊D3—52no—
94C3×D3—22no—
95C3×D3—102no—
96C2×D4—451noSylow
97C3×C5—21no—
98C3×C5D3×D411yescharacteristic, complemented
99C3×C5D3×D411yescharacteristic, complemented
100C22×C3—32no—
101C22×C3—152no—
102C2×D3—52no—
103C2×D3—151no—
104C2×D3—151no—
105C2×D3—301no—
106C2×D3—304no—
107C2×D3—304no—
108C2×D3C2×D1512yescomplemented
109C3×C4—21no—
110C3×C4D3×D511yescharacteristic, complemented
111C3×C4C2×D1511yescharacteristic, complemented
112Dic3—151no—
113Dic3—151no—
114Dic3—301no—
115C2×C5—62no—
116C2×C5C2×D3211yescharacteristic
117D5—62no—
118D5—92no—
119C32D4×D511yescharacteristic, complemented, Sylow
120C23—452no—
121D4—31no—
122D4—151no—
123D4—452no—
124C2×C4—451no—
125C2×C3—21no—
126C2×C3—62no—
127C2×C3—302no—
128C2×C3C2×D3×D511yescharacteristic
129C2×C3C22×D1511yescharacteristic
130D3—22no—
131D3—102no—
132D3—152no—
133D3—152no—
134D3—302no—
135C5#380211yescharacteristic, complemented, Sylow
136C22—32no—
137C22—152no—
138C22—451no—
139C22—904no—
140C4—451no—
141C4D3×D1511yescharacteristic, complemented
142C3—21no—
143C3#677711yescharacteristic, complemented
144C3D4×D1511yescharacteristic, complemented
145C2—62no—
146C2—302no—
147C2—452no—
148C2C2×D3×D1511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D3×D15
commutator, socle, cosocleC2×C32×C5C23
FittingC32×C4×C5C22
hypercenterC4D3×D15
hypocenterC32×C5C2×D4
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C2×D3×D15C2semidirect
2. C4×C3⋊D15C2semidirect
3. C3×D60C2semidirect
4. C3×C5×D12C2semidirect
5. #67150C2semidirect
9. C2×C3×D15C22semidirect
10. C2×C3×C5×D3C22semidirect
12. C32×C4×C5C22semidirect
13. C3⋊Dic15C22semidirect
19. D60D3semidirect
23. C5×D12D3semidirect
27. C3⋊D15D4semidirect
34. C3×D12D5semidirect
41. C2×D15C2×D3semidirect
42. C2×C5×D3C2×D3semidirect
46. C3×C4×C5C2×D3semidirect
47. C3×C4×C5C2×D3semidirect
53. C32×C5C2×D4semidirect
61. C2×C3×D3C2×D5semidirect
63. C32×C4C2×D5semidirect
82. D12D15semidirect
90. C4×C5D32semidirect
98. C3×C5D3×D4semidirect
99. C3×C5D3×D4semidirect
108. C2×D3C2×D15semidirect
110. C3×C4D3×D5semidirect
111. C3×C4C2×D15semidirect
119. C32D4×D5semidirect
135. C5#3802semidirect
141. C4D3×D15semidirect
143. C3#6777semidirect
144. C3D4×D15semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D152
C4×C3⋊D152
C3×D602
C3×C5×D122
#671502
D4×D153
#67773
#38025