Finite Group Databasebeta

← D12×D15 #10624203

Subgroups — D12×D15 #10624203

Order 720.

232 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 54 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⋊D15)C211yesmaximal, characteristic, complemented
3C3×C4×D15C211yesmaximal, characteristic, complemented
4C3×C5×D12C211yesmaximal, characteristic, complemented
5#67152C212yesmaximal, complemented
6D4×D15—31nomaximal
7D5×D12—31nomaximal
8C2×C3⋊D15C2212yescomplemented
9C2×C3×D15C2211yescharacteristic
10C2×C3×C5×D3C2212yescomplemented
11D3×D15—24no—
12C32×C4×C5C2211yescharacteristic, complemented
13C3×Dic15C2211yescharacteristic, complemented
14D3×D12—51nomaximal, Hall
15C22×D15—32no—
16C2×D3×D5—32no—
17C3×C5×D4—31no—
18#1220—32no—
19D60—31no—
20D60—31no—
21D60—61no—
22C4×D15D311yescharacteristic, complemented
23C5×D12D311yescharacteristic, complemented
24C3×C4×D5—31no—
25#1203—32no—
26C2×C32×C5C2311yescharacteristic
27C3⋊D15—22no—
28C3×D15D412yescomplemented
29C3×C5×D3—22no—
30D4×D5—91noHall
31C2×D32—52no—
32C4⋊(C3⋊D3)—51no—
33C3×D12D511yescharacteristic, complemented
34C3×C4×D3—51no—
35#626—52no—
36C22×C3×C5—32no—
37C2×D15—32no—
38C2×D15—32no—
39C2×D15—62no—
40C2×D15—64no—
41C2×D15C2×D311yescharacteristic
42C2×C5×D3C2×D312yescomplemented
43C2×C3×D5—31no—
44D3×D5—64no—
45C3×C4×C5—21no—
46C3×C4×C5C2×D311yescharacteristic, complemented
47C3×C4×C5C2×D311yescharacteristic, complemented
48Dic15C2×D311yescharacteristic, complemented
49C3×Dic5—31no—
50D3×D4—151no—
51C2×D12—151no—
52C32×C5C2×D411yescharacteristic, complemented, Hall
53C22×D5—92no—
54C5×D4—31no—
55C22⋊D5—92no—
56D20—91no—
57C4×D5—31no—
58C2×C3⋊D3—52no—
59C2×C3×D3—51no—
60C2×C3×D3C2×D512yescomplemented
61D32—104no—
62C32×C4C2×D511yescharacteristic, complemented
63C3×Dic3—51no—
64C2×C3×C5—21no—
65C2×C3×C5—62no—
66C2×C3×C5C22×D311yescharacteristic
67C2×C3×C5C22×D311yescharacteristic
68D15—62no—
69D15—62no—
70D15—122no—
71D15D1212yesdirect product factor
72C3×D5—32no—
73C5×D3—22no—
74C22×D3—152no—
75C22×D3—152no—
76C3×D4—31no—
77C2×C3×C4—151no—
78#67—152no—
79D12—152no—
80D12—151no—
81D12—151no—
82D12—301no—
83D12D1511yescharacteristic, direct product factor
84C4×D3—51no—
85C22×C5—32no—
86C2×D5—31no—
87C2×D5—92no—
88C2×D5—184no—
89C4×C5D3211yescharacteristic, complemented
90Dic5—31no—
91C2×C32C22×D511yescharacteristic
92C3⋊D3—102no—
93C3×D3—22no—
94C3×D3—52no—
95C2×D4—451noSylow
96C3×C5—21no—
97C3×C5C2×D1211yescharacteristic, complemented
98C3×C5D3×D411yescharacteristic, complemented
99C22×C3—32no—
100C22×C3—151no—
101C2×D3—51no—
102C2×D3—152no—
103C2×D3—152no—
104C2×D3—302no—
105C2×D3—304no—
106C2×D3—304no—
107C2×D3C2×D1512yescomplemented
108C3×C4—21no—
109C3×C4—151no—
110C3×C4D3×D511yescharacteristic, complemented
111C3×C4C2×D1511yescharacteristic, complemented
112Dic3—51no—
113C2×C5—62no—
114C2×C5C2×D3211yescharacteristic
115D5—32no—
116D5—182no—
117C32D4×D511yescharacteristic, complemented, Sylow
118C23—452no—
119D4—31no—
120D4—452no—
121D4—451no—
122C2×C4—151no—
123C2×C3—21no—
124C2×C3—62no—
125C2×C3—152no—
126C2×C3C2×D3×D511yescharacteristic
127C2×C3C22×D1511yescharacteristic
128D3—22no—
129D3—52no—
130D3—302no—
131D3—302no—
132D3—602no—
133C5D3×D1211yescharacteristic, complemented, Sylow
134C22—32no—
135C22—151no—
136C22—452no—
137C22—904no—
138C4—151no—
139C4D3×D1511yescharacteristic, complemented
140C3—21no—
141C3D5×D1211yescharacteristic, complemented
142C3D4×D1511yescharacteristic, complemented
143C2—62no—
144C2—152no—
145C2—902no—
146C2C2×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⋊D15)C2semidirect
3. C3×C4×D15C2semidirect
4. C3×C5×D12C2semidirect
5. #67152C2semidirect
8. C2×C3⋊D15C22semidirect
10. C2×C3×C5×D3C22semidirect
12. C32×C4×C5C22semidirect
13. C3×Dic15C22semidirect
22. C4×D15D3semidirect
23. C5×D12D3semidirect
28. C3×D15D4semidirect
33. C3×D12D5semidirect
42. C2×C5×D3C2×D3semidirect
46. C3×C4×C5C2×D3semidirect
47. C3×C4×C5C2×D3semidirect
48. Dic15C2×D3semidirect
52. C32×C5C2×D4semidirect
60. C2×C3×D3C2×D5semidirect
62. C32×C4C2×D5semidirect
71. D15D12direct (N × H)
83. D12D15direct (N × H)
89. C4×C5D32semidirect
97. C3×C5C2×D12semidirect
98. C3×C5D3×D4semidirect
107. C2×D3C2×D15semidirect
110. C3×C4D3×D5semidirect
111. C3×C4C2×D15semidirect
117. C32D4×D5semidirect
133. C5D3×D12semidirect
139. C4D3×D15semidirect
141. C3D5×D12semidirect
142. 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⋊D15)2
C3×C4×D152
C3×C5×D122
#671522
D4×D153
D5×D123
D3×D125