Finite Group Databasebeta

← D3×#1200 #10624021

Subgroups — D3×#1200 #10624021

Order 720.

232 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 64 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×D15C211yesmaximal, characteristic
2#67154C211yesmaximal, characteristic, complemented
3#67149C211yesmaximal, characteristic, complemented
4C5×D3×Dic3C211yesmaximal, characteristic, complemented
5#67137C211yesmaximal, characteristic, complemented
6C3×#1200C211yesmaximal, characteristic, complemented
7C3×D3×Dic5C211yesmaximal, characteristic, complemented
8C2×#1200—31nomaximal
9C4×D3×D5—31nomaximal
10C2×C3⋊D15C2211yescharacteristic
11C2×C3×D15C2211yescharacteristic
12C2×C3×C5×D3C2211yescharacteristic
13D3×D15C414yescomplemented
14C5×C3⋊Dic3C2211yescharacteristic, complemented
15C3×Dic15C2211yescharacteristic, complemented
16C3×C5×Dic3C2211yescharacteristic, complemented
17C32×Dic5C2211yescharacteristic, complemented
18C4×D32—51nomaximal, Hall
19C22×D15—31no—
20C2×D3×D5—31no—
21C4×D15—31no—
22C2×C5×Dic3—31no—
23C4×C5×D3—31no—
24C2×C3×Dic5—31no—
25C3×C4×D5—31no—
26#1200—31no—
27#1200—31no—
28#1200—31no—
29#1200—31no—
30#1200—61no—
31#1200D311yescharacteristic, direct product factor
32D3×Dic5D311yescharacteristic, complemented
33D5×Dic3—31no—
34C2×C32×C5C2311yescharacteristic
35C3⋊D15C2×C412yescomplemented
36C3×D15C2×C412yescomplemented
37C3×C5×D3C2×C412yescomplemented
38C2×C4×D5—91noHall
39C2×D32—51no—
40C4×C3⋊D3—51no—
41C3×C4×D3—51no—
42C3×C4×D3—51no—
43#624—51no—
44D3×Dic3—51no—
45D3×Dic3D511yescharacteristic, complemented
46C22×C3×C5—31no—
47C2×D15—31no—
48C2×D15—31no—
49C2×D15—34no—
50C2×D15—61no—
51C2×D15C2×D311yescharacteristic
52C2×C5×D3C2×D311yescharacteristic
53C2×C3×D5—31no—
54D3×D5—34no—
55C3×C4×C5—31no—
56Dic15C2×D311yescharacteristic, complemented
57C3×Dic5—21no—
58C3×Dic5—31no—
59C3×Dic5C2×D311yescharacteristic, complemented
60C3×Dic5C2×D311yescharacteristic, complemented
61C5×Dic3—31no—
62C5×Dic3—31no—
63C5×Dic3—61no—
64C5×Dic3C2×D311yescharacteristic, complemented
65C2×C4×D3—151no—
66C2×C4×D3—151no—
67C32×C5C22×C411yescharacteristic, complemented, Hall
68C22×D5—91no—
69C2×C4×C5—91no—
70C2×Dic5—31no—
71C4×D5—31no—
72C4×D5—91no—
73C4×D5—91no—
74C4×D5—91no—
75C2×C3⋊D3—51no—
76C2×C3×D3—51no—
77C2×C3×D3C2×D511yescharacteristic
78D32—54no—
79C32×C4—51no—
80C3⋊Dic3C2×D511yescharacteristic, complemented
81C3×Dic3—51no—
82C3×Dic3C2×D511yescharacteristic, complemented
83C2×C3×C5—21no—
84C2×C3×C5—32no—
85C2×C3×C5C22×D311yescharacteristic
86C2×C3×C5C22×D311yescharacteristic
87D15—32no—
88D15—32no—
89D15—62no—
90D15C4×D312yescomplemented
91C3×D5—32no—
92C5×D3C4×D312yescomplemented
93C22×D3—151no—
94C22×D3—151no—
95C2×C3×C4—151no—
96C2×C3×C4—151no—
97C2×Dic3—31no—
98C2×Dic3—151no—
99C4×D3—31no—
100C4×D3—51no—
101C4×D3—51no—
102C4×D3—151no—
103C4×D3—151no—
104C4×D3—151no—
105C4×D3—151no—
106C4×D3—151no—
107C4×D3—301no—
108C22×C5—31no—
109C2×D5—31no—
110C2×D5—94no—
111C2×D5—91no—
112C4×C5—31no—
113C4×C5—91no—
114Dic5—31no—
115Dic5D3211yescharacteristic, complemented
116C2×C32C22×D511yescharacteristic
117C3⋊D3—52no—
118C3×D3—52no—
119C3×D3C4×D512yescomplemented
120C22×C4—451noSylow
121C3×C5—21no—
122C3×C5C2×C4×D311yescharacteristic, complemented
123C3×C5C2×C4×D311yescharacteristic, complemented
124C22×C3—31no—
125C22×C3—151no—
126C2×D3—51no—
127C2×D3—154no—
128C2×D3—151no—
129C2×D3—151no—
130C2×D3—154no—
131C2×D3—301no—
132C2×D3D3×D511yescharacteristic
133C3×C4—31no—
134C3×C4—51no—
135C3×C4—51no—
136C3×C4—101no—
137C3×C4—151no—
138Dic3—31no—
139Dic3—31no—
140Dic3—51no—
141Dic3—61no—
142Dic3D3×D511yescharacteristic, complemented
143C2×C5—32no—
144C2×C5C2×D3211yescharacteristic
145D5—32no—
146D5—92no—
147C32C2×C4×D511yescharacteristic, complemented, Sylow
148C23—451no—
149C2×C4—91no—
150C2×C4—151no—
151C2×C4—151no—
152C2×C4—451no—
153C2×C4—451no—
154C2×C4—451no—
155C2×C3—21no—
156C2×C3—32no—
157C2×C3—152no—
158C2×C3C2×D3×D511yescharacteristic
159C2×C3C2×D3×D511yescharacteristic
160D3—52no—
161D3—152no—
162D3—152no—
163D3—302no—
164D3#120012yesdirect product factor
165C5C4×D3211yescharacteristic, complemented, Sylow
166C22—31no—
167C22—151no—
168C22—454no—
169C22—451no—
170C4—31no—
171C4—51no—
172C4—91no—
173C4—151no—
174C3—21no—
175C3C4×D3×D511yescharacteristic, complemented
176C3C2×#120011yescharacteristic, complemented
177C2—32no—
178C2—152no—
179C2—452no—
180C2D32×D511yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2D32×D5
commutator, hypocenterC32×C5C22×C4
Fitting, socle, cosocleC2×C32×C5C23
prime radicalC2×D3×D15C2
LCSG ▷ C32×C5
UCSC1 ◁ C2

Complements

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

NH ≅ G/Ntype
2. #67154C2semidirect
3. #67149C2semidirect
4. C5×D3×Dic3C2semidirect
5. #67137C2semidirect
6. C3×#1200C2semidirect
7. C3×D3×Dic5C2semidirect
13. D3×D15C4semidirect
14. C5×C3⋊Dic3C22semidirect
15. C3×Dic15C22semidirect
16. C3×C5×Dic3C22semidirect
17. C32×Dic5C22semidirect
31. #1200D3direct (N × H)
32. D3×Dic5D3semidirect
35. C3⋊D15C2×C4semidirect
36. C3×D15C2×C4semidirect
37. C3×C5×D3C2×C4semidirect
45. D3×Dic3D5semidirect
56. Dic15C2×D3semidirect
59. C3×Dic5C2×D3semidirect
60. C3×Dic5C2×D3semidirect
64. C5×Dic3C2×D3semidirect
67. C32×C5C22×C4semidirect
80. C3⋊Dic3C2×D5semidirect
82. C3×Dic3C2×D5semidirect
90. D15C4×D3semidirect
92. C5×D3C4×D3semidirect
115. Dic5D32semidirect
119. C3×D3C4×D5semidirect
122. C3×C5C2×C4×D3semidirect
123. C3×C5C2×C4×D3semidirect
142. Dic3D3×D5semidirect
147. C32C2×C4×D5semidirect
164. D3#1200direct (N × H)
165. C5C4×D32semidirect
175. C3C4×D3×D5semidirect
176. C3C2×#1200semidirect

Maximal subgroups

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

subgroupindex
C2×D3×D152
#671542
#671492
C5×D3×Dic32
#671372
C3×#12002
C3×D3×Dic52
C2×#12003
C4×D3×D53
C4×D325