Finite Group Databasebeta

← C2×C3×#6706 #13087999

Subgroups — C2×C3×#6706 #13087999

Order 1440.

294 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 110 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×C3×D60C211yesmaximal, characteristic
2C2×C3×Dic30C211yesmaximal, characteristic, complemented
3C2×C32×C5×C8C211yesmaximal, characteristic, complemented
4C3×#6706C214yesmaximal, direct product factor
5C2×#6706C311yesmaximal, characteristic, direct product factor
6C2×C3×#677—31nomaximal
7C22×C3×D15—21no—
8C2×C32×C4×C5C2211yescharacteristic
9C2×C3×Dic15—21no—
10C3×D60—21no—
11C3×D60C2212yes—
12C3×Dic30—21no—
13C3×Dic30C2212yescomplemented
14C32×C5×C8C2212yescomplemented
15C2×C3×#204—51nomaximal, Hall
16C2×D60C2×C311yescharacteristic
17C2×Dic30C2×C311yescharacteristic, complemented
18C2×C3×D20—31no—
19C2×C3×Dic10—31no—
20C2×C3×C5×C8—21no—
21C2×C3×C5×C8D311yescharacteristic, complemented
22C2×C3×C5×C8C2×C311yescharacteristic, complemented
23#6706C2×C314yesdirect product factor
24C3×#677—34no—
25C22×C32×C5D411yescharacteristic
26C2×C3×D15—22no—
27C2×C3×D15—42no—
28C32×C4×C5D411yescharacteristic
29C32×C4×C5C2311yescharacteristic
30C3×Dic15—22no—
31C2×#677—31noHall
32C2×C3×D12—51no—
33C2×C3×Dic6—51no—
34C2×C32×C8D511yescharacteristic, complemented
35C3×#204—54no—
36C22×D15—21no—
37C22×C3×D5—61no—
38C2×C3×C4×C5—21no—
39C2×C3×C4×C5C2×D311yescharacteristic
40C2×C3×C4×C5C22×C311yescharacteristic
41C2×Dic15—21no—
42D60—21no—
43D60C22×C312yes—
44Dic30—21no—
45Dic30C22×C312yescomplemented
46C2×C3×Dic5—61no—
47C3×D20—32no—
48C3×D20—61no—
49C3×Dic10—32no—
50C3×Dic10—61no—
51C3×C5×C8—22no—
52C3×C5×C8C2×D312yescomplemented
53C3×C5×C8C22×C312yescomplemented
54C2×C3×SD16—151no—
55C2×#204—51no—
56C2×C32×C5SD1612yescomplemented
57C2×C32×C5C2×D411yescharacteristic
58C3×D15—42no—
59C2×D20—31no—
60C2×Dic10—31no—
61C2×C5×C8C3×D311yescharacteristic, complemented
62#677—34no—
63C22×C3×D3—101no—
64C2×C32×C4C2×D511yescharacteristic
65C2×C3×Dic3—101no—
66C3×D12—52no—
67C3×D12—101no—
68C3×Dic6—52no—
69C3×Dic6—101no—
70C32×C8C2×D512yescomplemented
71C22×C3×C5—21no—
72C22×C3×C5D1211yescharacteristic
73C22×C3×C5C3×D411yescharacteristic
74C2×D15—22no—
75C2×D15—42no—
76C2×C3×D5—62no—
77C2×C3×D5—122no—
78C3×C4×C5—21no—
79C3×C4×C5—21no—
80C3×C4×C5D1211yescharacteristic
81C3×C4×C5C3×D411yescharacteristic
82C3×C4×C5C22×D311yescharacteristic
83C3×C4×C5C23×C311yescharacteristic
84Dic15—22no—
85C3×Dic5—62no—
86C2×C3×Q8—151no—
87C2×C3×D4—151no—
88C2×D12—51no—
89C2×Dic6—51no—
90C3×SD16—154no—
91C2×C3×C8—21no—
92C2×C3×C8C3×D511yescharacteristic, complemented
93C2×C3×C8D1511yescharacteristic, complemented
94#204—54no—
95C32×C5C2×SD1611yescharacteristic, complemented, Hall
96C22×D5—61no—
97C2×C4×C5C2×C3×D311yescharacteristic
98C2×Dic5—61no—
99D20—32no—
100D20—61no—
101Dic10—32no—
102Dic10—61no—
103C5×C8C2×C3×D312yescomplemented
104C22×C32D2011yescharacteristic
105C2×C3×D3—102no—
106C2×C3×D3—202no—
107C32×C4D2011yescharacteristic
108C32×C4C22×D511yescharacteristic
109C3×Dic3—102no—
110C2×SD16—151noSylow
111C2×C3×C5—22no—
112C2×C3×C5—21no—
113C2×C3×C5#20412yescomplemented
114C2×C3×C5C3×SD1612yescomplemented
115C2×C3×C5C2×D1211yescharacteristic
116C2×C3×C5C2×C3×D411yescharacteristic
117D15—42no—
118C3×D5—122no—
119C23×C3—301no—
120C22×D3—101no—
121C3×Q8—152no—
122C3×Q8—301no—
123C3×D4—152no—
124C3×D4—301no—
125C2×C3×C4—21no—
126C2×C3×C4—301no—
127C2×C3×C4C2×C3×D511yescharacteristic
128C2×C3×C4C2×D1511yescharacteristic
129C2×Dic3—101no—
130D12—52no—
131D12—101no—
132Dic6—52no—
133Dic6—101no—
134C3×C8—22no—
135C3×C8C2×C3×D512yescomplemented
136C3×C8C2×D1512yescomplemented
137C22×C5C3×D1211yescharacteristic
138C2×D5—62no—
139C2×D5—122no—
140C4×C5C3×D1211yescharacteristic
141C4×C5C22×C3×D311yescharacteristic
142Dic5—62no—
143C2×C32#67712yescomplemented
144C2×C32C2×D2011yescharacteristic
145C3×D3—202no—
146C2×Q8—151no—
147C2×D4—151no—
148SD16—154no—
149C2×C8C3×D1511yescharacteristic, complemented
150C3×C5—21no—
151C3×C5C2×#20411yescharacteristic, complemented
152C3×C5C2×C3×SD1611yescharacteristic, complemented
153C22×C3—21no—
154C22×C3—302no—
155C22×C3—602no—
156C22×C3C3×D2011yescharacteristic
157C22×C3D6011yescharacteristic
158C2×D3—102no—
159C2×D3—202no—
160C3×C4—21no—
161C3×C4—21no—
162C3×C4—302no—
163C3×C4C3×D2011yescharacteristic
164C3×C4D6011yescharacteristic
165C3×C4C22×C3×D511yescharacteristic
166C3×C4C22×D1511yescharacteristic
167Dic3—102no—
168C2×C5C3×#20412yescomplemented
169C2×C5C2×C3×D1211yescharacteristic
170D5—122no—
171C32C2×#67711yescharacteristic, complemented, Sylow
172C23—301no—
173Q8—152no—
174Q8—301no—
175D4—152no—
176D4—301no—
177C2×C4—301no—
178C2×C4C2×C3×D1511yescharacteristic
179C8C2×C3×D1512yescomplemented
180C2×C3—21no—
181C2×C3—22no—
182C2×C3—602no—
183C2×C3C3×#67712yescomplemented
184C2×C3#670612yesdirect product factor
185C2×C3C2×C3×D2011yescharacteristic
186C2×C3C2×D6011yescharacteristic
187D3—202no—
188C5C2×C3×#20411yescharacteristic, complemented, Sylow
189C22—302no—
190C22—602no—
191C22C3×D6011yescharacteristic
192C4—302no—
193C4C3×D6011yescharacteristic
194C4C22×C3×D1511yescharacteristic
195C3—21no—
196C3C2×C3×#67711yescharacteristic, complemented
197C3C2×#670611yescharacteristic, direct product factor
198C2—602no—
199C2C3×#670612yesdirect product factor
200C2C2×C3×D6011yescharacteristic
subgroupG/H
centerC22×C3D60
commutator, cosocleC3×C4×C5C23×C3
FrattiniC4C22×C3×D15
FittingC2×C32×C5×C8C2
socleC22×C32×C5D4
prime radicalC2×C3×D60C2
hypercenterC2×C3×C8D15
hypocenterC3×C5C2×C3×SD16
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C2×C3×Dic30C2semidirect
3. C2×C32×C5×C8C2semidirect
4. C3×#6706C2direct (N × H)
5. C2×#6706C3direct (N × H)
13. C3×Dic30C22semidirect
14. C32×C5×C8C22semidirect
17. C2×Dic30C2×C3semidirect
21. C2×C3×C5×C8D3semidirect
22. C2×C3×C5×C8C2×C3semidirect
23. #6706C2×C3direct (N × H)
34. C2×C32×C8D5semidirect
45. Dic30C22×C3semidirect
52. C3×C5×C8C2×D3semidirect
53. C3×C5×C8C22×C3semidirect
56. C2×C32×C5SD16semidirect
61. C2×C5×C8C3×D3semidirect
70. C32×C8C2×D5semidirect
92. C2×C3×C8C3×D5semidirect
93. C2×C3×C8D15semidirect
95. C32×C5C2×SD16semidirect
103. C5×C8C2×C3×D3semidirect
113. C2×C3×C5#204semidirect
114. C2×C3×C5C3×SD16semidirect
135. C3×C8C2×C3×D5semidirect
136. C3×C8C2×D15semidirect
143. C2×C32#677semidirect
149. C2×C8C3×D15semidirect
151. C3×C5C2×#204semidirect
152. C3×C5C2×C3×SD16semidirect
168. C2×C5C3×#204semidirect
171. C32C2×#677semidirect
179. C8C2×C3×D15semidirect
183. C2×C3C3×#677semidirect
184. C2×C3#6706direct (N × H)
188. C5C2×C3×#204semidirect
196. C3C2×C3×#677semidirect
197. C3C2×#6706direct (N × H)
199. C2C3×#6706direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D602
C2×C3×Dic302
C2×C32×C5×C82
C3×#67062
C2×#67063
C2×C3×#6773
C2×C3×#2045