Finite Group Databasebeta

← C2×C3×#6654 #13086501

Subgroups — C2×C3×#6654 #13086501

Order 1440.

326 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 104 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, complemented
2C2×C3×C5×D12C211yesmaximal, characteristic, complemented
3C2×C32×#168C211yesmaximal, characteristic, complemented
4C3×#6654C214yesmaximal, direct product factor
5C2×C3×#686—31nomaximal
6C2×#6654C311yesmaximal, characteristic, direct product factor
7C22×C3×D15—21no—
8C22×C3×C5×D3—21no—
9C2×C32×C4×C5C2211yescharacteristic
10C3×D60—21no—
11C3×D60C2212yescomplemented
12C3×C5×D12—21no—
13C3×C5×D12C2212yescomplemented
14C32×#168C2212yescomplemented
15C2×C3×D24—51nomaximal, Hall
16C2×C3×C5×D4—31no—
17C2×D60C2×C311yescharacteristic, complemented
18C2×C5×D12C2×C311yescharacteristic, complemented
19C2×C3×D20—31no—
20C3×#686—34no—
21C2×C3×#168—21no—
22C2×C3×#168D311yescharacteristic, complemented
23C2×C3×#168C2×C311yescharacteristic, complemented
24#6654C2×C314yesdirect product factor
25C22×C32×C5D411yescharacteristic
26C2×C3×D15—22no—
27C2×C3×D15—42no—
28C2×C3×C5×D3—22no—
29C2×C3×C5×D3—42no—
30C32×C4×C5D411yescharacteristic
31C32×C4×C5C2311yescharacteristic
32C2×#686—31noHall
33C2×C3×D12—51no—
34C2×C3×D12D511yescharacteristic, complemented
35C2×C32×C8—51no—
36C3×D24—54no—
37C23×C3×C5—61no—
38C22×D15—21no—
39C22×C5×D3—21no—
40C22×C3×D5—61no—
41C3×C5×D4—32no—
42C3×C5×D4—61no—
43C2×C3×C4×C5—21no—
44C2×C3×C4×C5C2×D311yescharacteristic
45C2×C3×C4×C5C22×C311yescharacteristic
46D60—21no—
47D60C22×C312yescomplemented
48C5×D12—21no—
49C5×D12C22×C312yescomplemented
50C3×D20—32no—
51C3×D20—61no—
52C3×#168—22no—
53C3×#168C2×D312yescomplemented
54C3×#168C22×C312yescomplemented
55C2×C3×D8—151no—
56C2×D24—51no—
57C2×C32×C5D812yescomplemented
58C2×C32×C5C2×D411yescharacteristic
59C3×D15—42no—
60C3×C5×D3—42no—
61C2×C5×D4—31no—
62C2×D20—31no—
63#686—34no—
64C2×#168C3×D311yescharacteristic, complemented
65C22×C3×D3—21no—
66C22×C3×D3—101no—
67C2×C32×C4C2×D511yescharacteristic
68C3×D12—21no—
69C3×D12—52no—
70C3×D12—101no—
71C3×D12C2×D512yescomplemented
72C32×C8—52no—
73C22×C3×C5—21no—
74C22×C3×C5—62no—
75C22×C3×C5—122no—
76C22×C3×C5D1211yescharacteristic
77C22×C3×C5C3×D411yescharacteristic
78C2×D15—22no—
79C2×D15—42no—
80C2×C5×D3—22no—
81C2×C5×D3—42no—
82C2×C3×D5—62no—
83C2×C3×D5—122no—
84C3×C4×C5—21no—
85C3×C4×C5—21no—
86C3×C4×C5D1211yescharacteristic
87C3×C4×C5C3×D411yescharacteristic
88C3×C4×C5C22×D311yescharacteristic
89C3×C4×C5C23×C311yescharacteristic
90C2×C3×D4—31no—
91C2×C3×D4—151no—
92C2×D12—51no—
93C2×D12C3×D511yescharacteristic, complemented
94C3×D8—154no—
95C2×C3×C8—51no—
96C2×C3×C8—51no—
97C2×C3×C8—101no—
98D24—54no—
99C32×C5C2×D811yescharacteristic, complemented, Hall
100C23×C5—61no—
101C22×D5—61no—
102C5×D4—32no—
103C5×D4—61no—
104C2×C4×C5C2×C3×D311yescharacteristic
105D20—32no—
106D20—61no—
107#168C2×C3×D312yescomplemented
108C22×C32C22⋊D511yescharacteristic
109C2×C3×D3—22no—
110C2×C3×D3—42no—
111C2×C3×D3—102no—
112C2×C3×D3—202no—
113C32×C4C22⋊D511yescharacteristic
114C32×C4C22×D511yescharacteristic
115C2×D8—151noSylow
116C2×C3×C5—22no—
117C2×C3×C5—21no—
118C2×C3×C5—122no—
119C2×C3×C5D2412yescomplemented
120C2×C3×C5C3×D812yescomplemented
121C2×C3×C5C2×D1211yescharacteristic
122C2×C3×C5C2×C3×D411yescharacteristic
123D15—42no—
124C3×D5—122no—
125C5×D3—42no—
126C23×C3—61no—
127C23×C3—301no—
128C22×D3—21no—
129C22×D3—101no—
130C3×D4—32no—
131C3×D4—61no—
132C3×D4—152no—
133C3×D4—301no—
134C2×C3×C4—21no—
135C2×C3×C4D3×D511yescharacteristic
136C2×C3×C4C2×C3×D511yescharacteristic
137D12—21no—
138D12—52no—
139D12—101no—
140D12C2×C3×D512yescomplemented
141C3×C8—52no—
142C3×C8—52no—
143C3×C8—102no—
144C22×C5—62no—
145C22×C5—122no—
146C22×C5C3×D1211yescharacteristic
147C2×D5—62no—
148C2×D5—122no—
149C4×C5C3×D1211yescharacteristic
150C4×C5C22×C3×D311yescharacteristic
151C2×C32#68612yescomplemented
152C2×C32C2×C22⋊D511yescharacteristic
153C3×D3—42no—
154C3×D3—202no—
155C2×D4—31no—
156C2×D4—151no—
157D8—154no—
158C2×C8—51no—
159C3×C5—21no—
160C3×C5C2×D2411yescharacteristic, complemented
161C3×C5C2×C3×D811yescharacteristic, complemented
162C22×C3—21no—
163C22×C3—62no—
164C22×C3—122no—
165C22×C3—302no—
166C22×C3—602no—
167C22×C3#120311yescharacteristic
168C22×C3C3×C22⋊D511yescharacteristic
169C2×D3—22no—
170C2×D3—42no—
171C2×D3—102no—
172C2×D3—202no—
173C3×C4—21no—
174C3×C4—21no—
175C3×C4#120311yescharacteristic
176C3×C4C3×C22⋊D511yescharacteristic
177C3×C4C2×D3×D511yescharacteristic
178C3×C4C22×C3×D511yescharacteristic
179C2×C5—122no—
180C2×C5C3×D2412yescomplemented
181C2×C5C2×C3×D1211yescharacteristic
182D5—122no—
183C32C2×#68611yescharacteristic, complemented, Sylow
184C23—61no—
185C23—301no—
186D4—32no—
187D4—61no—
188D4—152no—
189D4—301no—
190C2×C4C3×D3×D511yescharacteristic
191C8—52no—
192C2×C3—21no—
193C2×C3—22no—
194C2×C3—122no—
195C2×C3—602no—
196C2×C3#665412yesdirect product factor
197C2×C3C3×#68612yescomplemented
198C2×C3C2×#120311yescharacteristic
199C2×C3C2×C3×C22⋊D511yescharacteristic
200D3—42no—
201D3—202no—
202C5C2×C3×D2411yescharacteristic, complemented, Sylow
203C22—62no—
204C22—122no—
205C22—302no—
206C22—602no—
207C22C3×#120311yescharacteristic
208C4C3×#120311yescharacteristic
209C4C2×C3×D3×D511yescharacteristic
210C3—21no—
211C3C2×#665411yescharacteristic, direct product factor
212C3C2×C3×#68611yescharacteristic, complemented
213C2—122no—
214C2—602no—
215C2C3×#665412yesdirect product factor
216C2C2×C3×#120311yescharacteristic
subgroupG/H
centerC22×C3#1203
commutator, cosocleC3×C4×C5C23×C3
FrattiniC4C2×C3×D3×D5
FittingC2×C32×C4×C5C22
socleC22×C32×C5D4
hypercenterC2×C3×C4D3×D5
hypocenterC3×C5C2×C3×D8
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×D60C2semidirect
2. C2×C3×C5×D12C2semidirect
3. C2×C32×#168C2semidirect
4. C3×#6654C2direct (N × H)
6. C2×#6654C3direct (N × H)
11. C3×D60C22semidirect
13. C3×C5×D12C22semidirect
14. C32×#168C22semidirect
17. C2×D60C2×C3semidirect
18. C2×C5×D12C2×C3semidirect
22. C2×C3×#168D3semidirect
23. C2×C3×#168C2×C3semidirect
24. #6654C2×C3direct (N × H)
34. C2×C3×D12D5semidirect
47. D60C22×C3semidirect
49. C5×D12C22×C3semidirect
53. C3×#168C2×D3semidirect
54. C3×#168C22×C3semidirect
57. C2×C32×C5D8semidirect
64. C2×#168C3×D3semidirect
71. C3×D12C2×D5semidirect
93. C2×D12C3×D5semidirect
99. C32×C5C2×D8semidirect
107. #168C2×C3×D3semidirect
119. C2×C3×C5D24semidirect
120. C2×C3×C5C3×D8semidirect
140. D12C2×C3×D5semidirect
151. C2×C32#686semidirect
160. C3×C5C2×D24semidirect
161. C3×C5C2×C3×D8semidirect
180. C2×C5C3×D24semidirect
183. C32C2×#686semidirect
196. C2×C3#6654direct (N × H)
197. C2×C3C3×#686semidirect
202. C5C2×C3×D24semidirect
211. C3C2×#6654direct (N × H)
212. C3C2×C3×#686semidirect
215. C2C3×#6654direct (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×C5×D122
C2×C32×#1682
C3×#66542
C2×C3×#6863
C2×#66543
C2×C3×D245