Finite Group Databasebeta

← C2×C3×#6653 #13086499

Subgroups — C2×C3×#6653 #13086499

Order 1440.

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

normal subgroups only

This group has 358 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×C32×D20C211yesmaximal, characteristic, complemented
3C2×C3×C5×C3⋊C8C211yesmaximal, characteristic, complemented
4C3×#6653C214yesmaximal, direct product factor
5C2×C3×D40—31nomaximal
6C2×#6653C311yesmaximal, characteristic, direct product factor
7C22×C3×D15—21no—
8C22×C32×D5—21no—
9C2×C32×C4×C5C2211yescharacteristic
10C3×D60—21no—
11C3×D60C2212yescomplemented
12C32×D20—21no—
13C32×D20C2212yescomplemented
14C3×C5×C3⋊C8C2212yescomplemented
15C2×C3×#213—51nomaximal, Hall
16C2×D60C2×C311yescharacteristic, complemented
17C2×C3×D20—21no—
18C2×C3×D20—31no—
19C2×C3×D20D311yescharacteristic, complemented
20C2×C3×D20C2×C311yescharacteristic, complemented
21C2×C3×C5×C8—31no—
22C2×C5×C3⋊C8C2×C311yescharacteristic, complemented
23C3×D40—34no—
24#6653C2×C314yesdirect product factor
25C22×C32×C5D411yescharacteristic
26C2×C3×D15—22no—
27C2×C3×D15—42no—
28C2×C32×D5—22no—
29C2×C32×D5—42no—
30C32×C4×C5D411yescharacteristic
31C32×C4×C5C2311yescharacteristic
32C2×D40—31noHall
33C2×C32×D4—51no—
34C2×C3×D12—51no—
35C3×#213—54no—
36C2×C3×C3⋊C8D511yescharacteristic, complemented
37C22×D15—21no—
38C22×C3×D5—22no—
39C22×C3×D5—21no—
40C22×C3×D5—21no—
41C22×C3×D5—61no—
42C2×C3×C4×C5—21no—
43C2×C3×C4×C5C2×D311yescharacteristic
44C2×C3×C4×C5C22×C311yescharacteristic
45D60—21no—
46D60C22×C312yescomplemented
47C3×D20—22no—
48C3×D20—22no—
49C3×D20—21no—
50C3×D20—21no—
51C3×D20—32no—
52C3×D20—61no—
53C3×D20C2×D312yescomplemented
54C3×D20C22×C312yescomplemented
55C3×C5×C8—32no—
56C5×C3⋊C8C22×C312yescomplemented
57C2×C3×D8—151no—
58C2×#213—51no—
59C2×C32×C5D812yescomplemented
60C2×C32×C5C2×D411yescharacteristic
61C3×D15—42no—
62C32×D5—42no—
63C2×D20—31no—
64C2×D20C3×D311yescharacteristic, complemented
65C2×C5×C8—31no—
66D40—34no—
67C23×C32—101no—
68C22×C3×D3—101no—
69C32×D4—52no—
70C32×D4—101no—
71C2×C32×C4C2×D511yescharacteristic
72C3×D12—52no—
73C3×D12—101no—
74C3×C3⋊C8C2×D512yescomplemented
75C22×C3×C5—21no—
76C22×C3×C5#6711yescharacteristic
77C22×C3×C5C3×D411yescharacteristic
78C2×D15—22no—
79C2×D15—42no—
80C2×C3×D5—24no—
81C2×C3×D5—22no—
82C2×C3×D5—22no—
83C2×C3×D5—44no—
84C2×C3×D5—42no—
85C2×C3×D5—42no—
86C2×C3×D5—62no—
87C2×C3×D5—122no—
88C3×C4×C5—21no—
89C3×C4×C5—21no—
90C3×C4×C5#6711yescharacteristic
91C3×C4×C5C3×D411yescharacteristic
92C3×C4×C5C22×D311yescharacteristic
93C3×C4×C5C23×C311yescharacteristic
94C2×C3×D4—51no—
95C2×C3×D4—51no—
96C2×C3×D4—101no—
97C2×C3×D4—151no—
98C2×D12—51no—
99C3×D8—154no—
100C2×C3×C8—31no—
101#213—54no—
102C2×C3⋊C8C3×D511yescharacteristic, complemented
103C32×C5C2×D811yescharacteristic, complemented, Hall
104C22×D5—21no—
105C22×D5—61no—
106C2×C4×C5C2×C3×D311yescharacteristic
107D20—21no—
108D20—32no—
109D20—61no—
110D20C2×C3×D312yescomplemented
111C5×C8—32no—
112C22×C32—102no—
113C22×C32—202no—
114C22×C32D2011yescharacteristic
115C2×C3×D3—102no—
116C2×C3×D3—202no—
117C32×C4D2011yescharacteristic
118C32×C4C22×D511yescharacteristic
119C2×D8—151noSylow
120C2×C3×C5—22no—
121C2×C3×C5—21no—
122C2×C3×C5#21312yescomplemented
123C2×C3×C5C3×D812yescomplemented
124C2×C3×C5C2×#6711yescharacteristic
125C2×C3×C5C2×C3×D411yescharacteristic
126D15—42no—
127C3×D5—44no—
128C3×D5—42no—
129C3×D5—42no—
130C3×D5—122no—
131C23×C3—102no—
132C23×C3—101no—
133C23×C3—101no—
134C23×C3—301no—
135C22×D3—101no—
136C3×D4—52no—
137C3×D4—52no—
138C3×D4—102no—
139C3×D4—102no—
140C3×D4—101no—
141C3×D4—101no—
142C3×D4—152no—
143C3×D4—301no—
144C2×C3×C4—21no—
145C2×C3×C4D3×D511yescharacteristic
146C2×C3×C4C2×C3×D511yescharacteristic
147D12—52no—
148D12—101no—
149C3×C8—32no—
150C3⋊C8C2×C3×D512yescomplemented
151C22×C5C3×#6711yescharacteristic
152C2×D5—22no—
153C2×D5—42no—
154C2×D5—62no—
155C2×D5—122no—
156C4×C5C3×#6711yescharacteristic
157C4×C5C22×C3×D311yescharacteristic
158C2×C32—202no—
159C2×C32D4012yescomplemented
160C2×C32C2×D2011yescharacteristic
161C3×D3—202no—
162C2×D4—51no—
163C2×D4—151no—
164D8—154no—
165C2×C8—31no—
166C3×C5—21no—
167C3×C5C2×#21311yescharacteristic, complemented
168C3×C5C2×C3×D811yescharacteristic, complemented
169C22×C3—21no—
170C22×C3—104no—
171C22×C3—102no—
172C22×C3—102no—
173C22×C3—204no—
174C22×C3—202no—
175C22×C3—202no—
176C22×C3—302no—
177C22×C3—602no—
178C22×C3#120211yescharacteristic
179C22×C3C3×D2011yescharacteristic
180C2×D3—102no—
181C2×D3—202no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4#120211yescharacteristic
185C3×C4C3×D2011yescharacteristic
186C3×C4C2×D3×D511yescharacteristic
187C3×C4C22×C3×D511yescharacteristic
188C2×C5C3×#21312yescomplemented
189C2×C5C2×C3×#6711yescharacteristic
190D5—42no—
191D5—122no—
192C32C2×D4011yescharacteristic, complemented, Sylow
193C23—101no—
194C23—301no—
195D4—52no—
196D4—101no—
197D4—152no—
198D4—301no—
199C2×C4C3×D3×D511yescharacteristic
200C8—32no—
201C2×C3—21no—
202C2×C3—22no—
203C2×C3—204no—
204C2×C3—202no—
205C2×C3—202no—
206C2×C3—602no—
207C2×C3#665312yesdirect product factor
208C2×C3C3×D4012yescomplemented
209C2×C3C2×#120211yescharacteristic
210C2×C3C2×C3×D2011yescharacteristic
211D3—202no—
212C5C2×C3×#21311yescharacteristic, complemented, Sylow
213C22—102no—
214C22—202no—
215C22—302no—
216C22—602no—
217C22C3×#120211yescharacteristic
218C4C3×#120211yescharacteristic
219C4C2×C3×D3×D511yescharacteristic
220C3—21no—
221C3C2×#665311yescharacteristic, direct product factor
222C3C2×C3×D4011yescharacteristic, complemented
223C2—202no—
224C2—602no—
225C2C3×#665312yesdirect product factor
226C2C2×C3×#120211yescharacteristic
subgroupG/H
centerC22×C3#1202
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×C32×D20C2semidirect
3. C2×C3×C5×C3⋊C8C2semidirect
4. C3×#6653C2direct (N × H)
6. C2×#6653C3direct (N × H)
11. C3×D60C22semidirect
13. C32×D20C22semidirect
14. C3×C5×C3⋊C8C22semidirect
16. C2×D60C2×C3semidirect
19. C2×C3×D20D3semidirect
20. C2×C3×D20C2×C3semidirect
22. C2×C5×C3⋊C8C2×C3semidirect
24. #6653C2×C3direct (N × H)
36. C2×C3×C3⋊C8D5semidirect
46. D60C22×C3semidirect
53. C3×D20C2×D3semidirect
54. C3×D20C22×C3semidirect
56. C5×C3⋊C8C22×C3semidirect
59. C2×C32×C5D8semidirect
64. C2×D20C3×D3semidirect
74. C3×C3⋊C8C2×D5semidirect
102. C2×C3⋊C8C3×D5semidirect
103. C32×C5C2×D8semidirect
110. D20C2×C3×D3semidirect
122. C2×C3×C5#213semidirect
123. C2×C3×C5C3×D8semidirect
150. C3⋊C8C2×C3×D5semidirect
159. C2×C32D40semidirect
167. C3×C5C2×#213semidirect
168. C3×C5C2×C3×D8semidirect
188. C2×C5C3×#213semidirect
192. C32C2×D40semidirect
207. C2×C3#6653direct (N × H)
208. C2×C3C3×D40semidirect
212. C5C2×C3×#213semidirect
221. C3C2×#6653direct (N × H)
222. C3C2×C3×D40semidirect
225. C2C3×#6653direct (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×C32×D202
C2×C3×C5×C3⋊C82
C3×#66532
C2×C3×D403
C2×#66533
C2×C3×#2135