Finite Group Databasebeta

← C2×C3×#6652 #13086495

Subgroups — C2×C3×#6652 #13086495

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×C5×D12C211yesmaximal, characteristic, complemented
2C2×C32×D20C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊#168C211yesmaximal, characteristic, complemented
4C3×#6652C214yesmaximal, direct product factor
5C2×C3×#686—31nomaximal
6C2×#6652C311yesmaximal, characteristic, direct product factor
7C22×C3×C5×D3—21no—
8C22×C32×D5—21no—
9C2×C32×C4×C5C2211yescharacteristic
10C3×C5×D12—21no—
11C3×C5×D12C2212yescomplemented
12C32×D20—21no—
13C32×D20C2212yescomplemented
14C3×C3⋊#168C2212yescomplemented
15C2×C3×#213—51nomaximal, Hall
16C2×C3×C5×D4—31no—
17C2×C5×D12C2×C311yescharacteristic, complemented
18C2×C3×D20—21no—
19C2×C3×D20D311yescharacteristic, complemented
20C2×C3×D20C2×C311yescharacteristic, complemented
21C2×C3⋊#168C2×C311yescharacteristic, complemented
22C3×#686—34no—
23C2×C3×#168—31no—
24#6652C2×C314yesdirect product factor
25C22×C32×C5D411yescharacteristic
26C2×C3×C5×D3—22no—
27C2×C3×C5×D3—42no—
28C2×C32×D5—22no—
29C2×C32×D5—42no—
30C32×C4×C5D411yescharacteristic
31C32×C4×C5C2311yescharacteristic
32C2×#686—31noHall
33C2×C32×D4—51no—
34C2×C3×D12D511yescharacteristic, complemented
35C3×#213—54no—
36C2×C3×C3⋊C8—51no—
37C23×C3×C5—61no—
38C22×C5×D3—21no—
39C22×C3×D5—22no—
40C22×C3×D5—21no—
41C22×C3×D5—21no—
42C3×C5×D4—32no—
43C3×C5×D4—61no—
44C2×C3×C4×C5—21no—
45C2×C3×C4×C5C2×D311yescharacteristic
46C2×C3×C4×C5C22×C311yescharacteristic
47C5×D12—21no—
48C5×D12C22×C312yescomplemented
49C3×D20—22no—
50C3×D20—22no—
51C3×D20—21no—
52C3×D20—21no—
53C3×D20C2×D312yescomplemented
54C3×D20C22×C312yescomplemented
55C3⋊#168C22×C312yescomplemented
56C3×#168—32no—
57C2×C3×D8—151no—
58C2×#213—51no—
59C2×C32×C5D812yescomplemented
60C2×C32×C5C2×D411yescharacteristic
61C3×C5×D3—42no—
62C32×D5—42no—
63C2×C5×D4—31no—
64C2×D20C3×D311yescharacteristic, complemented
65#686—34no—
66C2×#168—31no—
67C23×C32—101no—
68C22×C3×D3—21no—
69C32×D4—52no—
70C32×D4—101no—
71C2×C32×C4C2×D511yescharacteristic
72C3×D12—21no—
73C3×D12C2×D512yescomplemented
74C3×C3⋊C8—52no—
75C22×C3×C5—21no—
76C22×C3×C5—62no—
77C22×C3×C5—122no—
78C22×C3×C5#6711yescharacteristic
79C22×C3×C5C3×D411yescharacteristic
80C2×C5×D3—22no—
81C2×C5×D3—42no—
82C2×C3×D5—24no—
83C2×C3×D5—22no—
84C2×C3×D5—22no—
85C2×C3×D5—44no—
86C2×C3×D5—42no—
87C2×C3×D5—42no—
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—31no—
95C2×C3×D4—51no—
96C2×C3×D4—51no—
97C2×C3×D4—101no—
98C2×D12C3×D511yescharacteristic, complemented
99C3×D8—154no—
100C2×C3×C8—151no—
101#213—54no—
102C2×C3⋊C8—51no—
103C32×C5C2×D811yescharacteristic, complemented, Hall
104C23×C5—61no—
105C22×D5—21no—
106C5×D4—32no—
107C5×D4—61no—
108C2×C4×C5C2×C3×D311yescharacteristic
109D20—21no—
110D20C2×C3×D312yescomplemented
111#168—32no—
112C22×C32—102no—
113C22×C32—202no—
114C22×C32C22⋊D511yescharacteristic
115C2×C3×D3—22no—
116C2×C3×D3—42no—
117C32×C4C22⋊D511yescharacteristic
118C32×C4C22×D511yescharacteristic
119C2×D8—151noSylow
120C2×C3×C5—22no—
121C2×C3×C5—21no—
122C2×C3×C5—122no—
123C2×C3×C5#21312yescomplemented
124C2×C3×C5C3×D812yescomplemented
125C2×C3×C5C2×#6711yescharacteristic
126C2×C3×C5C2×C3×D411yescharacteristic
127C3×D5—44no—
128C3×D5—42no—
129C3×D5—42no—
130C5×D3—42no—
131C23×C3—61no—
132C23×C3—102no—
133C23×C3—101no—
134C23×C3—101no—
135C22×D3—21no—
136C3×D4—32no—
137C3×D4—52no—
138C3×D4—52no—
139C3×D4—61no—
140C3×D4—102no—
141C3×D4—102no—
142C3×D4—101no—
143C3×D4—101no—
144C2×C3×C4—21no—
145C2×C3×C4D3×D511yescharacteristic
146C2×C3×C4C2×C3×D511yescharacteristic
147D12—21no—
148D12C2×C3×D512yescomplemented
149C3×C8—152no—
150C3⋊C8—52no—
151C22×C5—62no—
152C22×C5—122no—
153C22×C5C3×#6711yescharacteristic
154C2×D5—22no—
155C2×D5—42no—
156C4×C5C3×#6711yescharacteristic
157C4×C5C22×C3×D311yescharacteristic
158C2×C32—202no—
159C2×C32#68612yescomplemented
160C2×C32C2×C22⋊D511yescharacteristic
161C3×D3—42no—
162C2×D4—31no—
163C2×D4—51no—
164D8—154no—
165C2×C8—151no—
166C3×C5—21no—
167C3×C5C2×#21311yescharacteristic, complemented
168C3×C5C2×C3×D811yescharacteristic, complemented
169C22×C3—21no—
170C22×C3—62no—
171C22×C3—104no—
172C22×C3—102no—
173C22×C3—102no—
174C22×C3—122no—
175C22×C3—204no—
176C22×C3—202no—
177C22×C3—202no—
178C22×C3#120111yescharacteristic
179C22×C3C3×C22⋊D511yescharacteristic
180C2×D3—22no—
181C2×D3—42no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4#120111yescharacteristic
185C3×C4C3×C22⋊D511yescharacteristic
186C3×C4C2×D3×D511yescharacteristic
187C3×C4C22×C3×D511yescharacteristic
188C2×C5—122no—
189C2×C5C3×#21312yescomplemented
190C2×C5C2×C3×#6711yescharacteristic
191D5—42no—
192C32C2×#68611yescharacteristic, complemented, Sylow
193C23—61no—
194C23—101no—
195D4—32no—
196D4—52no—
197D4—61no—
198D4—101no—
199C2×C4C3×D3×D511yescharacteristic
200C8—152no—
201C2×C3—21no—
202C2×C3—22no—
203C2×C3—122no—
204C2×C3—204no—
205C2×C3—202no—
206C2×C3—202no—
207C2×C3#665212yesdirect product factor
208C2×C3C3×#68612yescomplemented
209C2×C3C2×#120111yescharacteristic
210C2×C3C2×C3×C22⋊D511yescharacteristic
211D3—42no—
212C5C2×C3×#21311yescharacteristic, complemented, Sylow
213C22—62no—
214C22—102no—
215C22—122no—
216C22—202no—
217C22C3×#120111yescharacteristic
218C4C3×#120111yescharacteristic
219C4C2×C3×D3×D511yescharacteristic
220C3—21no—
221C3C2×#665211yescharacteristic, direct product factor
222C3C2×C3×#68611yescharacteristic, complemented
223C2—122no—
224C2—202no—
225C2C3×#665212yesdirect product factor
226C2C2×C3×#120111yescharacteristic
subgroupG/H
centerC22×C3#1201
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×C5×D12C2semidirect
2. C2×C32×D20C2semidirect
3. C2×C3×C3⋊#168C2semidirect
4. C3×#6652C2direct (N × H)
6. C2×#6652C3direct (N × H)
11. C3×C5×D12C22semidirect
13. C32×D20C22semidirect
14. C3×C3⋊#168C22semidirect
17. C2×C5×D12C2×C3semidirect
19. C2×C3×D20D3semidirect
20. C2×C3×D20C2×C3semidirect
21. C2×C3⋊#168C2×C3semidirect
24. #6652C2×C3direct (N × H)
34. C2×C3×D12D5semidirect
48. C5×D12C22×C3semidirect
53. C3×D20C2×D3semidirect
54. C3×D20C22×C3semidirect
55. C3⋊#168C22×C3semidirect
59. C2×C32×C5D8semidirect
64. C2×D20C3×D3semidirect
73. C3×D12C2×D5semidirect
98. C2×D12C3×D5semidirect
103. C32×C5C2×D8semidirect
110. D20C2×C3×D3semidirect
123. C2×C3×C5#213semidirect
124. C2×C3×C5C3×D8semidirect
148. D12C2×C3×D5semidirect
159. C2×C32#686semidirect
167. C3×C5C2×#213semidirect
168. C3×C5C2×C3×D8semidirect
189. C2×C5C3×#213semidirect
192. C32C2×#686semidirect
207. C2×C3#6652direct (N × H)
208. C2×C3C3×#686semidirect
212. C5C2×C3×#213semidirect
221. C3C2×#6652direct (N × H)
222. C3C2×C3×#686semidirect
225. C2C3×#6652direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×D122
C2×C32×D202
C2×C3×C3⋊#1682
C3×#66522
C2×C3×#6863
C2×#66523
C2×C3×#2135