Finite Group Databasebeta

← C2×C3×#6715 #13088033

Subgroups — C2×C3×#6715 #13088033

Order 1440.

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

normal subgroups only

This group has 358 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×C32×C5×D4C211yesmaximal, characteristic, complemented
2C2×C3×D60C211yesmaximal, characteristic, complemented
3C3×#6715C214yesmaximal, direct product factor
4C2×C3×C3⋊#168C211yesmaximal, characteristic, complemented
5C2×#6715C311yesmaximal, characteristic, direct product factor
6C2×C3×#686—31nomaximal
7C23×C32×C5—21no—
8C22×C3×D15—21no—
9C32×C5×D4—21no—
10C32×C5×D4C2212yescomplemented
11C2×C32×C4×C5C2211yescharacteristic
12C3×D60—21no—
13C3×D60C2212yescomplemented
14C3×C3⋊#168C2212yescomplemented
15C2×C3×#213—51nomaximal, Hall
16C2×C3×C5×D4—21no—
17C2×C3×C5×D4D311yescharacteristic, complemented
18C2×C3×C5×D4C2×C311yescharacteristic, complemented
19C2×D60C2×C311yescharacteristic, complemented
20C2×C3×D20—31no—
21#6715C2×C314yesdirect product factor
22C2×C3⋊#168C2×C311yescharacteristic, complemented
23C3×#686—34no—
24C2×C3×#168—31no—
25C22×C32×C5—22no—
26C22×C32×C5—42no—
27C22×C32×C5D411yescharacteristic
28C2×C3×D15—22no—
29C2×C3×D15—42no—
30C32×C4×C5D411yescharacteristic
31C32×C4×C5C2311yescharacteristic
32C2×#686—31noHall
33C2×C32×D4D511yescharacteristic, complemented
34C2×C3×D12—51no—
35C3×#213—54no—
36C2×C3×C3⋊C8—51no—
37C23×C3×C5—22no—
38C23×C3×C5—21no—
39C23×C3×C5—21no—
40C22×D15—21no—
41C22×C3×D5—61no—
42C3×C5×D4—22no—
43C3×C5×D4—22no—
44C3×C5×D4—21no—
45C3×C5×D4—21no—
46C3×C5×D4C2×D312yescomplemented
47C3×C5×D4C22×C312yescomplemented
48C2×C3×C4×C5—21no—
49C2×C3×C4×C5C2×D311yescharacteristic
50C2×C3×C4×C5C22×C311yescharacteristic
51D60—21no—
52D60C22×C312yescomplemented
53C3×D20—32no—
54C3×D20—61no—
55C3⋊#168C22×C312yescomplemented
56C3×#168—32no—
57C2×C3×D8—151no—
58C2×#213—51no—
59C2×C32×C5—42no—
60C2×C32×C5D812yescomplemented
61C2×C32×C5C2×D411yescharacteristic
62C3×D15—42no—
63C2×C5×D4C3×D311yescharacteristic, complemented
64C2×D20—31no—
65#686—34no—
66C2×#168—31no—
67C23×C32—21no—
68C22×C3×D3—101no—
69C32×D4—21no—
70C32×D4C2×D512yescomplemented
71C2×C32×C4C2×D511yescharacteristic
72C3×D12—52no—
73C3×D12—101no—
74C3×C3⋊C8—52no—
75C22×C3×C5—24no—
76C22×C3×C5—22no—
77C22×C3×C5—22no—
78C22×C3×C5—21no—
79C22×C3×C5—44no—
80C22×C3×C5—42no—
81C22×C3×C5—42no—
82C22×C3×C5#6711yescharacteristic
83C22×C3×C5C3×D411yescharacteristic
84C2×D15—22no—
85C2×D15—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—21no—
95C2×C3×D4—151no—
96C2×C3×D4C3×D511yescharacteristic, complemented
97C2×C3×D4D1511yescharacteristic, complemented
98C2×D12—51no—
99C3×D8—154no—
100C2×C3×C8—151no—
101#213—54no—
102C2×C3⋊C8—51no—
103C32×C5C2×D811yescharacteristic, complemented, Hall
104C23×C5—21no—
105C22×D5—61no—
106C5×D4—21no—
107C5×D4C2×C3×D312yescomplemented
108C2×C4×C5C2×C3×D311yescharacteristic
109D20—32no—
110D20—61no—
111#168—32no—
112C22×C32—22no—
113C22×C32—42no—
114C22×C32C22⋊D511yescharacteristic
115C2×C3×D3—102no—
116C2×C3×D3—202no—
117C32×C4C22⋊D511yescharacteristic
118C32×C4C22×D511yescharacteristic
119C2×D8—151noSylow
120C2×C3×C5—22no—
121C2×C3×C5—21no—
122C2×C3×C5—44no—
123C2×C3×C5—42no—
124C2×C3×C5—42no—
125C2×C3×C5#21312yescomplemented
126C2×C3×C5C3×D812yescomplemented
127C2×C3×C5C2×#6711yescharacteristic
128C2×C3×C5C2×C3×D411yescharacteristic
129D15—42no—
130C3×D5—122no—
131C23×C3—22no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—301no—
135C22×D3—101no—
136C3×D4—22no—
137C3×D4—21no—
138C3×D4—21no—
139C3×D4—22no—
140C3×D4—152no—
141C3×D4—301no—
142C3×D4C2×C3×D512yescomplemented
143C3×D4C2×D1512yescomplemented
144C2×C3×C4—21no—
145C2×C3×C4C2×C3×D511yescharacteristic
146C2×C3×C4C2×D1511yescharacteristic
147D12—52no—
148D12—101no—
149C3×C8—152no—
150C3⋊C8—52no—
151C22×C5—22no—
152C22×C5—42no—
153C22×C5C3×#6711yescharacteristic
154C2×D5—62no—
155C2×D5—122no—
156C4×C5C3×#6711yescharacteristic
157C4×C5C22×C3×D311yescharacteristic
158C2×C32—42no—
159C2×C32#68612yescomplemented
160C2×C32C2×C22⋊D511yescharacteristic
161C3×D3—202no—
162C2×D4—151no—
163C2×D4C3×D1511yescharacteristic, complemented
164D8—154no—
165C2×C8—151no—
166C3×C5—21no—
167C3×C5C2×#21311yescharacteristic, complemented
168C3×C5C2×C3×D811yescharacteristic, complemented
169C22×C3—24no—
170C22×C3—22no—
171C22×C3—22no—
172C22×C3—21no—
173C22×C3—44no—
174C22×C3—42no—
175C22×C3—42no—
176C22×C3—302no—
177C22×C3—602no—
178C22×C3C3×C22⋊D511yescharacteristic
179C22×C3#122011yescharacteristic
180C2×D3—102no—
181C2×D3—202no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4C3×C22⋊D511yescharacteristic
185C3×C4#122011yescharacteristic
186C3×C4C22×C3×D511yescharacteristic
187C3×C4C22×D1511yescharacteristic
188C2×C5—42no—
189C2×C5C3×#21312yescomplemented
190C2×C5C2×C3×#6711yescharacteristic
191D5—122no—
192C32C2×#68611yescharacteristic, complemented, Sylow
193C23—21no—
194C23—301no—
195D4—21no—
196D4—152no—
197D4—301no—
198D4C2×C3×D1512yescomplemented
199C2×C4C2×C3×D1511yescharacteristic
200C8—152no—
201C2×C3—21no—
202C2×C3—22no—
203C2×C3—44no—
204C2×C3—42no—
205C2×C3—42no—
206C2×C3—602no—
207C2×C3C3×#68612yescomplemented
208C2×C3#671512yesdirect product factor
209C2×C3C2×C3×C22⋊D511yescharacteristic
210C2×C3C2×#122011yescharacteristic
211D3—202no—
212C5C2×C3×#21311yescharacteristic, complemented, Sylow
213C22—22no—
214C22—42no—
215C22—302no—
216C22—602no—
217C22C3×#122011yescharacteristic
218C4C3×#122011yescharacteristic
219C4C22×C3×D1511yescharacteristic
220C3—21no—
221C3C2×C3×#68611yescharacteristic, complemented
222C3C2×#671511yescharacteristic, direct product factor
223C2—42no—
224C2—602no—
225C2C3×#671512yesdirect product factor
226C2C2×C3×#122011yescharacteristic
subgroupG/H
centerC22×C3#1220
commutator, cosocleC3×C4×C5C23×C3
FrattiniC4C22×C3×D15
FittingC2×C32×C5×D4C2
socleC22×C32×C5D4
hypercenterC2×C3×D4D15
hypocenterC3×C5C2×C3×D8
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×D4

Complements

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

NH ≅ G/Ntype
1. C2×C32×C5×D4C2semidirect
2. C2×C3×D60C2semidirect
3. C3×#6715C2direct (N × H)
4. C2×C3×C3⋊#168C2semidirect
5. C2×#6715C3direct (N × H)
10. C32×C5×D4C22semidirect
13. C3×D60C22semidirect
14. C3×C3⋊#168C22semidirect
17. C2×C3×C5×D4D3semidirect
18. C2×C3×C5×D4C2×C3semidirect
19. C2×D60C2×C3semidirect
21. #6715C2×C3direct (N × H)
22. C2×C3⋊#168C2×C3semidirect
33. C2×C32×D4D5semidirect
46. C3×C5×D4C2×D3semidirect
47. C3×C5×D4C22×C3semidirect
52. D60C22×C3semidirect
55. C3⋊#168C22×C3semidirect
60. C2×C32×C5D8semidirect
63. C2×C5×D4C3×D3semidirect
70. C32×D4C2×D5semidirect
96. C2×C3×D4C3×D5semidirect
97. C2×C3×D4D15semidirect
103. C32×C5C2×D8semidirect
107. C5×D4C2×C3×D3semidirect
125. C2×C3×C5#213semidirect
126. C2×C3×C5C3×D8semidirect
142. C3×D4C2×C3×D5semidirect
143. C3×D4C2×D15semidirect
159. C2×C32#686semidirect
163. C2×D4C3×D15semidirect
167. C3×C5C2×#213semidirect
168. C3×C5C2×C3×D8semidirect
189. C2×C5C3×#213semidirect
192. C32C2×#686semidirect
198. D4C2×C3×D15semidirect
207. C2×C3C3×#686semidirect
208. C2×C3#6715direct (N × H)
212. C5C2×C3×#213semidirect
221. C3C2×C3×#686semidirect
222. C3C2×#6715direct (N × H)
225. C2C3×#6715direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C32×C5×D42
C2×C3×D602
C3×#67152
C2×C3×C3⋊#1682
C2×#67153
C2×C3×#6863
C2×C3×#2135