Finite Group Databasebeta

← C2×C3×#6705 #13087997

Subgroups — C2×C3×#6705 #13087997

Order 1440.

294 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 126 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×C4×D15C211yesmaximal, characteristic
2C2×C32×C5×C8C211yesmaximal, characteristic, complemented
3C2×C3×C3⋊#168C211yesmaximal, characteristic, complemented
4C3×#6705C214yesmaximal, direct product factor
5C2×#6705C311yesmaximal, characteristic, direct product factor
6C2×C3×#676—31nomaximal
7C22×C3×D15C411yescharacteristic
8C2×C32×C4×C5C2211yescharacteristic
9C2×C3×Dic15C411yescharacteristic
10C3×C4×D15C412yes—
11C3×C4×D15C2212yes—
12C32×C5×C8C2212yescomplemented
13C3×C3⋊#168C2212yescomplemented
14C2×C3×ΓL(1,𝔽25)—51nomaximal, Hall
15C2×C4×D15C2×C311yescharacteristic
16C2×C3×C4×D5—31no—
17C2×C3×C5×C8—21no—
18C2×C3×C5×C8D311yescharacteristic, complemented
19C2×C3×C5×C8C2×C311yescharacteristic, complemented
20C2×C3⋊#168C2×C311yescharacteristic, complemented
21#6705C2×C314yesdirect product factor
22C2×C3×#168—31no—
23C3×#676—34no—
24C22×C32×C5C2×C411yescharacteristic
25C2×C3×D15—22no—
26C2×C3×D15C2×C412yes—
27C32×C4×C5C2×C411yescharacteristic
28C32×C4×C5C2311yescharacteristic
29C3×Dic15C2×C412yes—
30C2×#676—31noHall
31C2×C3×C4×D3—51no—
32C2×C32×C8D511yescharacteristic, complemented
33C2×C3×C3⋊C8—51no—
34C3×ΓL(1,𝔽25)—54no—
35C22×D15C3×C411yescharacteristic
36C22×C3×D5—31no—
37C2×C3×C4×C5—21no—
38C2×C3×C4×C5C2×D311yescharacteristic
39C2×C3×C4×C5C22×C311yescharacteristic
40C2×Dic15C3×C411yescharacteristic
41C4×D15C3×C412yes—
42C4×D15C22×C312yes—
43C2×C3×Dic5—31no—
44C3×C4×D5—32no—
45C3×C4×D5—32no—
46C3×C5×C8—22no—
47C3×C5×C8C2×D312yescomplemented
48C3×C5×C8C22×C312yescomplemented
49C3⋊#168C22×C312yescomplemented
50C3×#168—32no—
51C2×C3×M16—151no—
52C2×ΓL(1,𝔽25)—51no—
53C2×C32×C5M1612yescomplemented
54C2×C32×C5C22×C411yescharacteristic
55C3×D15—22no—
56C2×C4×D5—31no—
57C2×C5×C8C3×D311yescharacteristic, complemented
58C2×#168—31no—
59#676—34no—
60C22×C3×D3—51no—
61C2×C32×C4C2×D511yescharacteristic
62C2×C3×Dic3—51no—
63C3×C4×D3—52no—
64C3×C4×D3—52no—
65C32×C8C2×D512yescomplemented
66C3×C3⋊C8—52no—
67C22×C3×C5—21no—
68C22×C3×C5C4×D311yescharacteristic
69C22×C3×C5C2×C3×C411yescharacteristic
70C2×D15—22no—
71C2×D15C2×C3×C412yes—
72C2×C3×D5—32no—
73C2×C3×D5—62no—
74C3×C4×C5—21no—
75C3×C4×C5—21no—
76C3×C4×C5C4×D311yescharacteristic
77C3×C4×C5C2×C3×C411yescharacteristic
78C3×C4×C5C22×D311yescharacteristic
79C3×C4×C5C23×C311yescharacteristic
80Dic15C2×C3×C412yes—
81C3×Dic5—32no—
82C22×C3×C4—151no—
83C2×C4×D3—51no—
84C3×M16—154no—
85C2×C3×C8—21no—
86C2×C3×C8—151no—
87C2×C3×C8C3×D511yescharacteristic, complemented
88C2×C3×C8D1511yescharacteristic, complemented
89C2×C3⋊C8—51no—
90ΓL(1,𝔽25)—54no—
91C32×C5C2×M1611yescharacteristic, complemented, Hall
92C22×D5—31no—
93C2×C4×C5C2×C3×D311yescharacteristic
94C2×Dic5—31no—
95C4×D5—32no—
96C4×D5—32no—
97C5×C8C2×C3×D312yescomplemented
98#168—32no—
99C22×C32C4×D511yescharacteristic
100C2×C3×D3—52no—
101C2×C3×D3—102no—
102C32×C4C4×D511yescharacteristic
103C32×C4C22×D511yescharacteristic
104C3×Dic3—52no—
105C2×M16—151noSylow
106C2×C3×C5—22no—
107C2×C3×C5—21no—
108C2×C3×C5ΓL(1,𝔽25)12yescomplemented
109C2×C3×C5C3×M1612yescomplemented
110C2×C3×C5C2×C4×D311yescharacteristic
111C2×C3×C5C22×C3×C411yescharacteristic
112D15—22no—
113C3×D5—62no—
114C23×C3—151no—
115C22×D3—51no—
116C2×C3×C4—21no—
117C2×C3×C4—151no—
118C2×C3×C4—152no—
119C2×C3×C4—152no—
120C2×C3×C4C2×C3×D511yescharacteristic
121C2×C3×C4C2×D1511yescharacteristic
122C2×Dic3—51no—
123C4×D3—52no—
124C4×D3—52no—
125C3×C8—22no—
126C3×C8—152no—
127C3×C8C2×C3×D512yescomplemented
128C3×C8C2×D1512yescomplemented
129C3⋊C8—52no—
130C22×C5C3×C4×D311yescharacteristic
131C2×D5—32no—
132C2×D5—62no—
133C4×C5C3×C4×D311yescharacteristic
134C4×C5C22×C3×D311yescharacteristic
135Dic5—32no—
136C2×C32#67612yescomplemented
137C2×C32C2×C4×D511yescharacteristic
138C3×D3—102no—
139C22×C4—151no—
140M16—154no—
141C2×C8—151no—
142C2×C8C3×D1511yescharacteristic, complemented
143C3×C5—21no—
144C3×C5C2×ΓL(1,𝔽25)11yescharacteristic, complemented
145C3×C5C2×C3×M1611yescharacteristic, complemented
146C22×C3—21no—
147C22×C3—152no—
148C22×C3—302no—
149C22×C3C3×C4×D511yescharacteristic
150C22×C3C4×D1511yescharacteristic
151C2×D3—52no—
152C2×D3—102no—
153C3×C4—21no—
154C3×C4—21no—
155C3×C4—152no—
156C3×C4C3×C4×D511yescharacteristic
157C3×C4C4×D1511yescharacteristic
158C3×C4C22×C3×D511yescharacteristic
159C3×C4C22×D1511yescharacteristic
160Dic3—52no—
161C2×C5C3×ΓL(1,𝔽25)12yescomplemented
162C2×C5C2×C3×C4×D311yescharacteristic
163D5—62no—
164C32C2×#67611yescharacteristic, complemented, Sylow
165C23—151no—
166C2×C4—151no—
167C2×C4—152no—
168C2×C4—152no—
169C2×C4C2×C3×D1511yescharacteristic
170C8—152no—
171C8C2×C3×D1512yescomplemented
172C2×C3—21no—
173C2×C3—22no—
174C2×C3—302no—
175C2×C3C3×#67612yescomplemented
176C2×C3#670512yesdirect product factor
177C2×C3C2×C3×C4×D511yescharacteristic
178C2×C3C2×C4×D1511yescharacteristic
179D3—102no—
180C5C2×C3×ΓL(1,𝔽25)11yescharacteristic, complemented, Sylow
181C22—152no—
182C22—302no—
183C22C3×C4×D1511yescharacteristic
184C4—152no—
185C4C3×C4×D1511yescharacteristic
186C4C22×C3×D1511yescharacteristic
187C3—21no—
188C3C2×C3×#67611yescharacteristic, complemented
189C3C2×#670511yescharacteristic, direct product factor
190C2—302no—
191C2C3×#670512yesdirect product factor
192C2C2×C3×C4×D1511yescharacteristic
subgroupG/H
centerC2×C3×C4C2×D15
commutatorC2×C3×C5C22×C3×C4
FrattiniC4C22×C3×D15
FittingC2×C32×C5×C8C2
socleC22×C32×C5C2×C4
prime radicalC22×C3×D15C4
cosocleC3×C4×C5C23×C3
hypercenterC2×C3×C8D15
hypocenterC3×C5C2×C3×M16
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C8

Complements

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

NH ≅ G/Ntype
2. C2×C32×C5×C8C2semidirect
3. C2×C3×C3⋊#168C2semidirect
4. C3×#6705C2direct (N × H)
5. C2×#6705C3direct (N × H)
12. C32×C5×C8C22semidirect
13. C3×C3⋊#168C22semidirect
18. C2×C3×C5×C8D3semidirect
19. C2×C3×C5×C8C2×C3semidirect
20. C2×C3⋊#168C2×C3semidirect
21. #6705C2×C3direct (N × H)
32. C2×C32×C8D5semidirect
47. C3×C5×C8C2×D3semidirect
48. C3×C5×C8C22×C3semidirect
49. C3⋊#168C22×C3semidirect
53. C2×C32×C5M16semidirect
57. C2×C5×C8C3×D3semidirect
65. C32×C8C2×D5semidirect
87. C2×C3×C8C3×D5semidirect
88. C2×C3×C8D15semidirect
91. C32×C5C2×M16semidirect
97. C5×C8C2×C3×D3semidirect
108. C2×C3×C5ΓL(1,𝔽25)semidirect
109. C2×C3×C5C3×M16semidirect
127. C3×C8C2×C3×D5semidirect
128. C3×C8C2×D15semidirect
136. C2×C32#676semidirect
142. C2×C8C3×D15semidirect
144. C3×C5C2×ΓL(1,𝔽25)semidirect
145. C3×C5C2×C3×M16semidirect
161. C2×C5C3×ΓL(1,𝔽25)semidirect
164. C32C2×#676semidirect
171. C8C2×C3×D15semidirect
175. C2×C3C3×#676semidirect
176. C2×C3#6705direct (N × H)
180. C5C2×C3×ΓL(1,𝔽25)semidirect
188. C3C2×C3×#676semidirect
189. C3C2×#6705direct (N × H)
191. C2C3×#6705direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×D152
C2×C32×C5×C82
C2×C3×C3⋊#1682
C3×#67052
C2×#67053
C2×C3×#6763
C2×C3×ΓL(1,𝔽25)5