Finite Group Databasebeta

← C2×C3×C5×#214 #13087949

Subgroups — C2×C3×C5×#214 #13087949

Order 1440.

326 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 132 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
2C2×C3×C5×Dic6C211yesmaximal, characteristic, complemented
3C3×C5×#214C214yesmaximal, direct product factor
4C2×C3×C5×C3⋊C8C211yesmaximal, characteristic, complemented
5C2×C3×C5×SD16—31nomaximal
6C2×C5×#214C311yesmaximal, characteristic, direct product factor
7C23×C32×C5—21no—
8C32×C5×D4—21no—
9C32×C5×D4C2212yes—
10C2×C32×C4×C5C2211yescharacteristic
11C2×C3×C5×Dic3—21no—
12C3×C5×Dic6—21no—
13C3×C5×Dic6C2212yescomplemented
14C3×C5×C3⋊C8C2212yescomplemented
15C2×C3×#214C511yesmaximal, characteristic, direct product factor, Hall
16C2×C3×C5×Q8—31no—
17C2×C3×C5×D4—21no—
18C2×C3×C5×D4D311yescharacteristic
19C2×C3×C5×D4C2×C311yescharacteristic
20C2×C5×Dic6C2×C311yescharacteristic, complemented
21C3×C5×SD16—34no—
22C2×C3×C5×C8—31no—
23C5×#214C2×C314yesdirect product factor
24C2×C5×C3⋊C8C2×C311yescharacteristic, complemented
25C22×C32×C5—22no—
26C22×C32×C5—42no—
27C22×C32×C5D411yescharacteristic
28C32×C4×C5D411yescharacteristic
29C32×C4×C5C2311yescharacteristic
30C3×C5×Dic3—22no—
31C2×C5×SD16—31noHall
32C2×C32×D4C2×C511yescharacteristic
33C2×C3×Dic6C2×C511yescharacteristic, complemented
34C3×#214C2×C514yesdirect product factor
35C2×C3×C3⋊C8C2×C511yescharacteristic, complemented
36C23×C3×C5—22no—
37C23×C3×C5—21no—
38C23×C3×C5—21no—
39C3×C5×Q8—32no—
40C3×C5×Q8—61no—
41C3×C5×D4—22no—
42C3×C5×D4—22no—
43C3×C5×D4—21no—
44C3×C5×D4—21no—
45C3×C5×D4C2×D312yes—
46C3×C5×D4C22×C312yes—
47C2×C3×C4×C5—21no—
48C2×C3×C4×C5—61no—
49C2×C3×C4×C5C2×D311yescharacteristic
50C2×C3×C4×C5C22×C311yescharacteristic
51C2×C5×Dic3—21no—
52C5×Dic6—21no—
53C5×Dic6C22×C312yescomplemented
54C3×C5×C8—32no—
55C5×C3⋊C8C22×C312yescomplemented
56C2×C3×SD16—31no—
57C2×#214C3×C511yescharacteristic, direct product factor
58C2×C32×C5—42no—
59C2×C32×C5SD1612yescomplemented
60C2×C32×C5C2×D411yescharacteristic
61C2×C5×Q8—31no—
62C2×C5×D4C3×D311yescharacteristic
63C5×SD16—34no—
64C2×C5×C8—31no—
65C23×C32—21no—
66C32×D4—21no—
67C32×D4C22×C512yes—
68C2×C32×C4C22×C511yescharacteristic
69C2×C3×Dic3—21no—
70C3×Dic6—21no—
71C3×Dic6C22×C512yescomplemented
72C3×C3⋊C8C22×C512yescomplemented
73C22×C3×C5—24no—
74C22×C3×C5—22no—
75C22×C3×C5—22no—
76C22×C3×C5—21no—
77C22×C3×C5—44no—
78C22×C3×C5—42no—
79C22×C3×C5—42no—
80C22×C3×C5#6711yescharacteristic
81C22×C3×C5C3×D411yescharacteristic
82C3×C4×C5—21no—
83C3×C4×C5—21no—
84C3×C4×C5—62no—
85C3×C4×C5#6711yescharacteristic
86C3×C4×C5C3×D411yescharacteristic
87C3×C4×C5C22×D311yescharacteristic
88C3×C4×C5C23×C311yescharacteristic
89C5×Dic3—22no—
90C2×C3×Q8—31no—
91C2×C3×D4—21no—
92C2×C3×D4C5×D311yescharacteristic
93C2×C3×D4C2×C3×C511yescharacteristic
94C2×Dic6C2×C3×C511yescharacteristic, complemented
95C3×SD16—34no—
96C2×C3×C8—31no—
97#214C2×C3×C514yesdirect product factor
98C2×C3⋊C8C2×C3×C511yescharacteristic, complemented
99C32×C5C2×SD1611yescharacteristic, complemented, Hall
100C23×C5—21no—
101C5×Q8—32no—
102C5×Q8—61no—
103C5×D4—21no—
104C5×D4C2×C3×D312yes—
105C2×C4×C5—61no—
106C2×C4×C5C2×C3×D311yescharacteristic
107C5×C8—32no—
108C22×C32—22no—
109C22×C32—42no—
110C22×C32C5×D411yescharacteristic
111C32×C4C5×D411yescharacteristic
112C32×C4C23×C511yescharacteristic
113C3×Dic3—22no—
114C2×SD16—31noSylow
115C2×C3×C5—22no—
116C2×C3×C5—21no—
117C2×C3×C5—44no—
118C2×C3×C5—42no—
119C2×C3×C5—42no—
120C2×C3×C5#21412yesdirect product factor
121C2×C3×C5C3×SD1612yescomplemented
122C2×C3×C5C2×#6711yescharacteristic
123C2×C3×C5C2×C3×D411yescharacteristic
124C23×C3—22no—
125C23×C3—21no—
126C23×C3—21no—
127C3×Q8—32no—
128C3×Q8—61no—
129C3×D4—22no—
130C3×D4—21no—
131C3×D4—21no—
132C3×D4—22no—
133C3×D4C2×C5×D312yes—
134C3×D4C22×C3×C512yes—
135C2×C3×C4—21no—
136C2×C3×C4—61no—
137C2×C3×C4C2×C5×D311yescharacteristic
138C2×C3×C4C22×C3×C511yescharacteristic
139C2×Dic3—21no—
140Dic6—21no—
141Dic6C22×C3×C512yescomplemented
142C3×C8—32no—
143C3⋊C8C22×C3×C512yescomplemented
144C22×C5—22no—
145C22×C5—42no—
146C22×C5C3×#6711yescharacteristic
147C4×C5—62no—
148C4×C5C3×#6711yescharacteristic
149C4×C5C22×C3×D311yescharacteristic
150C2×C32—42no—
151C2×C32C5×SD1612yescomplemented
152C2×C32C2×C5×D411yescharacteristic
153C2×Q8—31no—
154C2×D4C3×C5×D311yescharacteristic
155SD16—34no—
156C2×C8—31no—
157C3×C5—21no—
158C3×C5C2×#21411yescharacteristic, direct product factor
159C3×C5C2×C3×SD1611yescharacteristic, complemented
160C22×C3—24no—
161C22×C3—22no—
162C22×C3—22no—
163C22×C3—21no—
164C22×C3—44no—
165C22×C3—42no—
166C22×C3—42no—
167C22×C3C5×#6711yescharacteristic
168C22×C3C3×C5×D411yescharacteristic
169C3×C4—21no—
170C3×C4—21no—
171C3×C4—62no—
172C3×C4C5×#6711yescharacteristic
173C3×C4C3×C5×D411yescharacteristic
174C3×C4C22×C5×D311yescharacteristic
175C3×C4C23×C3×C511yescharacteristic
176Dic3—22no—
177C2×C5—42no—
178C2×C5C3×#21412yesdirect product factor
179C2×C5C2×C3×#6711yescharacteristic
180C32C2×C5×SD1611yescharacteristic, complemented, Sylow
181C23—21no—
182Q8—32no—
183Q8—61no—
184D4—21no—
185D4C2×C3×C5×D312yes—
186C2×C4—61no—
187C2×C4C2×C3×C5×D311yescharacteristic
188C8—32no—
189C2×C3—21no—
190C2×C3—22no—
191C2×C3—44no—
192C2×C3—42no—
193C2×C3—42no—
194C2×C3C5×#21412yesdirect product factor
195C2×C3C3×C5×SD1612yescomplemented
196C2×C3C2×C5×#6711yescharacteristic
197C2×C3C2×C3×C5×D411yescharacteristic
198C5C2×C3×#21411yescharacteristic, direct product factor, Sylow
199C22—22no—
200C22—42no—
201C22C3×C5×#6711yescharacteristic
202C4—62no—
203C4C3×C5×#6711yescharacteristic
204C4C22×C3×C5×D311yescharacteristic
205C3—21no—
206C3C2×C5×#21411yescharacteristic, direct product factor
207C3C2×C3×C5×SD1611yescharacteristic, complemented
208C2—42no—
209C2C3×C5×#21412yesdirect product factor
210C2C2×C3×C5×#6711yescharacteristic
subgroupG/H
centerC22×C3×C5#67
commutator, cosocleC3×C4C23×C3×C5
FrattiniC4C22×C3×C5×D3
Fitting, prime radicalC2×C32×C5×D4C2
socleC22×C32×C5D4
hypercenterC2×C3×C5×D4D3
hypocenterC3C2×C3×C5×SD16
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×D4

Complements

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

NH ≅ G/Ntype
2. C2×C3×C5×Dic6C2semidirect
3. C3×C5×#214C2direct (N × H)
4. C2×C3×C5×C3⋊C8C2semidirect
6. C2×C5×#214C3direct (N × H)
13. C3×C5×Dic6C22semidirect
14. C3×C5×C3⋊C8C22semidirect
15. C2×C3×#214C5direct (N × H)
20. C2×C5×Dic6C2×C3semidirect
23. C5×#214C2×C3direct (N × H)
24. C2×C5×C3⋊C8C2×C3semidirect
33. C2×C3×Dic6C2×C5semidirect
34. C3×#214C2×C5direct (N × H)
35. C2×C3×C3⋊C8C2×C5semidirect
53. C5×Dic6C22×C3semidirect
55. C5×C3⋊C8C22×C3semidirect
57. C2×#214C3×C5direct (N × H)
59. C2×C32×C5SD16semidirect
71. C3×Dic6C22×C5semidirect
72. C3×C3⋊C8C22×C5semidirect
94. C2×Dic6C2×C3×C5semidirect
97. #214C2×C3×C5direct (N × H)
98. C2×C3⋊C8C2×C3×C5semidirect
99. C32×C5C2×SD16semidirect
120. C2×C3×C5#214direct (N × H)
121. C2×C3×C5C3×SD16semidirect
141. Dic6C22×C3×C5semidirect
143. C3⋊C8C22×C3×C5semidirect
151. C2×C32C5×SD16semidirect
158. C3×C5C2×#214direct (N × H)
159. C3×C5C2×C3×SD16semidirect
178. C2×C5C3×#214direct (N × H)
180. C32C2×C5×SD16semidirect
194. C2×C3C5×#214direct (N × H)
195. C2×C3C3×C5×SD16semidirect
198. C5C2×C3×#214direct (N × H)
206. C3C2×C5×#214direct (N × H)
207. C3C2×C3×C5×SD16semidirect
209. C2C3×C5×#214direct (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×C5×Dic62
C3×C5×#2142
C2×C3×C5×C3⋊C82
C2×C3×C5×SD163
C2×C5×#2143
C2×C3×#2145