Finite Group Databasebeta

← C2×C3×C5×#215 #13087954

Subgroups — C2×C3×C5×#215 #13087954

Order 1440.

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

normal subgroups only

This group has 310 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×Q8C211yesmaximal, characteristic, complemented
2C2×C3×C5×D12C211yesmaximal, characteristic
3C3×C5×#215C214yesmaximal, direct product factor
4C2×C3×C5×C3⋊C8C211yesmaximal, characteristic, complemented
5C2×C3×C5×SD16—31nomaximal
6C2×C5×#215C311yesmaximal, characteristic, direct product factor
7C22×C3×C5×D3—21no—
8C32×C5×Q8—21no—
9C32×C5×Q8C2212yescomplemented
10C2×C32×C4×C5—21no—
11C2×C32×C4×C5C2211yescharacteristic
12C3×C5×D12—21no—
13C3×C5×D12C2212yes—
14C3×C5×C3⋊C8C2212yescomplemented
15C2×C3×#215C511yesmaximal, characteristic, direct product factor, Hall
16C2×C3×C5×Q8—21no—
17C2×C3×C5×Q8D311yescharacteristic, complemented
18C2×C3×C5×Q8C2×C311yescharacteristic, complemented
19C2×C3×C5×D4—31no—
20C2×C5×D12C2×C311yescharacteristic
21C3×C5×SD16—34no—
22C2×C3×C5×C8—31no—
23C5×#215C2×C314yesdirect product factor
24C2×C5×C3⋊C8C2×C311yescharacteristic, complemented
25C22×C32×C5D411yescharacteristic
26C2×C3×C5×D3—22no—
27C2×C3×C5×D3—42no—
28C32×C4×C5—22no—
29C32×C4×C5D411yescharacteristic
30C32×C4×C5C2311yescharacteristic
31C2×C5×SD16—31noHall
32C2×C32×Q8C2×C511yescharacteristic, complemented
33C2×C3×D12C2×C511yescharacteristic
34C3×#215C2×C514yesdirect product factor
35C2×C3×C3⋊C8C2×C511yescharacteristic, complemented
36C23×C3×C5—61no—
37C22×C5×D3—21no—
38C3×C5×Q8—22no—
39C3×C5×Q8—22no—
40C3×C5×Q8—21no—
41C3×C5×Q8—21no—
42C3×C5×Q8C2×D312yescomplemented
43C3×C5×Q8C22×C312yescomplemented
44C3×C5×D4—32no—
45C3×C5×D4—61no—
46C2×C3×C4×C5—22no—
47C2×C3×C4×C5—21no—
48C2×C3×C4×C5—21no—
49C2×C3×C4×C5—21no—
50C2×C3×C4×C5C2×D311yescharacteristic
51C2×C3×C4×C5C22×C311yescharacteristic
52C5×D12—21no—
53C5×D12C22×C312yes—
54C3×C5×C8—32no—
55C5×C3⋊C8C22×C312yescomplemented
56C2×C3×SD16—31no—
57C2×#215C3×C511yescharacteristic, direct product factor
58C2×C32×C5SD1612yescomplemented
59C2×C32×C5C2×D411yescharacteristic
60C3×C5×D3—42no—
61C2×C5×Q8C3×D311yescharacteristic, complemented
62C2×C5×D4—31no—
63C5×SD16—34no—
64C2×C5×C8—31no—
65C22×C3×D3—21no—
66C32×Q8—21no—
67C32×Q8C22×C512yescomplemented
68C2×C32×C4—21no—
69C2×C32×C4C22×C511yescharacteristic
70C3×D12—21no—
71C3×D12C22×C512yes—
72C3×C3⋊C8C22×C512yescomplemented
73C22×C3×C5—21no—
74C22×C3×C5—62no—
75C22×C3×C5—122no—
76C22×C3×C5#6711yescharacteristic
77C22×C3×C5C3×D411yescharacteristic
78C2×C5×D3—22no—
79C2×C5×D3—42no—
80C3×C4×C5—24no—
81C3×C4×C5—22no—
82C3×C4×C5—22no—
83C3×C4×C5—21no—
84C3×C4×C5—21no—
85C3×C4×C5#6711yescharacteristic
86C3×C4×C5C3×D411yescharacteristic
87C3×C4×C5C22×D311yescharacteristic
88C3×C4×C5C23×C311yescharacteristic
89C2×C3×Q8—21no—
90C2×C3×Q8C5×D311yescharacteristic, complemented
91C2×C3×Q8C2×C3×C511yescharacteristic, complemented
92C2×C3×D4—31no—
93C2×D12C2×C3×C511yescharacteristic
94C3×SD16—34no—
95C2×C3×C8—31no—
96#215C2×C3×C514yesdirect product factor
97C2×C3⋊C8C2×C3×C511yescharacteristic, complemented
98C32×C5C2×SD1611yescharacteristic, complemented, Hall
99C23×C5—61no—
100C5×Q8—21no—
101C5×Q8C2×C3×D312yescomplemented
102C5×D4—32no—
103C5×D4—61no—
104C2×C4×C5—21no—
105C2×C4×C5C2×C3×D311yescharacteristic
106C5×C8—32no—
107C22×C32C5×D411yescharacteristic
108C2×C3×D3—22no—
109C2×C3×D3—42no—
110C32×C4—22no—
111C32×C4C5×D411yescharacteristic
112C32×C4C23×C511yescharacteristic
113C2×SD16—31noSylow
114C2×C3×C5—22no—
115C2×C3×C5—21no—
116C2×C3×C5—122no—
117C2×C3×C5#21512yesdirect product factor
118C2×C3×C5C3×SD1612yescomplemented
119C2×C3×C5C2×#6711yescharacteristic
120C2×C3×C5C2×C3×D411yescharacteristic
121C5×D3—42no—
122C23×C3—61no—
123C22×D3—21no—
124C3×Q8—22no—
125C3×Q8—21no—
126C3×Q8—21no—
127C3×Q8—22no—
128C3×Q8C2×C5×D312yescomplemented
129C3×Q8C22×C3×C512yescomplemented
130C3×D4—32no—
131C3×D4—61no—
132C2×C3×C4—22no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—21no—
136C2×C3×C4C2×C5×D311yescharacteristic
137C2×C3×C4C22×C3×C511yescharacteristic
138D12—21no—
139D12C22×C3×C512yes—
140C3×C8—32no—
141C3⋊C8C22×C3×C512yescomplemented
142C22×C5—62no—
143C22×C5—122no—
144C22×C5C3×#6711yescharacteristic
145C4×C5—22no—
146C4×C5C3×#6711yescharacteristic
147C4×C5C22×C3×D311yescharacteristic
148C2×C32C5×SD1612yescomplemented
149C2×C32C2×C5×D411yescharacteristic
150C3×D3—42no—
151C2×Q8C3×C5×D311yescharacteristic, complemented
152C2×D4—31no—
153SD16—34no—
154C2×C8—31no—
155C3×C5—21no—
156C3×C5C2×#21511yescharacteristic, direct product factor
157C3×C5C2×C3×SD1611yescharacteristic, complemented
158C22×C3—21no—
159C22×C3—62no—
160C22×C3—122no—
161C22×C3C5×#6711yescharacteristic
162C22×C3C3×C5×D411yescharacteristic
163C2×D3—22no—
164C2×D3—42no—
165C3×C4—24no—
166C3×C4—22no—
167C3×C4—22no—
168C3×C4—21no—
169C3×C4—21no—
170C3×C4C5×#6711yescharacteristic
171C3×C4C3×C5×D411yescharacteristic
172C3×C4C22×C5×D311yescharacteristic
173C3×C4C23×C3×C511yescharacteristic
174C2×C5—122no—
175C2×C5C3×#21512yesdirect product factor
176C2×C5C2×C3×#6711yescharacteristic
177C32C2×C5×SD1611yescharacteristic, complemented, Sylow
178C23—61no—
179Q8—21no—
180Q8C2×C3×C5×D312yescomplemented
181D4—32no—
182D4—61no—
183C2×C4—21no—
184C2×C4C2×C3×C5×D311yescharacteristic
185C8—32no—
186C2×C3—21no—
187C2×C3—22no—
188C2×C3—122no—
189C2×C3C5×#21512yesdirect product factor
190C2×C3C3×C5×SD1612yescomplemented
191C2×C3C2×C5×#6711yescharacteristic
192C2×C3C2×C3×C5×D411yescharacteristic
193D3—42no—
194C5C2×C3×#21511yescharacteristic, direct product factor, Sylow
195C22—62no—
196C22—122no—
197C22C3×C5×#6711yescharacteristic
198C4—22no—
199C4C3×C5×#6711yescharacteristic
200C4C22×C3×C5×D311yescharacteristic
201C3—21no—
202C3C2×C5×#21511yescharacteristic, direct product factor
203C3C2×C3×C5×SD1611yescharacteristic, complemented
204C2—122no—
205C2C3×C5×#21512yesdirect product factor
206C2C2×C3×C5×#6711yescharacteristic
subgroupG/H
centerC22×C3×C5#67
commutator, cosocleC3×C4C23×C3×C5
FrattiniC4C22×C3×C5×D3
FittingC2×C32×C5×Q8C2
socleC22×C32×C5D4
prime radicalC2×C3×C5×D12C2
hypercenterC2×C3×C5×Q8D3
hypocenterC3C2×C3×C5×SD16
LCSG ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C32×C5×Q8C2semidirect
3. C3×C5×#215C2direct (N × H)
4. C2×C3×C5×C3⋊C8C2semidirect
6. C2×C5×#215C3direct (N × H)
9. C32×C5×Q8C22semidirect
14. C3×C5×C3⋊C8C22semidirect
15. C2×C3×#215C5direct (N × H)
17. C2×C3×C5×Q8D3semidirect
18. C2×C3×C5×Q8C2×C3semidirect
23. C5×#215C2×C3direct (N × H)
24. C2×C5×C3⋊C8C2×C3semidirect
32. C2×C32×Q8C2×C5semidirect
34. C3×#215C2×C5direct (N × H)
35. C2×C3×C3⋊C8C2×C5semidirect
42. C3×C5×Q8C2×D3semidirect
43. C3×C5×Q8C22×C3semidirect
55. C5×C3⋊C8C22×C3semidirect
57. C2×#215C3×C5direct (N × H)
58. C2×C32×C5SD16semidirect
61. C2×C5×Q8C3×D3semidirect
67. C32×Q8C22×C5semidirect
72. C3×C3⋊C8C22×C5semidirect
90. C2×C3×Q8C5×D3semidirect
91. C2×C3×Q8C2×C3×C5semidirect
96. #215C2×C3×C5direct (N × H)
97. C2×C3⋊C8C2×C3×C5semidirect
98. C32×C5C2×SD16semidirect
101. C5×Q8C2×C3×D3semidirect
117. C2×C3×C5#215direct (N × H)
118. C2×C3×C5C3×SD16semidirect
128. C3×Q8C2×C5×D3semidirect
129. C3×Q8C22×C3×C5semidirect
141. C3⋊C8C22×C3×C5semidirect
148. C2×C32C5×SD16semidirect
151. C2×Q8C3×C5×D3semidirect
156. C3×C5C2×#215direct (N × H)
157. C3×C5C2×C3×SD16semidirect
175. C2×C5C3×#215direct (N × H)
177. C32C2×C5×SD16semidirect
180. Q8C2×C3×C5×D3semidirect
189. C2×C3C5×#215direct (N × H)
190. C2×C3C3×C5×SD16semidirect
194. C5C2×C3×#215direct (N × H)
202. C3C2×C5×#215direct (N × H)
203. C3C2×C3×C5×SD16semidirect
205. C2C3×C5×#215direct (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×Q82
C2×C3×C5×D122
C3×C5×#2152
C2×C3×C5×C3⋊C82
C2×C3×C5×SD163
C2×C5×#2153
C2×C3×#2155