Finite Group Databasebeta

← #13089411

Subgroups — #13089411

Order 1440.

718 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 718 conjugacy classes of subgroups — too many to draw. Only its 143 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×C5×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#6817C212yesmaximal, complemented
3#10624236C214yesmaximal, complemented
4#10624226C214yesmaximal, complemented
5#10624224C214yesmaximal, complemented
6#92018—32nomaximal
7C22×C5×C3⋊D3C2212yescomplemented
8C2×(C3⋊D3)⋊D5C2214yescomplemented
9C2×C32×C4×C5C2211yescharacteristic, complemented
10C5×C4⋊(C3⋊D3)C2214yescomplemented
11C3×#1220C2214yescomplemented
12C3×D60C2212yescomplemented
13C3×C4×D15C2214yescomplemented
14C3×Dic30C2212yescomplemented
15#67156C2218yescomplemented
16#67155C2214yescomplemented
17#64308—51nomaximal, Hall
18#6817D312yescomplemented
19C2×C5×D12—32no—
20C2×C5×D12—61no—
21C3×#709—32no—
22D3×C22⋊D5—38no—
23#6777—34no—
24D5×D12—34no—
25#6772—34no—
26#6768—34no—
27C22×C32×C5C2311yescharacteristic
28C2×C5×C3⋊D3—42no—
29C2×C5×C3⋊D3C2314yes—
30C2×C3×D15C2314yes—
31(C3⋊D3)⋊D5—44no—
32C32×C4×C5C2312yes—
33C3×Dic15C2314yes—
34#4144—91noHall
35C2×C4⋊(C3⋊D3)D511yescharacteristic, complemented
36C3×#235—52no—
37#3811—54no—
38D3×D12—54no—
39#3799—54no—
40C22×C5×D3—34no—
41C22×C5×D3—62no—
42C2×D3×D5—38no—
43C2×C3×C4×C5—21no—
44C2×C3×C4×C5C2×D312yescomplemented
45#1220C2×D314yescomplemented
46D60C2×D312yescomplemented
47C4×D15C2×D314yescomplemented
48Dic30C2×D312yescomplemented
49C5×D12—34no—
50C5×D12—34no—
51C5×D12—64no—
52C3×C22⋊D5—34no—
53C3×D20—32no—
54C3×C4×D5—34no—
55C3×Dic10—32no—
56#1203—38no—
57#1201—38no—
58D3×Dic5—38no—
59#1009—152no—
60C2×C32×C5—21no—
61C2×C32×C5C2411yescharacteristic
62C5×C3⋊D3—24no—
63C3×D15—24no—
64C2×C5×D4—91no—
65C2×C22⋊D5—94no—
66#711—94no—
67D4×D5—94no—
68#709—32no—
69C22×C3⋊D3C2×D512yescomplemented
70C2×D32—54no—
71C2×C32×C4C2×D511yescharacteristic, complemented
72C4⋊(C3⋊D3)C2×D514yescomplemented
73C3×#67—54no—
74C3×D12—52no—
75C3×C4×D3—54no—
76C3×Dic6—52no—
77#626—58no—
78#624—54no—
79C22×C3×C5—21no—
80C22×C3×C5C22×D312yes—
81C2×D15C22×D314yes—
82C2×C5×D3—38no—
83C2×C5×D3—64no—
84C2×C5×D3—124no—
85C2×C5×D3—124no—
86C2×C3×D5—34no—
87D3×D5—128no—
88C3×C4×C5—22no—
89C3×C4×C5C22×D312yes—
90C3×C4×C5C22×D312yes—
91Dic15C22×D314yes—
92C3×Dic5—34no—
93C3×#41—152no—
94#239—154no—
95D3×D4—154no—
96D3×D4—158no—
97#235—52no—
98#235—154no—
99C2×D12—32no—
100C2×D12—61no—
101C2×D12—154no—
102C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
103C23×C5—92no—
104C22×D5—94no—
105C5×D4—94no—
106C2×C4×C5D3211yescharacteristic, complemented
107C22⋊D5—34no—
108C22⋊D5—98no—
109C2×Dic5—94no—
110D20—32no—
111C4×D5—34no—
112Dic10—32no—
113C22×C32C22×D511yescharacteristic
114C2×C3⋊D3—42no—
115C2×C3⋊D3C22×D514yes—
116C2×C3×D3—54no—
117D32—204no—
118C32×C4C22×D512yes—
119C3×Dic3—54no—
120H(5,𝔽2)—451noSylow
121C2×C3×C5—22no—
122C2×C3×C5—22no—
123C2×C3×C5—21no—
124C2×C3×C5C23×D312yes—
125D15—24no—
126C3×D5—64no—
127C5×D3—68no—
128C5×D3—124no—
129C22×D3—34no—
130C22×D3—62no—
131C22×D3—158no—
132C3×Q8—152no—
133C3×D4—154no—
134C3×D4—152no—
135C2×C3×C4—21no—
136C2×C3×C4—154no—
137C2×C3×C4D3×D512yescomplemented
138#67—54no—
139#67—158no—
140D12—34no—
141D12—34no—
142D12—52no—
143D12—64no—
144D12—158no—
145C4×D3—54no—
146C4×D3—158no—
147Dic6—52no—
148C22×C5—94no—
149C22×C5—362no—
150C22×C5C2×D3211yescharacteristic
151C2×D5—34no—
152C2×D5—364no—
153C4×C5C2×D3212yes—
154Dic5—34no—
155C2×C32—21no—
156C2×C32C23×D511yescharacteristic
157C3⋊D3—24no—
158C3×D3—104no—
159#41—152no—
160#41—454no—
161C2×D4—91no—
162C2×D4—454no—
163C2×D4—454no—
164C3×C5—21no—
165C3×C5#100912yescomplemented
166C22×C3—21no—
167C22×C3—154no—
168C22×C3C2×D3×D512yes—
169C2×D3—38no—
170C2×D3—54no—
171C2×D3—64no—
172C2×D3—124no—
173C2×D3—124no—
174C2×D3—608no—
175C3×C4—22no—
176C3×C4—154no—
177C3×C4C2×D3×D512yes—
178C3×C4C2×D3×D512yes—
179Dic3—54no—
180C2×C5—21no—
181C2×C5—184no—
182C2×C5C22×D3211yescharacteristic
183D5—64no—
184C32#414411yescharacteristic, complemented, Sylow
185C23—92no—
186C23—454no—
187Q8—152no—
188D4—94no—
189D4—154no—
190D4—152no—
191D4—458no—
192C2×C4—154no—
193C2×C4—454no—
194C2×C4(C3⋊D3)⋊D511yescharacteristic, complemented
195C2×C3—22no—
196C2×C3—22no—
197C2×C3—21no—
198C2×C3—304no—
199C2×C3C22×D3×D512yes—
200D3—68no—
201D3—104no—
202D3—124no—
203C5#6430811yescharacteristic, complemented, Sylow
204C22—94no—
205C22—154no—
206C22—362no—
207C22—1804no—
208C22C2×(C3⋊D3)⋊D511yescharacteristic
209C4—154no—
210C4C2×(C3⋊D3)⋊D512yes—
211C3—21no—
212C3#9201812yescomplemented
213C2—21no—
214C2—184no—
215C2—304no—
216C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
center, FrattiniC2C22×(C3⋊D3)⋊D5
commutator, socle, cosocleC2×C32×C5C24
FittingC2×C32×C4×C5C22
hypercenterC2×C4(C3⋊D3)⋊D5
hypocenterC32×C5H(5,𝔽2)
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C5×C4⋊(C3⋊D3)C2semidirect
2. C3×#6817C2semidirect
3. #10624236C2semidirect
4. #10624226C2semidirect
5. #10624224C2semidirect
7. C22×C5×C3⋊D3C22semidirect
8. C2×(C3⋊D3)⋊D5C22semidirect
9. C2×C32×C4×C5C22semidirect
10. C5×C4⋊(C3⋊D3)C22semidirect
11. C3×#1220C22semidirect
12. C3×D60C22semidirect
13. C3×C4×D15C22semidirect
14. C3×Dic30C22semidirect
15. #67156C22semidirect
16. #67155C22semidirect
18. #6817D3semidirect
35. C2×C4⋊(C3⋊D3)D5semidirect
44. C2×C3×C4×C5C2×D3semidirect
45. #1220C2×D3semidirect
46. D60C2×D3semidirect
47. C4×D15C2×D3semidirect
48. Dic30C2×D3semidirect
69. C22×C3⋊D3C2×D5semidirect
71. C2×C32×C4C2×D5semidirect
72. C4⋊(C3⋊D3)C2×D5semidirect
102. C32×C5H(5,𝔽2)semidirect
106. C2×C4×C5D32semidirect
137. C2×C3×C4D3×D5semidirect
165. C3×C5#1009semidirect
184. C32#4144semidirect
194. C2×C4(C3⋊D3)⋊D5semidirect
203. C5#64308semidirect
212. C3#92018semidirect

Maximal subgroups

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

subgroupindex
C2×C5×C4⋊(C3⋊D3)2
C3×#68172
#106242362
#106242262
#106242242
#920183
#643085