Finite Group Databasebeta

← #13089423

Subgroups — #13089423

Order 1440.

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

normal subgroups only

This group has 704 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
1C5×#3832C211yesmaximal, characteristic, complemented
2C3×#6821C212yesmaximal, complemented
3#10624227C213yesmaximal, complemented
4#10624226C216yesmaximal, complemented
5#10624223C213yesmaximal, complemented
6#92035—32nomaximal
7C2×(C3⋊D3)⋊D5C2216yescomplemented
8C32×C5×Q8C2211yescharacteristic, complemented
9C5×C4⋊(C3⋊D3)C2213yescomplemented
10C4×C5×C3⋊D3C2213yescomplemented
11C3×D60C2216yescomplemented
12C3×C4×D15C2216yescomplemented
13#67158C2211yescharacteristic, complemented
14#67157C2213yescomplemented
15#67156C2216yescomplemented
16#64320—51nomaximal, Hall
17#6821D312yescomplemented
18C5×#239—32no—
19C5×#239—61no—
20C3×#713—32no—
21#6777—36no—
22D3×D20—36no—
23D5×D12—36no—
24#6771—36no—
25C2×C5×C3⋊D3C2313yes—
26C2×C3×D15C2316yes—
27(C3⋊D3)⋊D5—46no—
28C32×C4×C5C2313yes—
29C5×C3⋊Dic3C2311yescharacteristic
30C3×Dic15C2312yes—
31#4149—91noHall
32#3832D511yescharacteristic, complemented
33C3×#239—52no—
34#3802—53no—
35D3×D12—56no—
36#3798—53no—
37C2×D3×D5—312no—
38C3×C5×Q8—21no—
39C3×C5×Q8C2×D312yescomplemented
40D60C2×D316yescomplemented
41C4×D15C2×D316yescomplemented
42C5×D12—36no—
43C5×D12—63no—
44C4×C5×D3—36no—
45C4×C5×D3—63no—
46C3×D20—36no—
47C3×C4×D5—36no—
48#1204—32no—
49#1203—36no—
50#1202—36no—
51#1201—36no—
52#1009—152no—
53C2×C32×C5C2411yescharacteristic
54C5×C3⋊D3—23no—
55C3×D15—26no—
56C5×#41—91no—
57#713—32no—
58D4×D5—96no—
59#709—93no—
60C2×D20—93no—
61C2×D32—56no—
62C32×Q8C2×D511yescharacteristic, complemented
63C4⋊(C3⋊D3)C2×D513yescomplemented
64C4×C3⋊D3C2×D513yescomplemented
65C3×D12—56no—
66C3×C4×D3—56no—
67#627—51no—
68#626—56no—
69#625—53no—
70C2×D15C22×D316yes—
71C2×C5×D3—36no—
72C2×C5×D3—63no—
73C2×C3×D5—36no—
74D3×D5—1212no—
75C3×C4×C5—23no—
76C3×C4×C5C22×D316yes—
77Dic15C22×D312yes—
78C3×Dic5—32no—
79C5×Dic3—32no—
80C5×Dic3—61no—
81C3×#41—152no—
82#239—32no—
83#239—52no—
84#239—61no—
85D3×D4—156no—
86D3×D4—156no—
87#235—156no—
88C2×D12—156no—
89C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
90C22×D5—96no—
91C5×Q8D3211yescharacteristic, complemented
92C5×D4—93no—
93C2×C4×C5—93no—
94C22⋊D5—96no—
95D20—36no—
96D20—93no—
97C4×D5—36no—
98Dic10—91no—
99C2×C3⋊D3C22×D513yes—
100C2×C3×D3—56no—
101D32—206no—
102C32×C4C22×D513yes—
103C3⋊Dic3C22×D511yescharacteristic
104C3×Dic3—52no—
105H(5,𝔽2)—451noSylow
106C2×C3×C5—21no—
107C2×C3×C5C23×D312yes—
108D15—26no—
109C3×D5—66no—
110C5×D3—66no—
111C5×D3—123no—
112C22×D3—1512no—
113C3×Q8—21no—
114C3×Q8D3×D512yescomplemented
115C3×D4—156no—
116C2×C3×C4—156no—
117#67—156no—
118#67—156no—
119D12—36no—
120D12—56no—
121D12—63no—
122D12—156no—
123C4×D3—36no—
124C4×D3—56no—
125C4×D3—63no—
126Dic6—152no—
127C22×C5—93no—
128C2×D5—36no—
129C2×D5—366no—
130C4×C5—91no—
131C4×C5C2×D3213yes—
132Dic5—32no—
133C2×C32C23×D511yescharacteristic
134C3⋊D3—23no—
135C3×D3—106no—
136#41—91no—
137#41—152no—
138#41—453no—
139C2×D4—453no—
140C2×D4—456no—
141C3×C5—21no—
142C3×C5#100912yescomplemented
143C22×C3—156no—
144C2×D3—36no—
145C2×D3—56no—
146C2×D3—63no—
147C2×D3—6012no—
148C3×C4—23no—
149C3×C4—152no—
150C3×C4C2×D3×D516yes—
151Dic3—32no—
152Dic3—52no—
153Dic3—61no—
154C2×C5—183no—
155C2×C5C22×D3211yescharacteristic
156D5—66no—
157C32#414911yescharacteristic, complemented, Sylow
158C23—456no—
159Q8—451no—
160Q8(C3⋊D3)⋊D511yescharacteristic, complemented
161D4—93no—
162D4—156no—
163D4—453no—
164D4—456no—
165C2×C4—93no—
166C2×C4—156no—
167C2×C3—21no—
168C2×C3—306no—
169C2×C3C22×D3×D512yes—
170D3—66no—
171D3—106no—
172D3—123no—
173C5#6432011yescharacteristic, complemented, Sylow
174C22—93no—
175C22—156no—
176C22—1806no—
177C4—91no—
178C4—152no—
179C4C2×(C3⋊D3)⋊D513yes—
180C3—21no—
181C3#9203512yescomplemented
182C2—183no—
183C2—306no—
184C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
center, FrattiniC2C22×(C3⋊D3)⋊D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×Q8C22
hypercenterQ8(C3⋊D3)⋊D5
hypocenterC32×C5H(5,𝔽2)
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C5×#3832C2semidirect
2. C3×#6821C2semidirect
3. #10624227C2semidirect
4. #10624226C2semidirect
5. #10624223C2semidirect
7. C2×(C3⋊D3)⋊D5C22semidirect
8. C32×C5×Q8C22semidirect
9. C5×C4⋊(C3⋊D3)C22semidirect
10. C4×C5×C3⋊D3C22semidirect
11. C3×D60C22semidirect
12. C3×C4×D15C22semidirect
13. #67158C22semidirect
14. #67157C22semidirect
15. #67156C22semidirect
17. #6821D3semidirect
32. #3832D5semidirect
39. C3×C5×Q8C2×D3semidirect
40. D60C2×D3semidirect
41. C4×D15C2×D3semidirect
62. C32×Q8C2×D5semidirect
63. C4⋊(C3⋊D3)C2×D5semidirect
64. C4×C3⋊D3C2×D5semidirect
89. C32×C5H(5,𝔽2)semidirect
91. C5×Q8D32semidirect
114. C3×Q8D3×D5semidirect
142. C3×C5#1009semidirect
157. C32#4149semidirect
160. Q8(C3⋊D3)⋊D5semidirect
173. C5#64320semidirect
181. C3#92035semidirect

Maximal subgroups

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

subgroupindex
C5×#38322
C3×#68212
#106242272
#106242262
#106242232
#920353
#643205