Finite Group Databasebeta

← #13089275

Subgroups — #13089275

Order 1440.

996 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 996 conjugacy classes of subgroups — too many to draw. Only its 200 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
1#10624283C211yesmaximal, characteristic, complemented
2C2×C5×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×#709C211yesmaximal, characteristic, complemented
4C3⋊D3×C22⋊D5C214yesmaximal, complemented
5#10624160C212yesmaximal, complemented
6D5×C4⋊(C3⋊D3)C212yesmaximal, complemented
7#10624155C212yesmaximal, complemented
8#10624151C212yesmaximal, complemented
9#92018—34nomaximal
10C22×C5×C3⋊D3C2212yescomplemented
11C2×D5×C3⋊D3C2214yescomplemented
12C2×C32×C4×C5C2211yescharacteristic, complemented
13#67184C2212yescomplemented
14C4⋊(C3⋊D15)C2211yescharacteristic, complemented
15C4×C3⋊D15C2212yescomplemented
16#67180C2211yescharacteristic, complemented
17C5×C4⋊(C3⋊D3)C2212yescomplemented
18C5×C4⋊(C3⋊D3)C2212yescomplemented
19C32×C22⋊D5C2212yescomplemented
20C32×D20C2211yescharacteristic, complemented
21C32×C4×D5C2212yescomplemented
22C32×Dic10C2211yescharacteristic, complemented
23#67140C2214yescomplemented
24#67138C2214yescomplemented
25C3⋊D3×Dic5C2214yescomplemented
26#64366—51nomaximal, Hall
27#6817—34no—
28C2×C5×D12—34no—
29C3×#709D314yescomplemented
30D3×C22⋊D5—316no—
31#6777—38no—
32D5×D12—38no—
33#6772—38no—
34#6768—38no—
35C22×C32×C5C2311yescharacteristic
36C2×C3⋊D15C2312yes—
37C2×C5×C3⋊D3—42no—
38C2×C5×C3⋊D3C2314yes—
39C2×C32×D5C2312yes—
40D5×C3⋊D3—44no—
41C32×C4×C5C2311yescharacteristic
42C32×C4×C5C2311yescharacteristic
43C3⋊Dic15C2312yes—
44C32×Dic5C2312yes—
45#4144—91noHall
46C32×#41—51no—
47#3832—52no—
48D4×C3⋊D3—54no—
49D4×C3⋊D3—52no—
50#3828—51no—
51#3828—52no—
52C2×C4⋊(C3⋊D3)—52no—
53C2×C4⋊(C3⋊D3)D511yescharacteristic, complemented
54C22×C5×D3—38no—
55C2×D3×D5—316no—
56C2×C3×C4×C5C2×D314yescomplemented
57#1220—38no—
58D60—34no—
59C4×D15—38no—
60Dic30—34no—
61C5×D12—38no—
62C5×D12—38no—
63C3×C22⋊D5C2×D318yescomplemented
64C3×D20C2×D314yescomplemented
65C3×C4×D5C2×D318yescomplemented
66C3×Dic10C2×D314yescomplemented
67#1203—316no—
68#1201—316no—
69D3×Dic5—316no—
70#1009—154no—
71C2×C32×C5—21no—
72C2×C32×C5C2411yescharacteristic
73C3⋊D15—22no—
74C5×C3⋊D3—24no—
75C32×D5—22no—
76C2×C5×D4—91no—
77C2×C22⋊D5—94no—
78#711—92no—
79#711—92no—
80D4×D5—92no—
81D4×D5—92no—
82#709—91no—
83#709C3⋊D311yescharacteristic, complemented
84C22×C3⋊D3—54no—
85C22×C3⋊D3C2×D512yescomplemented
86C32×Q8—51no—
87C32×D4—52no—
88C32×D4—51no—
89C2×C32×C4—52no—
90C2×C32×C4C2×D511yescharacteristic, complemented
91#638—52no—
92#638—54no—
93C4⋊(C3⋊D3)—54no—
94C4⋊(C3⋊D3)—51no—
95C4⋊(C3⋊D3)C2×D512yescomplemented
96C4⋊(C3⋊D3)C2×D512yescomplemented
97C4×C3⋊D3—54no—
98C4×C3⋊D3—52no—
99#634—51no—
100C22×C3×C5C22×D314yes—
101C2×D15—38no—
102C2×C5×D3—316no—
103C2×C5×D3—128no—
104C2×C3×D5C22×D318yes—
105D3×D5—1216no—
106C3×C4×C5C22×D314yes—
107C3×C4×C5C22×D314yes—
108Dic15—38no—
109C3×Dic5C22×D318yes—
110C3×#41—54no—
111#239—158no—
112D3×D4—158no—
113D3×D4—1516no—
114#235—158no—
115#235—154no—
116C2×D12—34no—
117C2×D12—158no—
118C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
119C23×C5—92no—
120C22×D5—94no—
121C5×D4—92no—
122C5×D4—92no—
123C2×C4×C5C2×C3⋊D311yescharacteristic, complemented
124C22⋊D5—92no—
125C22⋊D5—94no—
126C22⋊D5—94no—
127C22⋊D5C2×C3⋊D312yescomplemented
128C2×Dic5—94no—
129D20—91no—
130D20C2×C3⋊D311yescharacteristic, complemented
131C4×D5—92no—
132C4×D5C2×C3⋊D312yescomplemented
133Dic10—91no—
134Dic10C2×C3⋊D311yescharacteristic, complemented
135C22×C32—52no—
136C22×C32C22×D511yescharacteristic
137C2×C3⋊D3—42no—
138C2×C3⋊D3—52no—
139C2×C3⋊D3—204no—
140C2×C3⋊D3C22×D514yes—
141C32×C4—52no—
142C32×C4C22×D511yescharacteristic
143C32×C4C22×D511yescharacteristic
144C3⋊Dic3—52no—
145H(5,𝔽2)—451noSylow
146C2×C3×C5—24no—
147C2×C3×C5C23×D314yes—
148D15—68no—
149C3×D5—28no—
150C5×D3—616no—
151C22×D3—38no—
152C22×D3—1516no—
153C3×Q8—54no—
154C3×D4—58no—
155C3×D4—54no—
156C2×C3×C4—58no—
157C2×C3×C4D3×D514yescomplemented
158#67—1516no—
159#67—158no—
160D12—38no—
161D12—38no—
162D12—1516no—
163D12—154no—
164C4×D3—1516no—
165C4×D3—158no—
166Dic6—154no—
167C22×C5—94no—
168C22×C5—362no—
169C22×C5C22×C3⋊D311yescharacteristic
170C2×D5—92no—
171C2×D5—364no—
172C2×D5C22×C3⋊D312yes—
173C4×C5C22×C3⋊D311yescharacteristic
174C4×C5C22×C3⋊D311yescharacteristic
175Dic5—92no—
176Dic5C22×C3⋊D312yes—
177C2×C32—21no—
178C2×C32—102no—
179C2×C32C23×D511yescharacteristic
180C3⋊D3—24no—
181C3⋊D3—102no—
182#41—51no—
183#41—452no—
184#41—452no—
185#41—451no—
186C2×D4—91no—
187C2×D4—452no—
188C2×D4—452no—
189C2×D4—454no—
190C3×C5#100914yescomplemented
191C22×C3—58no—
192C22×C3C2×D3×D514yes—
193C2×D3—316no—
194C2×D3—128no—
195C2×D3—158no—
196C2×D3—6016no—
197C3×C4—58no—
198C3×C4C2×D3×D514yes—
199C3×C4C2×D3×D514yes—
200Dic3—158no—
201C2×C5—21no—
202C2×C5—184no—
203C2×C5C23×C3⋊D311yescharacteristic
204D5—22no—
205D5—182no—
206C32#414411yescharacteristic, complemented, Sylow
207C23—92no—
208C23—454no—
209Q8—51no—
210Q8—451no—
211D4—52no—
212D4—51no—
213D4—92no—
214D4—92no—
215D4—454no—
216D4—454no—
217D4—452no—
218D4—451no—
219C2×C4—52no—
220C2×C4—454no—
221C2×C4—452no—
222C2×C4D5×C3⋊D311yescharacteristic, complemented
223C2×C3—24no—
224C2×C3—108no—
225C2×C3C22×D3×D514yes—
226D3—616no—
227D3—308no—
228C5#6436611yescharacteristic, complemented, Sylow
229C22—52no—
230C22—94no—
231C22—362no—
232C22—452no—
233C22—1804no—
234C22C2×D5×C3⋊D311yescharacteristic
235C4—52no—
236C4—452no—
237C4C2×D5×C3⋊D311yescharacteristic
238C4C2×D5×C3⋊D311yescharacteristic
239C3#9201814yescomplemented
240C2—21no—
241C2—102no—
242C2—184no—
243C2—902no—
244C2C22×D5×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×D5×C3⋊D3
commutator, socle, cosocleC2×C32×C5C24
FittingC2×C32×C4×C5C22
hypercenterC2×C4D5×C3⋊D3
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. #10624283C2semidirect
2. C2×C5×C4⋊(C3⋊D3)C2semidirect
3. C32×#709C2semidirect
4. C3⋊D3×C22⋊D5C2semidirect
5. #10624160C2semidirect
6. D5×C4⋊(C3⋊D3)C2semidirect
7. #10624155C2semidirect
8. #10624151C2semidirect
10. C22×C5×C3⋊D3C22semidirect
11. C2×D5×C3⋊D3C22semidirect
12. C2×C32×C4×C5C22semidirect
13. #67184C22semidirect
14. C4⋊(C3⋊D15)C22semidirect
15. C4×C3⋊D15C22semidirect
16. #67180C22semidirect
17. C5×C4⋊(C3⋊D3)C22semidirect
18. C5×C4⋊(C3⋊D3)C22semidirect
19. C32×C22⋊D5C22semidirect
20. C32×D20C22semidirect
21. C32×C4×D5C22semidirect
22. C32×Dic10C22semidirect
23. #67140C22semidirect
24. #67138C22semidirect
25. C3⋊D3×Dic5C22semidirect
29. C3×#709D3semidirect
53. C2×C4⋊(C3⋊D3)D5semidirect
56. C2×C3×C4×C5C2×D3semidirect
63. C3×C22⋊D5C2×D3semidirect
64. C3×D20C2×D3semidirect
65. C3×C4×D5C2×D3semidirect
66. C3×Dic10C2×D3semidirect
83. #709C3⋊D3semidirect
85. C22×C3⋊D3C2×D5semidirect
90. C2×C32×C4C2×D5semidirect
95. C4⋊(C3⋊D3)C2×D5semidirect
96. C4⋊(C3⋊D3)C2×D5semidirect
118. C32×C5H(5,𝔽2)semidirect
123. C2×C4×C5C2×C3⋊D3semidirect
127. C22⋊D5C2×C3⋊D3semidirect
130. D20C2×C3⋊D3semidirect
132. C4×D5C2×C3⋊D3semidirect
134. Dic10C2×C3⋊D3semidirect
157. C2×C3×C4D3×D5semidirect
190. C3×C5#1009semidirect
206. C32#4144semidirect
222. C2×C4D5×C3⋊D3semidirect
228. C5#64366semidirect
239. C3#92018semidirect

Maximal subgroups

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

subgroupindex
#106242832
C2×C5×C4⋊(C3⋊D3)2
C32×#7092
C3⋊D3×C22⋊D52
#106241602
D5×C4⋊(C3⋊D3)2
#106241552
#106241512
#920183
#643665