Finite Group Databasebeta

← #13088961

Subgroups — #13088961

Order 1440.

704 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 140 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#10624227C211yesmaximal, characteristic, complemented
2D3×D60C212yesmaximal, complemented
3C5×#3798C211yesmaximal, characteristic, complemented
4#10624152C211yesmaximal, characteristic, complemented
5C3×#6769C212yesmaximal, complemented
6#10624052C212yesmaximal, complemented
7#10624035C212yesmaximal, complemented
8#10624033C214yesmaximal, complemented
9#92019—32nomaximal
10C2×(C3⋊D3)⋊D5C2212yescomplemented
11C2×D3×D15C2214yescomplemented
12C4⋊(C3⋊D15)C2211yescharacteristic, complemented
13C4×C5×C3⋊D3C2211yescharacteristic, complemented
14C3×D60C2212yescomplemented
15C3×C4×C5×D3C2212yescomplemented
16C32×Dic10C2211yescharacteristic, complemented
17#67157C2212yescomplemented
18#67151C2214yescomplemented
19C5×#627C2211yescharacteristic, complemented
20C5×#626C2212yescomplemented
21C5×#625C2211yescharacteristic, complemented
22#67140C2212yescomplemented
23#67137C2212yescomplemented
24C3×#1203C2214yescomplemented
25C3×#1200C2214yescomplemented
26#64320—51nomaximal, Hall
27C2×D60—32no—
28C5×#235—32no—
29C3×#709—32no—
30#6790—34no—
31#6790—34no—
32D3×D20—32no—
33D3×D20—32no—
34D5×D12—34no—
35#6773—34no—
36#6769—32no—
37#6769—61no—
38#6769D312yescomplemented
39C2×C3⋊D15C2312yes—
40C2×C5×C3⋊D3C2311yescharacteristic
41C2×C3×D15C2314yes—
42C2×C3×C5×D3C2312yes—
43(C3⋊D3)⋊D5—42no—
44D3×D15—44no—
45C32×C4×C5C2311yescharacteristic
46C5×C3⋊Dic3C2311yescharacteristic
47C3×C5×Dic3C2312yes—
48C32×Dic5C2312yes—
49#4149—91noHall
50#3832—51no—
51C3×#239—52no—
52#3802—51no—
53#3802—52no—
54D3×D12—52no—
55D3×D12—54no—
56#3798—52no—
57#3798D511yescharacteristic, complemented
58C22×D15—34no—
59C2×D3×D5—34no—
60C2×D3×D5—34no—
61C2×C3×C4×C5—32no—
62D60—34no—
63D60—32no—
64D60—61no—
65D60C2×D312yescomplemented
66C5×#67—32no—
67C5×#67—32no—
68C5×D12—32no—
69C4×C5×D3—32no—
70C4×C5×D3—61no—
71C4×C5×D3C2×D312yescomplemented
72C5×Dic6—32no—
73C3×C22⋊D5—34no—
74C3×D20—32no—
75C3×C4×D5—34no—
76C3×Dic10—21no—
77C3×Dic10C2×D312yescomplemented
78#1203—34no—
79#1203—62no—
80#1203C2×D314yescomplemented
81#1202—34no—
82#1202—34no—
83#1200—34no—
84#1200—62no—
85#1200C2×D314yescomplemented
86#1009—152no—
87C2×C32×C5C2411yescharacteristic
88C3⋊D15—22no—
89C5×C3⋊D3—21no—
90C3×D15—24no—
91C3×C5×D3—22no—
92C5×#41—91no—
93#713—92no—
94D4×D5—94no—
95D4×D5—92no—
96#709—32no—
97#709—91no—
98C2×D20—91no—
99C2×D20—92no—
100C2×D32—52no—
101C2×D32—54no—
102C32×Q8—51no—
103C4⋊(C3⋊D3)—52no—
104C4⋊(C3⋊D3)—51no—
105C4×C3⋊D3—52no—
106C4×C3⋊D3C2×D511yescharacteristic, complemented
107C3×D12—54no—
108C3×D12—52no—
109C3×C4×D3—54no—
110C3×C4×D3C2×D512yescomplemented
111#627C2×D511yescharacteristic, complemented
112#626—54no—
113#626C2×D512yescomplemented
114#625—52no—
115#625C2×D511yescharacteristic, complemented
116C22×C3×C5—32no—
117C2×D15—34no—
118C2×D15—62no—
119C2×D15—124no—
120C2×D15C22×D314yes—
121C2×C5×D3—32no—
122C2×C5×D3—61no—
123C2×C5×D3C22×D312yes—
124C2×C3×D5—34no—
125D3×D5—124no—
126D3×D5—124no—
127C3×C4×C5—21no—
128C3×C4×C5—32no—
129C3×C4×C5C22×D312yes—
130C3×Dic5—22no—
131C3×Dic5C22×D314yes—
132C5×Dic3—32no—
133C5×Dic3—61no—
134C5×Dic3C22×D312yes—
135C3×#41—152no—
136#239—52no—
137#239—152no—
138#239—301no—
139D3×D4—152no—
140D3×D4—154no—
141D3×D4—154no—
142D3×D4—152no—
143#235—32no—
144#235—154no—
145C2×D12—154no—
146C2×D12—152no—
147C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
148C22×D5—92no—
149C22×D5—94no—
150C5×Q8—91no—
151C5×D4—92no—
152C5×D4—91no—
153C2×C4×C5—32no—
154C2×C4×C5—91no—
155C22⋊D5—34no—
156C22⋊D5—92no—
157D20—32no—
158D20—94no—
159D20—92no—
160D20—91no—
161C4×D5—34no—
162C4×D5—92no—
163Dic10D3211yescharacteristic, complemented
164C2×C3⋊D3—52no—
165C2×C3⋊D3C22×D511yescharacteristic
166C2×C3×D3—54no—
167C2×C3×D3C22×D512yes—
168D32—204no—
169D32—202no—
170C32×C4—52no—
171C32×C4C22×D511yescharacteristic
172C3⋊Dic3C22×D511yescharacteristic
173C3×Dic3C22×D512yes—
174H(5,𝔽2)—451noSylow
175C2×C3×C5—21no—
176C2×C3×C5—62no—
177C2×C3×C5C23×D312yes—
178D15—24no—
179D15—64no—
180D15—122no—
181C3×D5—64no—
182C5×D3—22no—
183C5×D3—62no—
184C5×D3—121no—
185C22×D3—154no—
186C22×D3—154no—
187C22×D3—154no—
188C3×Q8—52no—
189C3×Q8—101no—
190C3×D4—154no—
191C3×D4—152no—
192C2×C3×C4—32no—
193C2×C3×C4—154no—
194#67—32no—
195#67—32no—
196#67—154no—
197#67—154no—
198D12—32no—
199D12—54no—
200D12—52no—
201D12—154no—
202D12—154no—
203D12—152no—
204D12—301no—
205D12—302no—
206C4×D3—32no—
207C4×D3—54no—
208C4×D3—61no—
209C4×D3—154no—
210C4×D3—302no—
211C4×D3D3×D512yescomplemented
212Dic6—32no—
213C22×C5—32no—
214C22×C5—91no—
215C2×D5—34no—
216C2×D5—92no—
217C2×D5—362no—
218C2×D5—364no—
219C4×C5—32no—
220C4×C5—91no—
221C4×C5C2×D3211yescharacteristic
222Dic5C2×D3212yes—
223C2×C32C23×D511yescharacteristic
224C3⋊D3—21no—
225C3⋊D3—102no—
226C3×D3—22no—
227C3×D3—104no—
228#41—91no—
229#41—152no—
230#41—451no—
231#41—452no—
232C2×D4—452no—
233C2×D4—451no—
234C2×D4—454no—
235C2×D4—452no—
236C3×C5—21no—
237C3×C5#100912yescomplemented
238C22×C3—32no—
239C22×C3—154no—
240C2×D3—32no—
241C2×D3—54no—
242C2×D3—61no—
243C2×D3—154no—
244C2×D3—302no—
245C2×D3—604no—
246C2×D3—604no—
247C2×D3—604no—
248C2×D3C2×D3×D512yes—
249C3×C4—21no—
250C3×C4—32no—
251C3×C4—54no—
252C3×C4—102no—
253C3×C4C2×D3×D512yes—
254Dic3—32no—
255Dic3—61no—
256Dic3C2×D3×D512yes—
257C2×C5—62no—
258C2×C5—181no—
259C2×C5C22×D3211yescharacteristic
260D5—64no—
261D5—182no—
262C32#414911yescharacteristic, complemented, Sylow
263C23—454no—
264C23—452no—
265Q8—51no—
266Q8—91no—
267D4—92no—
268D4—91no—
269D4—154no—
270D4—152no—
271D4—454no—
272D4—452no—
273D4—452no—
274D4—451no—
275C2×C4—32no—
276C2×C4—91no—
277C2×C4—154no—
278C2×C4—452no—
279C2×C3—21no—
280C2×C3—62no—
281C2×C3—304no—
282C2×C3C22×D3×D512yes—
283D3—22no—
284D3—62no—
285D3—104no—
286D3—121no—
287D3—304no—
288D3—602no—
289C5#6432011yescharacteristic, complemented, Sylow
290C22—32no—
291C22—91no—
292C22—154no—
293C22—452no—
294C22—1804no—
295C22—1802no—
296C4—32no—
297C4—52no—
298C4—91no—
299C4D32×D511yescharacteristic
300C3—21no—
301C3#9201912yescomplemented
302C2—62no—
303C2—181no—
304C2—304no—
305C2—902no—
306C2C2×D32×D511yescharacteristic
subgroupG/H
center, FrattiniC2C2×D32×D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C4×C5C23
hypercenterC4D32×D5
hypocenterC32×C5H(5,𝔽2)
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #10624227C2semidirect
2. D3×D60C2semidirect
3. C5×#3798C2semidirect
4. #10624152C2semidirect
5. C3×#6769C2semidirect
6. #10624052C2semidirect
7. #10624035C2semidirect
8. #10624033C2semidirect
10. C2×(C3⋊D3)⋊D5C22semidirect
11. C2×D3×D15C22semidirect
12. C4⋊(C3⋊D15)C22semidirect
13. C4×C5×C3⋊D3C22semidirect
14. C3×D60C22semidirect
15. C3×C4×C5×D3C22semidirect
16. C32×Dic10C22semidirect
17. #67157C22semidirect
18. #67151C22semidirect
19. C5×#627C22semidirect
20. C5×#626C22semidirect
21. C5×#625C22semidirect
22. #67140C22semidirect
23. #67137C22semidirect
24. C3×#1203C22semidirect
25. C3×#1200C22semidirect
38. #6769D3semidirect
57. #3798D5semidirect
65. D60C2×D3semidirect
71. C4×C5×D3C2×D3semidirect
77. C3×Dic10C2×D3semidirect
80. #1203C2×D3semidirect
85. #1200C2×D3semidirect
106. C4×C3⋊D3C2×D5semidirect
110. C3×C4×D3C2×D5semidirect
111. #627C2×D5semidirect
113. #626C2×D5semidirect
115. #625C2×D5semidirect
147. C32×C5H(5,𝔽2)semidirect
163. Dic10D32semidirect
211. C4×D3D3×D5semidirect
237. C3×C5#1009semidirect
262. C32#4149semidirect
289. C5#64320semidirect
301. C3#92019semidirect

Maximal subgroups

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

subgroupindex
#106242272
D3×D602
C5×#37982
#106241522
C3×#67692
#106240522
#106240352
#106240332
#920193
#643205