Finite Group Databasebeta

← #13088956

Subgroups — #13088956

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#10624223C211yesmaximal, characteristic, complemented
2D12×D15C212yesmaximal, complemented
3C5×#3802C211yesmaximal, characteristic, complemented
4#10624152C211yesmaximal, characteristic, complemented
5C3×#6768C212yesmaximal, complemented
6#10624048C212yesmaximal, complemented
7#10624035C212yesmaximal, complemented
8D3×#1203C214yesmaximal, complemented
9#92027—32nomaximal
10C2×D3×D15C2214yescomplemented
11C2×C5×D32C2212yescomplemented
12C4⋊(C3⋊D15)C2211yescharacteristic, complemented
13C4×C5×C3⋊D3C2211yescharacteristic, complemented
14C3×C4×D15C2212yescomplemented
15C3×C5×D12C2212yescomplemented
16C32×Dic10C2211yescharacteristic, complemented
17#67158C2211yescharacteristic, complemented
18#67157C2211yescharacteristic, complemented
19#67156C2212yescomplemented
20#67152C2214yescomplemented
21C5×#625C2212yescomplemented
22#67140C2212yescomplemented
23#67137C2212yescomplemented
24C3×#1203C2214yescomplemented
25C3×D3×Dic5C2214yescomplemented
26#64320—51nomaximal, Hall
27D4×D15—32no—
28C5×D3×D4—32no—
29C3×#711—32no—
30#6790—34no—
31D3×C22⋊D5—34no—
32C2×#1203—34no—
33#6782—34no—
34D5×D12—32no—
35#6771—32no—
36#6769—32no—
37#6769—61no—
38#6768D312yescomplemented
39C2×C3⋊D15C2312yes—
40C2×C5×C3⋊D3C2311yescharacteristic
41C2×C3×D15C2312yes—
42C2×C3×C5×D3C2314yes—
43D3×D15—44no—
44C5×D32—42no—
45C32×C4×C5C2311yescharacteristic
46C5×C3⋊Dic3C2311yescharacteristic
47C3×Dic15C2312yes—
48C32×Dic5C2312yes—
49#4144—91noHall
50#3832—51no—
51C3×#239—52no—
52#3802—52no—
53#3802D511yescharacteristic, complemented
54D3×D12—54no—
55D3×D12—52no—
56#3798—51no—
57#3798—52no—
58C22×D15—34no—
59C22×C5×D3—34no—
60C2×D3×D5—34no—
61C3×C5×D4—32no—
62#1220—34no—
63D60—32no—
64D60—61no—
65C4×D15C2×D312yescomplemented
66C5×#67—34no—
67C5×D12C2×D312yescomplemented
68C4×C5×D3—32no—
69C4×C5×D3—61no—
70C3×C22⋊D5—34no—
71C2×C3×Dic5—34no—
72C3×C4×D5—32no—
73C3×Dic10—21no—
74C3×Dic10C2×D312yescomplemented
75#1204—32no—
76#1203—32no—
77#1203—34no—
78#1203—34no—
79#1203—62no—
80#1203C2×D314yescomplemented
81#1202—32no—
82#1201—32no—
83#1200—34no—
84#1200—62no—
85D3×Dic5C2×D314yescomplemented
86#1009—152no—
87C2×C32×C5C2411yescharacteristic
88C3⋊D15—22no—
89C5×C3⋊D3—21no—
90C3×D15—22no—
91C3×C5×D3—24no—
92C2×C5×D4—91no—
93C2×C22⋊D5—94no—
94#711—32no—
95#711—92no—
96D4×D5—92no—
97D4×D5—92no—
98#709—91no—
99#709—91no—
100C2×D32—54no—
101C2×D32C2×D512yescomplemented
102C32×Q8—51no—
103C4⋊(C3⋊D3)—52no—
104C4⋊(C3⋊D3)—51no—
105C4×C3⋊D3—52no—
106C4×C3⋊D3C2×D511yescharacteristic, complemented
107C3×D12—54no—
108C3×D12C2×D512yescomplemented
109C3×C4×D3—54no—
110C3×C4×D3—52no—
111#627—51no—
112#626—52no—
113#626—54no—
114#625—51no—
115#625C2×D512yescomplemented
116C22×C3×C5—34no—
117C2×D15—34no—
118C2×D15—62no—
119C2×D15—124no—
120C2×D15C22×D312yes—
121C2×C5×D3—32no—
122C2×C5×D3—61no—
123C2×C5×D3—124no—
124C2×C5×D3C22×D314yes—
125C2×C3×D5—32no—
126D3×D5—124no—
127C3×C4×C5—21no—
128C3×C4×C5C22×D312yes—
129Dic15C22×D312yes—
130C3×Dic5—22no—
131C3×Dic5—32no—
132C3×Dic5C22×D314yes—
133C5×Dic3—32no—
134C5×Dic3—61no—
135C3×#41—152no—
136#239—52no—
137#239—152no—
138#239—301no—
139D3×D4—32no—
140D3×D4—152no—
141D3×D4—154no—
142D3×D4—154no—
143#235—152no—
144#235—154no—
145C2×D12—154no—
146C2×D12—152no—
147C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
148C23×C5—92no—
149C22×D5—94no—
150C5×D4—32no—
151C5×D4—92no—
152C2×C4×C5—91no—
153C22⋊D5—34no—
154C22⋊D5—92no—
155C22⋊D5—94no—
156C22⋊D5—92no—
157C2×Dic5—34no—
158D20—91no—
159D20—91no—
160C4×D5—32no—
161C4×D5—92no—
162Dic10—91no—
163Dic10D3211yescharacteristic, complemented
164C2×C3⋊D3—52no—
165C2×C3⋊D3C22×D511yescharacteristic
166C2×C3×D3—52no—
167C2×C3×D3C22×D514yes—
168D32—42no—
169D32—204no—
170C32×C4—52no—
171C32×C4C22×D511yescharacteristic
172C3⋊Dic3C22×D511yescharacteristic
173C3×Dic3—52no—
174H(5,𝔽2)—451noSylow
175C2×C3×C5—21no—
176C2×C3×C5—64no—
177C2×C3×C5C23×D312yes—
178D15—22no—
179D15—64no—
180D15—122no—
181C3×D5—62no—
182C5×D3—24no—
183C5×D3—62no—
184C5×D3—121no—
185C22×D3—34no—
186C22×D3—154no—
187C22×D3—154no—
188C3×Q8—52no—
189C3×Q8—101no—
190C3×D4—32no—
191C3×D4—154no—
192C2×C3×C4—154no—
193C2×C3×C4—152no—
194#67—34no—
195#67—152no—
196#67—152no—
197#67—154no—
198D12—54no—
199D12—154no—
200D12—152no—
201D12—154no—
202D12—152no—
203D12—301no—
204D12—302no—
205D12D3×D512yescomplemented
206C4×D3—32no—
207C4×D3—54no—
208C4×D3—52no—
209C4×D3—61no—
210C4×D3—154no—
211C4×D3—302no—
212Dic6—152no—
213C22×C5—34no—
214C22×C5—91no—
215C22×C5—362no—
216C2×D5—32no—
217C2×D5—92no—
218C2×D5—364no—
219C4×C5—91no—
220C4×C5C2×D3211yescharacteristic
221Dic5—32no—
222Dic5C2×D3212yes—
223C2×C32C23×D511yescharacteristic
224C3⋊D3—21no—
225C3⋊D3—102no—
226C3×D3—24no—
227C3×D3—102no—
228#41—152no—
229#41—451no—
230#41—451no—
231#41—452no—
232C2×D4—91no—
233C2×D4—452no—
234C2×D4—454no—
235C2×D4—452no—
236C3×C5—21no—
237C3×C5#100912yescomplemented
238C22×C3—34no—
239C22×C3—152no—
240C2×D3—32no—
241C2×D3—52no—
242C2×D3—61no—
243C2×D3—124no—
244C2×D3—154no—
245C2×D3—302no—
246C2×D3—604no—
247C2×D3—604no—
248C2×D3C2×D3×D514yes—
249C3×C4—21no—
250C3×C4—54no—
251C3×C4—102no—
252C3×C4—152no—
253C3×C4C2×D3×D512yes—
254Dic3—32no—
255Dic3—52no—
256Dic3—61no—
257C2×C5—64no—
258C2×C5—181no—
259C2×C5C22×D3211yescharacteristic
260D5—62no—
261D5—182no—
262C32#414411yescharacteristic, complemented, Sylow
263C23—92no—
264C23—454no—
265Q8—51no—
266Q8—451no—
267D4—32no—
268D4—92no—
269D4—154no—
270D4—452no—
271D4—451no—
272D4—454no—
273D4—452no—
274D4—451no—
275C2×C4—91no—
276C2×C4—154no—
277C2×C4—152no—
278C2×C4—452no—
279C2×C3—21no—
280C2×C3—64no—
281C2×C3—302no—
282C2×C3C22×D3×D512yes—
283D3—24no—
284D3—62no—
285D3—102no—
286D3—121no—
287D3—304no—
288D3—602no—
289C5#6432011yescharacteristic, complemented, Sylow
290C22—34no—
291C22—91no—
292C22—152no—
293C22—362no—
294C22—452no—
295C22—1804no—
296C4—52no—
297C4—91no—
298C4—152no—
299C4D32×D511yescharacteristic
300C3—21no—
301C3#9202712yescomplemented
302C2—64no—
303C2—181no—
304C2—302no—
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. #10624223C2semidirect
2. D12×D15C2semidirect
3. C5×#3802C2semidirect
4. #10624152C2semidirect
5. C3×#6768C2semidirect
6. #10624048C2semidirect
7. #10624035C2semidirect
8. D3×#1203C2semidirect
10. C2×D3×D15C22semidirect
11. C2×C5×D32C22semidirect
12. C4⋊(C3⋊D15)C22semidirect
13. C4×C5×C3⋊D3C22semidirect
14. C3×C4×D15C22semidirect
15. C3×C5×D12C22semidirect
16. C32×Dic10C22semidirect
17. #67158C22semidirect
18. #67157C22semidirect
19. #67156C22semidirect
20. #67152C22semidirect
21. C5×#625C22semidirect
22. #67140C22semidirect
23. #67137C22semidirect
24. C3×#1203C22semidirect
25. C3×D3×Dic5C22semidirect
38. #6768D3semidirect
53. #3802D5semidirect
65. C4×D15C2×D3semidirect
67. C5×D12C2×D3semidirect
74. C3×Dic10C2×D3semidirect
80. #1203C2×D3semidirect
85. D3×Dic5C2×D3semidirect
101. C2×D32C2×D5semidirect
106. C4×C3⋊D3C2×D5semidirect
108. C3×D12C2×D5semidirect
115. #625C2×D5semidirect
147. C32×C5H(5,𝔽2)semidirect
163. Dic10D32semidirect
205. D12D3×D5semidirect
237. C3×C5#1009semidirect
262. C32#4144semidirect
289. C5#64320semidirect
301. C3#92027semidirect

Maximal subgroups

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

subgroupindex
#106242232
D12×D152
C5×#38022
#106241522
C3×#67682
#106240482
#106240352
D3×#12032
#920273
#643205