Finite Group Databasebeta

← #13089417

Subgroups — #13089417

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×#3830C211yesmaximal, characteristic, complemented
2C3×D4×D15C212yesmaximal, complemented
3#10624235C214yesmaximal, complemented
4C2×#67157C212yesmaximal, complemented
5#10624229C212yesmaximal, complemented
6#10624227C211yesmaximal, characteristic, complemented
7#10624223C211yesmaximal, characteristic, complemented
8#10624221C212yesmaximal, complemented
9#92026—32nomaximal
10C22×C3×D15C2214yescomplemented
11C2×(C3⋊D3)⋊D5C2212yescomplemented
12C32×C5×D4C2211yescharacteristic, complemented
13C5×#638C2212yescomplemented
14C2×C5×C3⋊Dic3C2212yescomplemented
15C4×C5×C3⋊D3C2211yescharacteristic, complemented
16C5×#634C2211yescharacteristic, complemented
17C3×#1220C2214yescomplemented
18C3×D60C2212yescomplemented
19C3×C4×D15C2212yescomplemented
20#67158C2211yescharacteristic, complemented
21#67157C2214yescomplemented
22#67157C2211yescharacteristic, complemented
23#67157C2212yescomplemented
24#67156C2212yescomplemented
25#67154C2214yescomplemented
26#64313—51nomaximal, Hall
27D4×D15D312yescomplemented
28C5×#237—32no—
29C5×#237—61no—
30C3×D4×D5—32no—
31#6790—34no—
32D5×#67—34no—
33C2×#1202—34no—
34#6779—34no—
35D3×D20—32no—
36#6773—32no—
37#6771—32no—
38#6766—32no—
39C22×C32×C5C2312yes—
40C2×C5×C3⋊D3C2311yescharacteristic
41C2×C3×D15—44no—
42C2×C3×D15C2314yes—
43C2×C3×D15C2312yes—
44(C3⋊D3)⋊D5—42no—
45C32×C4×C5C2311yescharacteristic
46C5×C3⋊Dic3C2312yes—
47C5×C3⋊Dic3C2311yescharacteristic
48C3×Dic15C2312yes—
49#4149—91noHall
50#3830D511yescharacteristic, complemented
51C3×D3×D4—52no—
52D3×#67—54no—
53C2×#625—52no—
54#3805—52no—
55#3802—51no—
56#3798—51no—
57#3795—52no—
58C22×D15C2×D314yescomplemented
59C22×C3×D5—34no—
60C2×D3×D5—34no—
61C3×C5×D4—21no—
62C3×C5×D4C2×D312yescomplemented
63#1220C2×D314yescomplemented
64D60C2×D312yescomplemented
65C4×D15C2×D312yescomplemented
66C5×#67—34no—
67C5×#67—62no—
68C2×C5×Dic3—34no—
69C2×C5×Dic3—62no—
70C4×C5×D3—32no—
71C4×C5×D3—61no—
72C5×Dic6—32no—
73C5×Dic6—61no—
74C3×C22⋊D5—34no—
75C3×D20—32no—
76C3×C4×D5—32no—
77#1204—32no—
78#1203—32no—
79#1202—34no—
80#1202—32no—
81#1202—34no—
82#1202—34no—
83#1201—32no—
84#1200—34no—
85D5×Dic3—34no—
86#1004—152no—
87C2×C32×C5—22no—
88C2×C32×C5C2411yescharacteristic
89C5×C3⋊D3—21no—
90C3×D15—24no—
91C3×D15—22no—
92C5×#41—91no—
93#713—92no—
94D4×D5—32no—
95D4×D5—94no—
96#709—91no—
97#709—92no—
98C2×D20—91no—
99C2×D20—92no—
100C22×C3×D3—54no—
101C2×D32—52no—
102C32×D4C2×D511yescharacteristic, complemented
103#638C2×D512yescomplemented
104C2×C3⋊Dic3C2×D512yescomplemented
105C4×C3⋊D3C2×D511yescharacteristic, complemented
106#634C2×D511yescharacteristic, complemented
107C3×#67—54no—
108C3×D12—52no—
109C3×C4×D3—52no—
110#627—51no—
111#626—52no—
112#625—54no—
113#625—51no—
114#625—52no—
115D3×Dic3—54no—
116C22×C3×C5—22no—
117C22×C3×C5C22×D314yes—
118C2×D15—44no—
119C2×D15C22×D314yes—
120C2×D15C22×D312yes—
121C2×C5×D3—32no—
122C2×C5×D3—61no—
123C2×C3×D5—34no—
124C2×C3×D5—32no—
125C2×C3×D5—124no—
126D3×D5—124no—
127C3×C4×C5—21no—
128C3×C4×C5C22×D312yes—
129Dic15C22×D312yes—
130C3×Dic5—32no—
131C5×Dic3—34no—
132C5×Dic3—32no—
133C5×Dic3—62no—
134C5×Dic3—61no—
135C2×C3×D4—152no—
136C2×#67—154no—
137C2×#67—154no—
138#237—32no—
139#237—61no—
140#237—152no—
141#237—154no—
142D3×D4—52no—
143D3×D4—152no—
144D3×D4—154no—
145#235—152no—
146#235—152no—
147C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
148C22×D5—34no—
149C22×D5—92no—
150C5×Q8—91no—
151C5×D4—92no—
152C5×D4D3211yescharacteristic, complemented
153C2×C4×C5—92no—
154C2×C4×C5—91no—
155C22⋊D5—34no—
156C22⋊D5—92no—
157D20—32no—
158D20—91no—
159D20—92no—
160D20—94no—
161C4×D5—32no—
162C4×D5—94no—
163Dic10—91no—
164C22×C32C22×D512yes—
165C2×C3⋊D3C22×D511yescharacteristic
166C2×C3×D3—54no—
167C2×C3×D3—52no—
168C2×C3×D3—204no—
169D32—202no—
170C32×C4C22×D511yescharacteristic
171C3⋊Dic3C22×D512yes—
172C3⋊Dic3C22×D511yescharacteristic
173C3×Dic3—52no—
174H(5,𝔽2)—451noSylow
175C2×C3×C5—24no—
176C2×C3×C5—21no—
177C2×C3×C5—42no—
178C2×C3×C5C23×D312yes—
179D15—24no—
180D15—22no—
181C3×D5—64no—
182C3×D5—62no—
183C5×D3—62no—
184C5×D3—121no—
185C23×C3—154no—
186C22×D3—54no—
187C22×D3—154no—
188C3×D4—21no—
189C3×D4—154no—
190C3×D4—152no—
191C3×D4D3×D512yescomplemented
192C2×C3×C4—152no—
193#67—34no—
194#67—54no—
195#67—62no—
196#67—154no—
197#67—152no—
198#67—154no—
199#67—154no—
200#67—152no—
201C2×Dic3—34no—
202C2×Dic3—62no—
203C2×Dic3—154no—
204D12—52no—
205D12—152no—
206C4×D3—32no—
207C4×D3—52no—
208C4×D3—61no—
209C4×D3—154no—
210Dic6—32no—
211Dic6—61no—
212Dic6—152no—
213C22×C5—91no—
214C22×C5C2×D3212yes—
215C2×D5—32no—
216C2×D5—34no—
217C2×D5—124no—
218C2×D5—362no—
219C4×C5—91no—
220C4×C5—92no—
221C4×C5C2×D3211yescharacteristic
222Dic5—32no—
223C2×C32—22no—
224C2×C32C23×D511yescharacteristic
225C3⋊D3—21no—
226C3×D3—104no—
227C3×D3—102no—
228#41—91no—
229#41—451no—
230#41—452no—
231#41—452no—
232C2×D4—152no—
233C2×D4—451no—
234C2×D4—452no—
235C2×D4—454no—
236C3×C5—21no—
237C3×C5#100412yescomplemented
238C22×C3—22no—
239C22×C3—154no—
240C22×C3—152no—
241C22×C3—604no—
242C22×C3C2×D3×D514yes—
243C2×D3—32no—
244C2×D3—52no—
245C2×D3—54no—
246C2×D3—61no—
247C2×D3—204no—
248C2×D3—604no—
249C3×C4—21no—
250C3×C4—152no—
251C3×C4C2×D3×D512yes—
252Dic3—34no—
253Dic3—32no—
254Dic3—52no—
255Dic3—62no—
256Dic3—61no—
257C2×C5—22no—
258C2×C5—181no—
259C2×C5C22×D3211yescharacteristic
260D5—62no—
261D5—64no—
262C32#414911yescharacteristic, complemented, Sylow
263C23—154no—
264C23—452no—
265Q8—91no—
266Q8—451no—
267D4—92no—
268D4—154no—
269D4—152no—
270D4—451no—
271D4—452no—
272D4—454no—
273D4—452no—
274D4(C3⋊D3)⋊D511yescharacteristic, complemented
275C2×C4—92no—
276C2×C4—91no—
277C2×C4—152no—
278C2×C4—454no—
279C2×C3—24no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3—304no—
283C2×C3—302no—
284C2×C3C22×D3×D512yes—
285D3—62no—
286D3—102no—
287D3—104no—
288D3—121no—
289C5#6431311yescharacteristic, complemented, Sylow
290C22—91no—
291C22—152no—
292C22—154no—
293C22—604no—
294C22—1802no—
295C22C2×(C3⋊D3)⋊D512yes—
296C4—91no—
297C4—92no—
298C4—152no—
299C4C2×(C3⋊D3)⋊D511yescharacteristic
300C3—21no—
301C3#9202612yescomplemented
302C2—22no—
303C2—181no—
304C2—302no—
305C2—304no—
306C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
center, FrattiniC2C22×(C3⋊D3)⋊D5
commutator, socle, cosocleC2×C32×C5C24
FittingC32×C5×D4C22
hypercenterD4(C3⋊D3)⋊D5
hypocenterC32×C5H(5,𝔽2)
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C5×#3830C2semidirect
2. C3×D4×D15C2semidirect
3. #10624235C2semidirect
4. C2×#67157C2semidirect
5. #10624229C2semidirect
6. #10624227C2semidirect
7. #10624223C2semidirect
8. #10624221C2semidirect
10. C22×C3×D15C22semidirect
11. C2×(C3⋊D3)⋊D5C22semidirect
12. C32×C5×D4C22semidirect
13. C5×#638C22semidirect
14. C2×C5×C3⋊Dic3C22semidirect
15. C4×C5×C3⋊D3C22semidirect
16. C5×#634C22semidirect
17. C3×#1220C22semidirect
18. C3×D60C22semidirect
19. C3×C4×D15C22semidirect
20. #67158C22semidirect
21. #67157C22semidirect
22. #67157C22semidirect
23. #67157C22semidirect
24. #67156C22semidirect
25. #67154C22semidirect
27. D4×D15D3semidirect
50. #3830D5semidirect
58. C22×D15C2×D3semidirect
62. C3×C5×D4C2×D3semidirect
63. #1220C2×D3semidirect
64. D60C2×D3semidirect
65. C4×D15C2×D3semidirect
102. C32×D4C2×D5semidirect
103. #638C2×D5semidirect
104. C2×C3⋊Dic3C2×D5semidirect
105. C4×C3⋊D3C2×D5semidirect
106. #634C2×D5semidirect
147. C32×C5H(5,𝔽2)semidirect
152. C5×D4D32semidirect
191. C3×D4D3×D5semidirect
237. C3×C5#1004semidirect
262. C32#4149semidirect
274. D4(C3⋊D3)⋊D5semidirect
289. C5#64313semidirect
301. C3#92026semidirect

Maximal subgroups

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

subgroupindex
C5×#38302
C3×D4×D152
#106242352
C2×#671572
#106242292
#106242272
#106242232
#106242212
#920263
#643135