Finite Group Databasebeta

← #13089306

Subgroups — #13089306

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
1C2×#67184C211yesmaximal, characteristic, complemented
2C2×C5×#638C211yesmaximal, characteristic, complemented
3C2×C32×C22⋊D5C211yesmaximal, characteristic, complemented
4#10624173C212yesmaximal, complemented
5C3⋊D3×C22⋊D5C212yesmaximal, complemented
6D5×#638C212yesmaximal, complemented
7#10624165C212yesmaximal, complemented
8#10624163C212yesmaximal, complemented
9#10624162C212yesmaximal, complemented
10#92049—34nomaximal
11C23×C32×C5C2211yescharacteristic, complemented
12C22×C3⋊D15C2211yescharacteristic, complemented
13C22×C5×C3⋊D3C2211yescharacteristic, complemented
14C22×C32×D5C2211yescharacteristic, complemented
15C2×D5×C3⋊D3C2212yescomplemented
16#67184C2212yescomplemented
17#67184C2212yescomplemented
18C2×C3⋊Dic15C2211yescharacteristic, complemented
19C5×#638C2212yescomplemented
20C5×#638C2212yescomplemented
21C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
22C32×C22⋊D5C2212yescomplemented
23C32×C22⋊D5C2212yescomplemented
24C2×C32×Dic5C2211yescharacteristic, complemented
25#67141C2212yescomplemented
26#67140C2212yescomplemented
27#67139C2212yescomplemented
28#67138C2212yescomplemented
29#67137C2212yescomplemented
30C3⋊D3×Dic5C2212yescomplemented
31D5×C3⋊Dic3C2212yescomplemented
32#64361—51nomaximal, Hall
33C2×#1220—34no—
34C2×C5×#67—34no—
35C2×C3×C22⋊D5D314yescomplemented
36#6790—38no—
37D3×C22⋊D5—38no—
38D5×#67—38no—
39#6782—38no—
40#6780—38no—
41#6779—38no—
42C22×C32×C5—41no—
43C22×C32×C5C2311yescharacteristic
44C22×C32×C5C2311yescharacteristic
45C22×C32×C5C2311yescharacteristic
46C2×C3⋊D15—41no—
47C2×C3⋊D15C2312yes—
48C2×C5×C3⋊D3—41no—
49C2×C5×C3⋊D3C2312yes—
50C2×C32×D5—41no—
51C2×C32×D5C2312yes—
52D5×C3⋊D3—42no—
53C3⋊Dic15C2312yes—
54C5×C3⋊Dic3C2312yes—
55C32×Dic5C2312yes—
56#4144—91noHall
57C2×C32×D4—51no—
58C2×#638—51no—
59C2×#638—52no—
60C2×#638D511yescharacteristic, complemented
61#3830—52no—
62#3830—52no—
63D4×C3⋊D3—52no—
64D4×C3⋊D3—52no—
65#3828—52no—
66C23×C3×C5C2×D314yescomplemented
67C22×D15—34no—
68C22×C5×D3—34no—
69C22×C3×D5C2×D314yescomplemented
70C2×D3×D5—38no—
71#1220—38no—
72#1220—38no—
73C2×Dic15—34no—
74C5×#67—38no—
75C5×#67—38no—
76C2×C5×Dic3—34no—
77C3×C22⋊D5C2×D318yescomplemented
78C3×C22⋊D5C2×D318yescomplemented
79C2×C3×Dic5C2×D314yescomplemented
80#1204—38no—
81#1203—38no—
82#1202—38no—
83#1201—38no—
84#1200—38no—
85D3×Dic5—38no—
86D5×Dic3—38no—
87#1004—154no—
88C2×C32×C5—21no—
89C2×C32×C5—21no—
90C2×C32×C5—21no—
91C2×C32×C5C2411yescharacteristic
92C3⋊D15—22no—
93C5×C3⋊D3—22no—
94C32×D5—22no—
95C2×C5×D4—91no—
96C2×C22⋊D5—91no—
97C2×C22⋊D5—92no—
98C2×C22⋊D5C3⋊D311yescharacteristic, complemented
99#711—92no—
100#711—92no—
101D4×D5—92no—
102D4×D5—92no—
103#709—92no—
104C23×C32—51no—
105C23×C32C2×D511yescharacteristic, complemented
106C22×C3⋊D3—51no—
107C22×C3⋊D3—52no—
108C22×C3⋊D3C2×D511yescharacteristic, complemented
109C32×D4—52no—
110C32×D4—52no—
111C2×C32×C4—51no—
112#638—52no—
113#638—52no—
114#638—52no—
115#638—52no—
116#638C2×D512yescomplemented
117#638C2×D512yescomplemented
118C2×C3⋊Dic3—51no—
119C2×C3⋊Dic3—52no—
120C2×C3⋊Dic3C2×D511yescharacteristic, complemented
121C4⋊(C3⋊D3)—52no—
122C4×C3⋊D3—52no—
123C4×C3⋊D3—52no—
124#634—52no—
125C22×C3×C5—44no—
126C22×C3×C5C22×D314yes—
127C22×C3×C5C22×D314yes—
128C22×C3×C5C22×D314yes—
129C2×D15—38no—
130C2×D15—124no—
131C2×C5×D3—38no—
132C2×C5×D3—124no—
133C2×C3×D5—44no—
134C2×C3×D5C22×D318yes—
135D3×D5—128no—
136Dic15—38no—
137C3×Dic5C22×D318yes—
138C5×Dic3—38no—
139C2×C3×D4—54no—
140C2×#67—34no—
141C2×#67—158no—
142C2×#67—154no—
143#237—158no—
144#237—158no—
145D3×D4—158no—
146D3×D4—158no—
147#235—158no—
148C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
149C23×C5—91no—
150C23×C5C2×C3⋊D311yescharacteristic, complemented
151C22×D5—91no—
152C22×D5—92no—
153C22×D5C2×C3⋊D311yescharacteristic, complemented
154C5×D4—92no—
155C5×D4—92no—
156C2×C4×C5—91no—
157C22⋊D5—92no—
158C22⋊D5—92no—
159C22⋊D5—92no—
160C22⋊D5—92no—
161C22⋊D5C2×C3⋊D312yescomplemented
162C22⋊D5C2×C3⋊D312yescomplemented
163C2×Dic5—91no—
164C2×Dic5—92no—
165C2×Dic5C2×C3⋊D311yescharacteristic, complemented
166D20—92no—
167C4×D5—92no—
168C4×D5—92no—
169Dic10—92no—
170C22×C32—41no—
171C22×C32—52no—
172C22×C32—201no—
173C22×C32C22×D511yescharacteristic
174C22×C32C22×D511yescharacteristic
175C22×C32C22×D511yescharacteristic
176C2×C3⋊D3—41no—
177C2×C3⋊D3—52no—
178C2×C3⋊D3—202no—
179C2×C3⋊D3—201no—
180C2×C3⋊D3C22×D512yes—
181C32×C4—52no—
182C3⋊Dic3—52no—
183C3⋊Dic3C22×D512yes—
184H(5,𝔽2)—451noSylow
185C2×C3×C5—24no—
186C2×C3×C5—24no—
187C2×C3×C5—24no—
188C2×C3×C5C23×D314yes—
189D15—68no—
190C3×D5—28no—
191C5×D3—68no—
192C23×C3—54no—
193C23×C3D3×D514yescomplemented
194C22×D3—34no—
195C22×D3—158no—
196C22×D3—154no—
197C3×D4—58no—
198C3×D4—58no—
199C2×C3×C4—54no—
200#67—38no—
201#67—38no—
202#67—158no—
203#67—158no—
204#67—158no—
205#67—158no—
206C2×Dic3—34no—
207C2×Dic3—158no—
208C2×Dic3—154no—
209D12—158no—
210C4×D3—158no—
211C4×D3—158no—
212Dic6—158no—
213C22×C5—41no—
214C22×C5—92no—
215C22×C5—361no—
216C22×C5C22×C3⋊D311yescharacteristic
217C22×C5C22×C3⋊D311yescharacteristic
218C22×C5C22×C3⋊D311yescharacteristic
219C2×D5—41no—
220C2×D5—92no—
221C2×D5—362no—
222C2×D5—361no—
223C2×D5C22×C3⋊D312yes—
224C4×C5—92no—
225Dic5—92no—
226Dic5C22×C3⋊D312yes—
227C2×C32—21no—
228C2×C32—21no—
229C2×C32—21no—
230C2×C32—102no—
231C2×C32C23×D511yescharacteristic
232C3⋊D3—22no—
233C3⋊D3—102no—
234#41—452no—
235#41—452no—
236#41—452no—
237C2×D4—51no—
238C2×D4—91no—
239C2×D4—452no—
240C2×D4—452no—
241C2×D4—452no—
242C2×D4—451no—
243C3×C5#100414yescomplemented
244C22×C3—44no—
245C22×C3—58no—
246C22×C3—204no—
247C22×C3C2×D3×D514yes—
248C22×C3C2×D3×D514yes—
249C22×C3C2×D3×D514yes—
250C2×D3—38no—
251C2×D3—124no—
252C2×D3—158no—
253C2×D3—608no—
254C2×D3—604no—
255C3×C4—58no—
256Dic3—38no—
257Dic3—158no—
258C2×C5—21no—
259C2×C5—21no—
260C2×C5—21no—
261C2×C5—182no—
262C2×C5C23×C3⋊D311yescharacteristic
263D5—22no—
264D5—182no—
265C32#414411yescharacteristic, complemented, Sylow
266C23—51no—
267C23—91no—
268C23—452no—
269C23—451no—
270C23D5×C3⋊D311yescharacteristic, complemented
271Q8—452no—
272D4—52no—
273D4—52no—
274D4—92no—
275D4—92no—
276D4—452no—
277D4—452no—
278D4—452no—
279D4—452no—
280D4—452no—
281C2×C4—51no—
282C2×C4—91no—
283C2×C4—452no—
284C2×C4—452no—
285C2×C4—452no—
286C2×C4—451no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—24no—
290C2×C3—108no—
291C2×C3C22×D3×D514yes—
292D3—68no—
293D3—308no—
294C5#6436111yescharacteristic, complemented, Sylow
295C22—41no—
296C22—52no—
297C22—92no—
298C22—201no—
299C22—361no—
300C22—452no—
301C22—1801no—
302C22—1802no—
303C22C2×D5×C3⋊D311yescharacteristic
304C22C2×D5×C3⋊D311yescharacteristic
305C22C2×D5×C3⋊D311yescharacteristic
306C4—52no—
307C4—92no—
308C4—452no—
309C3#9204914yescomplemented
310C2—21no—
311C2—21no—
312C2—21no—
313C2—102no—
314C2—182no—
315C2—902no—
316C2C22×D5×C3⋊D311yescharacteristic
subgroupG/H
center, FrattiniC2C22×D5×C3⋊D3
commutator, socle, cosocleC2×C32×C5C24
FittingC23×C32×C5C22
hypercenterC23D5×C3⋊D3
hypocenterC32×C5H(5,𝔽2)
LCSG ▷ C2×C32×C5 ▷ C32×C5
UCSC1 ◁ C2 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#67184C2semidirect
2. C2×C5×#638C2semidirect
3. C2×C32×C22⋊D5C2semidirect
4. #10624173C2semidirect
5. C3⋊D3×C22⋊D5C2semidirect
6. D5×#638C2semidirect
7. #10624165C2semidirect
8. #10624163C2semidirect
9. #10624162C2semidirect
11. C23×C32×C5C22semidirect
12. C22×C3⋊D15C22semidirect
13. C22×C5×C3⋊D3C22semidirect
14. C22×C32×D5C22semidirect
15. C2×D5×C3⋊D3C22semidirect
16. #67184C22semidirect
17. #67184C22semidirect
18. C2×C3⋊Dic15C22semidirect
19. C5×#638C22semidirect
20. C5×#638C22semidirect
21. C2×C5×C3⋊Dic3C22semidirect
22. C32×C22⋊D5C22semidirect
23. C32×C22⋊D5C22semidirect
24. C2×C32×Dic5C22semidirect
25. #67141C22semidirect
26. #67140C22semidirect
27. #67139C22semidirect
28. #67138C22semidirect
29. #67137C22semidirect
30. C3⋊D3×Dic5C22semidirect
31. D5×C3⋊Dic3C22semidirect
35. C2×C3×C22⋊D5D3semidirect
60. C2×#638D5semidirect
66. C23×C3×C5C2×D3semidirect
69. C22×C3×D5C2×D3semidirect
77. C3×C22⋊D5C2×D3semidirect
78. C3×C22⋊D5C2×D3semidirect
79. C2×C3×Dic5C2×D3semidirect
98. C2×C22⋊D5C3⋊D3semidirect
105. C23×C32C2×D5semidirect
108. C22×C3⋊D3C2×D5semidirect
116. #638C2×D5semidirect
117. #638C2×D5semidirect
120. C2×C3⋊Dic3C2×D5semidirect
148. C32×C5H(5,𝔽2)semidirect
150. C23×C5C2×C3⋊D3semidirect
153. C22×D5C2×C3⋊D3semidirect
161. C22⋊D5C2×C3⋊D3semidirect
162. C22⋊D5C2×C3⋊D3semidirect
165. C2×Dic5C2×C3⋊D3semidirect
193. C23×C3D3×D5semidirect
243. C3×C5#1004semidirect
265. C32#4144semidirect
270. C23D5×C3⋊D3semidirect
294. C5#64361semidirect
309. C3#92049semidirect

Maximal subgroups

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

subgroupindex
C2×#671842
C2×C5×#6382
C2×C32×C22⋊D52
#106241732
C3⋊D3×C22⋊D52
D5×#6382
#106241652
#106241632
#106241622
#920493
#643615