Finite Group Databasebeta

← #13089433

Subgroups — #13089433

Order 1440.

718 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 718 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
1C2×C5×#638C211yesmaximal, characteristic, complemented
2C2×C3×#1220C212yesmaximal, complemented
3#10624236C212yesmaximal, complemented
4#10624235C214yesmaximal, complemented
5#10624230C214yesmaximal, complemented
6#10624229C212yesmaximal, complemented
7#92049—32nomaximal
8C23×C32×C5C2211yescharacteristic, complemented
9C22×C5×C3⋊D3C2211yescharacteristic, complemented
10C22×C3×D15C2212yescomplemented
11C2×(C3⋊D3)⋊D5C2212yescomplemented
12C5×#638C2214yescomplemented
13C2×C5×C3⋊Dic3C2211yescharacteristic, complemented
14C3×#1220C2214yescomplemented
15C3×#1220C2214yescomplemented
16C2×C3×Dic15C2212yescomplemented
17#67158C2212yescomplemented
18#67157C2212yescomplemented
19#67156C2214yescomplemented
20#67155C2212yescomplemented
21#67154C2214yescomplemented
22#64330—51nomaximal, Hall
23C2×#1220D312yescomplemented
24C2×C5×#67—32no—
25C2×C5×#67—61no—
26C2×C3×C22⋊D5—32no—
27#6790—34no—
28D3×C22⋊D5—34no—
29D5×#67—34no—
30#6782—34no—
31#6780—34no—
32#6779—34no—
33C22×C32×C5—41no—
34C22×C32×C5C2312yes—
35C22×C32×C5C2311yescharacteristic
36C2×C5×C3⋊D3—41no—
37C2×C5×C3⋊D3C2312yes—
38C2×C3×D15—42no—
39C2×C3×D15C2314yes—
40(C3⋊D3)⋊D5—42no—
41C5×C3⋊Dic3C2312yes—
42C3×Dic15C2314yes—
43#4144—91noHall
44C2×#638D511yescharacteristic, complemented
45C2×C3×#67—52no—
46#3811—52no—
47D3×#67—54no—
48#3805—52no—
49#3804—54no—
50C23×C3×C5—21no—
51C23×C3×C5C2×D312yescomplemented
52C22×D15C2×D312yescomplemented
53C22×C5×D3—32no—
54C22×C5×D3—61no—
55C22×C3×D5—32no—
56C2×D3×D5—34no—
57#1220C2×D314yescomplemented
58#1220C2×D314yescomplemented
59C2×Dic15C2×D312yescomplemented
60C5×#67—34no—
61C5×#67—34no—
62C5×#67—64no—
63C2×C5×Dic3—32no—
64C2×C5×Dic3—61no—
65C3×C22⋊D5—34no—
66C3×C22⋊D5—34no—
67C2×C3×Dic5—32no—
68#1204—34no—
69#1203—34no—
70#1202—34no—
71#1201—34no—
72#1200—34no—
73D3×Dic5—34no—
74D5×Dic3—34no—
75#1004—152no—
76C2×C32×C5—21no—
77C2×C32×C5—22no—
78C2×C32×C5C2411yescharacteristic
79C5×C3⋊D3—22no—
80C3×D15—24no—
81C2×C5×D4—91no—
82C2×C22⋊D5—32no—
83C2×C22⋊D5—92no—
84#711—94no—
85D4×D5—94no—
86#709—92no—
87C23×C32C2×D511yescharacteristic, complemented
88C22×C3⋊D3C2×D511yescharacteristic, complemented
89C22×C3×D3—52no—
90C2×D32—52no—
91#638C2×D514yescomplemented
92C2×C3⋊Dic3C2×D511yescharacteristic, complemented
93C3×#67—54no—
94C3×#67—54no—
95C2×C3×Dic3—52no—
96#627—52no—
97#626—54no—
98#625—52no—
99#624—52no—
100D3×Dic3—54no—
101C22×C3×C5—21no—
102C22×C3×C5—22no—
103C22×C3×C5—42no—
104C22×C3×C5—42no—
105C22×C3×C5C22×D312yes—
106C22×C3×C5C22×D312yes—
107C22×C3×C5C22×D312yes—
108C2×D15—42no—
109C2×D15C22×D314yes—
110C2×C5×D3—34no—
111C2×C5×D3—62no—
112C2×C5×D3—122no—
113C2×C5×D3—122no—
114C2×C3×D5—34no—
115C2×C3×D5—122no—
116D3×D5—124no—
117Dic15C22×D314yes—
118C3×Dic5—34no—
119C5×Dic3—34no—
120C5×Dic3—62no—
121C2×C3×D4—152no—
122C2×#67—32no—
123C2×#67—52no—
124C2×#67—61no—
125C2×#67—154no—
126#237—154no—
127#237—154no—
128D3×D4—154no—
129D3×D4—154no—
130#235—154no—
131C32×C5H(5,𝔽2)11yescharacteristic, complemented, Hall
132C23×C5—91no—
133C23×C5D3211yescharacteristic, complemented
134C22×D5—32no—
135C22×D5—92no—
136C5×D4—94no—
137C2×C4×C5—91no—
138C22⋊D5—34no—
139C22⋊D5—34no—
140C22⋊D5—94no—
141C2×Dic5—32no—
142C2×Dic5—92no—
143D20—92no—
144C4×D5—94no—
145Dic10—92no—
146C22×C32—41no—
147C22×C32C22×D512yes—
148C22×C32C22×D511yescharacteristic
149C2×C3⋊D3—41no—
150C2×C3⋊D3C22×D512yes—
151C2×C3×D3—54no—
152C2×C3×D3—202no—
153D32—202no—
154C3⋊Dic3C22×D512yes—
155C3×Dic3—54no—
156H(5,𝔽2)—451noSylow
157C2×C3×C5—22no—
158C2×C3×C5—22no—
159C2×C3×C5—22no—
160C2×C3×C5—22no—
161C2×C3×C5—21no—
162C2×C3×C5—42no—
163C2×C3×C5C23×D312yes—
164D15—24no—
165C3×D5—64no—
166C5×D3—64no—
167C5×D3—122no—
168C23×C3—21no—
169C23×C3—152no—
170C23×C3D3×D512yescomplemented
171C22×D3—32no—
172C22×D3—52no—
173C22×D3—61no—
174C22×D3—154no—
175C3×D4—154no—
176C3×D4—154no—
177C2×C3×C4—152no—
178#67—34no—
179#67—34no—
180#67—54no—
181#67—54no—
182#67—64no—
183#67—154no—
184#67—154no—
185C2×Dic3—32no—
186C2×Dic3—52no—
187C2×Dic3—61no—
188C2×Dic3—154no—
189D12—154no—
190C4×D3—154no—
191C4×D3—154no—
192Dic6—154no—
193C22×C5—41no—
194C22×C5—92no—
195C22×C5—361no—
196C22×C5C2×D3212yes—
197C22×C5C2×D3211yescharacteristic
198C2×D5—34no—
199C2×D5—122no—
200C2×D5—362no—
201C4×C5—92no—
202Dic5—34no—
203C2×C32—21no—
204C2×C32—22no—
205C2×C32C23×D511yescharacteristic
206C3⋊D3—22no—
207C3×D3—104no—
208#41—452no—
209#41—454no—
210C2×D4—91no—
211C2×D4—152no—
212C2×D4—454no—
213C2×D4—452no—
214C3×C5—21no—
215C3×C5#100412yescomplemented
216C22×C3—21no—
217C22×C3—22no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—154no—
221C22×C3—602no—
222C22×C3C2×D3×D512yes—
223C22×C3C2×D3×D512yes—
224C22×C3C2×D3×D512yes—
225C2×D3—34no—
226C2×D3—54no—
227C2×D3—62no—
228C2×D3—122no—
229C2×D3—122no—
230C2×D3—202no—
231C2×D3—604no—
232C3×C4—154no—
233Dic3—34no—
234Dic3—54no—
235Dic3—62no—
236C2×C5—22no—
237C2×C5—21no—
238C2×C5—182no—
239C2×C5C22×D3211yescharacteristic
240D5—64no—
241C32#414411yescharacteristic, complemented, Sylow
242C23—91no—
243C23—152no—
244C23—452no—
245C23(C3⋊D3)⋊D511yescharacteristic, complemented
246Q8—452no—
247D4—94no—
248D4—154no—
249D4—154no—
250D4—452no—
251D4—454no—
252C2×C4—91no—
253C2×C4—152no—
254C2×C4—454no—
255C2×C4—452no—
256C2×C3—22no—
257C2×C3—22no—
258C2×C3—22no—
259C2×C3—22no—
260C2×C3—21no—
261C2×C3—42no—
262C2×C3—304no—
263C2×C3C22×D3×D512yes—
264D3—64no—
265D3—104no—
266D3—122no—
267C5#6433011yescharacteristic, complemented, Sylow
268C22—41no—
269C22—92no—
270C22—154no—
271C22—361no—
272C22—602no—
273C22—1802no—
274C22C2×(C3⋊D3)⋊D511yescharacteristic
275C22C2×(C3⋊D3)⋊D512yes—
276C4—92no—
277C4—154no—
278C3—21no—
279C3#9204912yescomplemented
280C2—21no—
281C2—22no—
282C2—182no—
283C2—304no—
284C2C22×(C3⋊D3)⋊D511yescharacteristic
subgroupG/H
center, FrattiniC2C22×(C3⋊D3)⋊D5
commutator, socle, cosocleC2×C32×C5C24
FittingC23×C32×C5C22
hypercenterC23(C3⋊D3)⋊D5
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×C5×#638C2semidirect
2. C2×C3×#1220C2semidirect
3. #10624236C2semidirect
4. #10624235C2semidirect
5. #10624230C2semidirect
6. #10624229C2semidirect
8. C23×C32×C5C22semidirect
9. C22×C5×C3⋊D3C22semidirect
10. C22×C3×D15C22semidirect
11. C2×(C3⋊D3)⋊D5C22semidirect
12. C5×#638C22semidirect
13. C2×C5×C3⋊Dic3C22semidirect
14. C3×#1220C22semidirect
15. C3×#1220C22semidirect
16. C2×C3×Dic15C22semidirect
17. #67158C22semidirect
18. #67157C22semidirect
19. #67156C22semidirect
20. #67155C22semidirect
21. #67154C22semidirect
23. C2×#1220D3semidirect
44. C2×#638D5semidirect
51. C23×C3×C5C2×D3semidirect
52. C22×D15C2×D3semidirect
57. #1220C2×D3semidirect
58. #1220C2×D3semidirect
59. C2×Dic15C2×D3semidirect
87. C23×C32C2×D5semidirect
88. C22×C3⋊D3C2×D5semidirect
91. #638C2×D5semidirect
92. C2×C3⋊Dic3C2×D5semidirect
131. C32×C5H(5,𝔽2)semidirect
133. C23×C5D32semidirect
170. C23×C3D3×D5semidirect
215. C3×C5#1004semidirect
241. C32#4144semidirect
245. C23(C3⋊D3)⋊D5semidirect
267. C5#64330semidirect
279. C3#92049semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#6382
C2×C3×#12202
#106242362
#106242352
#106242302
#106242292
#920493
#643305