Finite Group Databasebeta

← C3×#11717495 #72231619

Subgroups — C3×#11717495 #72231619

Order 2400.

772 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 216 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×C3×C22⋊(C5⋊D5)C211yesmaximal, characteristic, complemented
2C2×C3×C5×C22⋊D5C212yesmaximal, complemented
3C3×#87886C212yesmaximal, complemented
4C3×D5×C22⋊D5C214yesmaximal, complemented
5C3×#87880C212yesmaximal, complemented
6C3×#87879C214yesmaximal, complemented
7#11717495C311yesmaximal, characteristic, direct product factor, Hall
8C23×C3×C52C2211yescharacteristic, complemented
9C22×C3×C5⋊D5C2211yescharacteristic, complemented
10C22×C3×C5×D5C2212yescomplemented
11C2×C3×D52C2212yescomplemented
12C3×C22⋊(C5⋊D5)C2214yescomplemented
13C2×C3×#1039C2211yescharacteristic, complemented
14C3×C5×C22⋊D5C2214yescomplemented
15C3×C5×C22⋊D5C2214yescomplemented
16C2×C3×C5×Dic5C2212yescomplemented
17C3×#6039C2212yescomplemented
18C3×#6038C2214yescomplemented
19C3×#6037C2212yescomplemented
20C3×#6036C2212yescomplemented
21C3×D5×Dic5C2214yescomplemented
22C3×#4144—52nomaximal
23C2×C22⋊(C5⋊D5)C2×C311yescharacteristic, complemented
24C2×C5×C22⋊D5C2×C312yescomplemented
25#87886C2×C312yescomplemented
26D5×C22⋊D5C2×C314yescomplemented
27#87880C2×C312yescomplemented
28#87879C2×C314yescomplemented
29C22×C3×C52—41no—
30C22×C3×C52C2312yes—
31C22×C3×C52C2311yescharacteristic
32C2×C3×C5⋊D5—41no—
33C2×C3×C5⋊D5C2312yes—
34C2×C3×C5×D5—42no—
35C2×C3×C5×D5C2314yes—
36C3×D52—42no—
37C3×#1039C2312yes—
38C3×C5×Dic5C2314yes—
39C2×C3×C5×D4—52no—
40C2×C3×C22⋊D5—52no—
41C2×C3×C22⋊D5—54no—
42C2×C3×C22⋊D5—102no—
43C2×C3×C22⋊D5D512yescomplemented
44C3×#711—54no—
45C3×#711—54no—
46C3×D4×D5—54no—
47C3×D4×D5—54no—
48C3×#709—54no—
49C23×C52C22×C311yescharacteristic, complemented
50C22×C5⋊D5C22×C311yescharacteristic, complemented
51C22×C5×D5C22×C312yescomplemented
52C2×D52C22×C312yescomplemented
53C22⋊(C5⋊D5)C22×C314yescomplemented
54C2×#1039C22×C311yescharacteristic, complemented
55C5×C22⋊D5C22×C314yescomplemented
56C5×C22⋊D5C22×C314yescomplemented
57C2×C5×Dic5C22×C312yescomplemented
58#6039C22×C312yescomplemented
59#6038C22×C314yescomplemented
60#6037C22×C312yescomplemented
61#6036C22×C312yescomplemented
62D5×Dic5C22×C314yescomplemented
63#4144—52no—
64C2×C3×C52—21no—
65C2×C3×C52—22no—
66C2×C3×C52C2411yescharacteristic
67C3×C5⋊D5—22no—
68C3×C5×D5—24no—
69C23×C3×C5—22no—
70C23×C3×C5—52no—
71C23×C3×C5C2×D512yescomplemented
72C22×C3×D5—52no—
73C22×C3×D5—54no—
74C22×C3×D5—102no—
75C22×C3×D5C2×D512yescomplemented
76C3×C5×D4—54no—
77C3×C5×D4—54no—
78C2×C3×C4×C5—52no—
79C3×C22⋊D5—54no—
80C3×C22⋊D5—54no—
81C3×C22⋊D5—54no—
82C3×C22⋊D5—54no—
83C3×C22⋊D5—108no—
84C3×C22⋊D5C2×D514yescomplemented
85C3×C22⋊D5C2×D514yescomplemented
86C2×C3×Dic5—52no—
87C2×C3×Dic5—54no—
88C2×C3×Dic5—102no—
89C2×C3×Dic5C2×D512yescomplemented
90C3×D20—54no—
91C3×C4×D5—54no—
92C3×C4×D5—54no—
93C3×Dic10—54no—
94C22×C52—41no—
95C22×C52C23×C312yes—
96C22×C52C23×C311yescharacteristic
97C2×C5⋊D5—41no—
98C2×C5⋊D5C23×C312yes—
99C2×C5×D5—42no—
100C2×C5×D5C23×C314yes—
101D52—42no—
102#1039C23×C312yes—
103C5×Dic5C23×C314yes—
104C3×H(5,𝔽2)—251noHall
105C2×C5×D4—52no—
106C2×C22⋊D5—52no—
107C2×C22⋊D5—54no—
108C2×C22⋊D5—102no—
109C2×C22⋊D5C3×D512yescomplemented
110#711—54no—
111#711—54no—
112D4×D5—54no—
113D4×D5—54no—
114#709—54no—
115C3×C52H(5,𝔽2)11yescharacteristic, complemented, Hall
116C22×C3×C5—22no—
117C22×C3×C5—24no—
118C22×C3×C5—42no—
119C22×C3×C5—42no—
120C22×C3×C5—42no—
121C22×C3×C5—54no—
122C22×C3×C5—202no—
123C22×C3×C5C22×D512yes—
124C22×C3×C5C22×D512yes—
125C22×C3×C5C22×D512yes—
126C2×C3×D5—42no—
127C2×C3×D5—54no—
128C2×C3×D5—104no—
129C2×C3×D5—204no—
130C2×C3×D5—202no—
131C2×C3×D5—202no—
132C2×C3×D5—202no—
133C2×C3×D5C22×D514yes—
134C3×C4×C5—54no—
135C3×Dic5—54no—
136C3×Dic5—104no—
137C3×Dic5C22×D514yes—
138C2×C52—21no—
139C2×C52—22no—
140C2×C52C24×C311yescharacteristic
141C5⋊D5—22no—
142C5×D5—24no—
143C3×#41—254no—
144C3×#41—252no—
145C2×C3×D4—52no—
146C2×C3×D4—252no—
147C2×C3×D4—254no—
148C2×C3×D4—251no—
149C23×C5—22no—
150C23×C5—52no—
151C23×C5C2×C3×D512yescomplemented
152C22×D5—52no—
153C22×D5—54no—
154C22×D5—102no—
155C22×D5C2×C3×D512yescomplemented
156C5×D4—54no—
157C5×D4—54no—
158C2×C4×C5—52no—
159C22⋊D5—54no—
160C22⋊D5—54no—
161C22⋊D5—54no—
162C22⋊D5—54no—
163C22⋊D5—108no—
164C22⋊D5C2×C3×D514yescomplemented
165C22⋊D5C2×C3×D514yescomplemented
166C2×Dic5—52no—
167C2×Dic5—54no—
168C2×Dic5—102no—
169C2×Dic5C2×C3×D512yescomplemented
170D20—54no—
171C4×D5—54no—
172C4×D5—54no—
173Dic10—54no—
174H(5,𝔽2)—251noSylow
175C2×C3×C5—22no—
176C2×C3×C5—22no—
177C2×C3×C5—22no—
178C2×C3×C5—22no—
179C2×C3×C5—22no—
180C2×C3×C5—22no—
181C2×C3×C5—44no—
182C2×C3×C5—104no—
183C2×C3×C5C23×D512yes—
184C3×D5—24no—
185C3×D5—104no—
186C3×D5—204no—
187C52C3×H(5,𝔽2)11yescharacteristic, complemented, Sylow
188C23×C3—52no—
189C23×C3—252no—
190C23×C3—251no—
191C23×C3D5211yescharacteristic, complemented
192C3×Q8—252no—
193C3×D4—54no—
194C3×D4—54no—
195C3×D4—254no—
196C3×D4—252no—
197C3×D4—254no—
198C2×C3×C4—52no—
199C2×C3×C4—252no—
200C2×C3×C4—254no—
201C2×C3×C4—251no—
202C22×C5—22no—
203C22×C5—24no—
204C22×C5—42no—
205C22×C5—42no—
206C22×C5—42no—
207C22×C5—54no—
208C22×C5—202no—
209C22×C5C22×C3×D512yes—
210C22×C5C22×C3×D512yes—
211C22×C5C22×C3×D512yes—
212C2×D5—42no—
213C2×D5—54no—
214C2×D5—104no—
215C2×D5—204no—
216C2×D5—202no—
217C2×D5—202no—
218C2×D5—202no—
219C2×D5C22×C3×D514yes—
220C4×C5—54no—
221Dic5—54no—
222Dic5—104no—
223Dic5C22×C3×D514yes—
224#41—254no—
225#41—252no—
226C2×D4—52no—
227C2×D4—252no—
228C2×D4—254no—
229C2×D4—251no—
230C3×C5—22no—
231C3×C5#414412yescomplemented
232C22×C3—41no—
233C22×C3—54no—
234C22×C3—202no—
235C22×C3—252no—
236C22×C3—1001no—
237C22×C3—1002no—
238C22×C3C2×D5212yes—
239C22×C3C2×D5211yescharacteristic
240C3×C4—54no—
241C3×C4—252no—
242C2×C5—22no—
243C2×C5—22no—
244C2×C5—22no—
245C2×C5—22no—
246C2×C5—22no—
247C2×C5—22no—
248C2×C5—44no—
249C2×C5—104no—
250C2×C5C23×C3×D512yes—
251D5—24no—
252D5—104no—
253D5—204no—
254C23—52no—
255C23—252no—
256C23—251no—
257C23C3×D5211yescharacteristic, complemented
258Q8—252no—
259D4—54no—
260D4—54no—
261D4—254no—
262D4—252no—
263D4—254no—
264C2×C4—52no—
265C2×C4—252no—
266C2×C4—254no—
267C2×C4—251no—
268C2×C3—22no—
269C2×C3—21no—
270C2×C3—104no—
271C2×C3—502no—
272C2×C3C22×D5211yescharacteristic
273C5—22no—
274C5C3×#414412yescomplemented
275C22—41no—
276C22—54no—
277C22—202no—
278C22—252no—
279C22—1001no—
280C22—1002no—
281C22C2×C3×D5212yes—
282C22C2×C3×D5211yescharacteristic
283C4—54no—
284C4—252no—
285C3#1171749511yescharacteristic, direct product factor, Sylow
286C2—22no—
287C2—21no—
288C2—104no—
289C2—502no—
290C2C22×C3×D5211yescharacteristic
subgroupG/H
centerC2×C3C22×D52
commutator, cosocleC2×C52C24×C3
FrattiniC2C22×C3×D52
FittingC23×C3×C52C22
socleC2×C3×C52C24
hypercenterC23×C3D52
hypocenterC52C3×H(5,𝔽2)
LCSG ▷ C2×C52 ▷ C52
UCSC1 ◁ C2×C3 ◁ C23×C3

Complements

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

NH ≅ G/Ntype
1. C2×C3×C22⋊(C5⋊D5)C2semidirect
2. C2×C3×C5×C22⋊D5C2semidirect
3. C3×#87886C2semidirect
4. C3×D5×C22⋊D5C2semidirect
5. C3×#87880C2semidirect
6. C3×#87879C2semidirect
7. #11717495C3direct (N × H)
8. C23×C3×C52C22semidirect
9. C22×C3×C5⋊D5C22semidirect
10. C22×C3×C5×D5C22semidirect
11. C2×C3×D52C22semidirect
12. C3×C22⋊(C5⋊D5)C22semidirect
13. C2×C3×#1039C22semidirect
14. C3×C5×C22⋊D5C22semidirect
15. C3×C5×C22⋊D5C22semidirect
16. C2×C3×C5×Dic5C22semidirect
17. C3×#6039C22semidirect
18. C3×#6038C22semidirect
19. C3×#6037C22semidirect
20. C3×#6036C22semidirect
21. C3×D5×Dic5C22semidirect
23. C2×C22⋊(C5⋊D5)C2×C3semidirect
24. C2×C5×C22⋊D5C2×C3semidirect
25. #87886C2×C3semidirect
26. D5×C22⋊D5C2×C3semidirect
27. #87880C2×C3semidirect
28. #87879C2×C3semidirect
43. C2×C3×C22⋊D5D5semidirect
49. C23×C52C22×C3semidirect
50. C22×C5⋊D5C22×C3semidirect
51. C22×C5×D5C22×C3semidirect
52. C2×D52C22×C3semidirect
53. C22⋊(C5⋊D5)C22×C3semidirect
54. C2×#1039C22×C3semidirect
55. C5×C22⋊D5C22×C3semidirect
56. C5×C22⋊D5C22×C3semidirect
57. C2×C5×Dic5C22×C3semidirect
58. #6039C22×C3semidirect
59. #6038C22×C3semidirect
60. #6037C22×C3semidirect
61. #6036C22×C3semidirect
62. D5×Dic5C22×C3semidirect
71. C23×C3×C5C2×D5semidirect
75. C22×C3×D5C2×D5semidirect
84. C3×C22⋊D5C2×D5semidirect
85. C3×C22⋊D5C2×D5semidirect
89. C2×C3×Dic5C2×D5semidirect
109. C2×C22⋊D5C3×D5semidirect
115. C3×C52H(5,𝔽2)semidirect
151. C23×C5C2×C3×D5semidirect
155. C22×D5C2×C3×D5semidirect
164. C22⋊D5C2×C3×D5semidirect
165. C22⋊D5C2×C3×D5semidirect
169. C2×Dic5C2×C3×D5semidirect
187. C52C3×H(5,𝔽2)semidirect
191. C23×C3D52semidirect
231. C3×C5#4144semidirect
257. C23C3×D52semidirect
274. C5C3×#4144semidirect
285. C3#11717495direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C22⋊(C5⋊D5)2
C2×C3×C5×C22⋊D52
C3×#878862
C3×D5×C22⋊D52
C3×#878802
C3×#878792
#117174953
C3×#41445