Subgroups — C₅×#64309 #13089322

Order 1440.

624 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3830	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#237	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#627	C₂	1	2	yes	maximal, complemented
4	C₅×#3805	C₂	1	2	yes	maximal, complemented
5	C₅×#3804	C₂	1	4	yes	maximal, complemented
6	C₅×#3798	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#3797	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×D₃×Dic₆	C₂	1	2	yes	maximal, complemented
9	C₅×#1010	—	3	2	no	maximal
10	C₃²×C₅×D₄	C₂²	1	1	yes	characteristic
11	C₅×#638	C₂²	1	2	yes	—
12	C₂×C₅×C₃⋊Dic₃	C₂²	1	2	yes	—
13	C₄×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
14	C₅×#634	C₂²	1	1	yes	characteristic
15	C₃×C₅×#67	C₂²	1	4	yes	—
16	C₂×C₃×C₅×Dic₃	C₂²	1	4	yes	—
17	C₃×C₄×C₅×D₃	C₂²	1	2	yes	—
18	C₃×C₅×Dic₆	C₂²	1	2	yes	—
19	C₅×#627	C₂²	1	4	yes	—
20	C₅×#627	C₂²	1	2	yes	—
21	C₅×#627	C₂²	1	1	yes	characteristic
22	C₅×#626	C₂²	1	2	yes	—
23	C₅×#625	C₂²	1	1	yes	characteristic
24	C₅×#624	C₂²	1	2	yes	—
25	C₅×D₃×Dic₃	C₂²	1	4	yes	—
26	#64309	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₃×C₅×#41	—	3	2	no	—
28	C₅×D₃×Q₈	—	3	2	no	—
29	C₅×D₃×Q₈	—	3	2	no	—
30	C₅×#237	—	3	2	no	—
31	C₅×#237	—	3	4	no	—
32	C₅×#237	—	3	4	no	—
33	C₅×#237	—	6	1	no	—
34	C₅×#237	D₃	1	2	yes	complemented
35	C₅×#235	—	3	4	no	—
36	C₅×#235	—	3	2	no	—
37	C₂×C₅×Dic₆	—	3	4	no	—
38	C₂×C₅×Dic₆	—	3	2	no	—
39	C₂²×C₃²×C₅	C₂³	1	2	yes	—
40	C₂×C₅×C₃⋊D₃	C₂³	1	1	yes	characteristic
41	C₂×C₃×C₅×D₃	C₂³	1	2	yes	—
42	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₅×C₃⋊Dic₃	C₂³	1	2	yes	—
44	C₅×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
45	C₃×C₅×Dic₃	C₂³	1	4	yes	—
46	C₃×C₅×Dic₃	C₂³	1	2	yes	—
47	C₅×2₋¹⁺⁴	—	9	1	no	Hall
48	#3830	C₂×C₅	1	1	yes	characteristic, complemented
49	C₃×#237	C₂×C₅	1	2	yes	complemented
50	C₂×#627	C₂×C₅	1	2	yes	complemented
51	#3805	C₂×C₅	1	2	yes	complemented
52	#3804	C₂×C₅	1	4	yes	complemented
53	#3798	C₂×C₅	1	1	yes	characteristic, complemented
54	#3797	C₂×C₅	1	1	yes	characteristic, complemented
55	D₃×Dic₆	C₂×C₅	1	2	yes	complemented
56	C₃×C₅×Q₈	—	3	2	no	—
57	C₃×C₅×D₄	—	2	1	no	—
58	C₃×C₅×D₄	—	3	4	no	—
59	C₃×C₅×D₄	C₂×D₃	1	2	yes	—
60	C₂×C₃×C₄×C₅	—	3	4	no	—
61	C₂×C₃×C₄×C₅	—	3	2	no	—
62	C₅×#67	—	3	4	no	—
63	C₅×#67	—	3	2	no	—
64	C₅×#67	—	3	2	no	—
65	C₅×#67	—	6	2	no	—
66	C₅×#67	C₂×D₃	1	4	yes	—
67	C₂×C₅×Dic₃	—	3	4	no	—
68	C₂×C₅×Dic₃	—	3	4	no	—
69	C₂×C₅×Dic₃	—	6	2	no	—
70	C₂×C₅×Dic₃	C₂×D₃	1	4	yes	—
71	C₅×D₁₂	—	3	2	no	—
72	C₄×C₅×D₃	—	3	4	no	—
73	C₄×C₅×D₃	—	3	2	no	—
74	C₄×C₅×D₃	—	3	4	no	—
75	C₄×C₅×D₃	—	6	1	no	—
76	C₄×C₅×D₃	C₂×D₃	1	2	yes	—
77	C₅×Dic₆	—	3	2	no	—
78	C₅×Dic₆	—	3	2	no	—
79	C₅×Dic₆	—	3	4	no	—
80	C₅×Dic₆	—	3	4	no	—
81	C₅×Dic₆	—	3	4	no	—
82	C₅×Dic₆	—	6	1	no	—
83	C₅×Dic₆	C₂×D₃	1	2	yes	—
84	#1010	—	3	2	no	—
85	C₂×C₃²×C₅	—	2	2	no	—
86	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
87	C₅×C₃⋊D₃	—	2	1	no	—
88	C₃×C₅×D₃	—	2	2	no	—
89	C₅×#41	—	3	2	no	—
90	C₅×#41	—	9	1	no	—
91	C₅×#41	—	9	4	no	—
92	C₅×#41	—	9	2	no	—
93	C₅×#41	—	9	1	no	—
94	C₂×C₅×Q₈	—	9	2	no	—
95	C₂×C₅×Q₈	—	9	2	no	—
96	C₂×C₅×Q₈	—	9	1	no	—
97	C₃²×D₄	C₂²×C₅	1	1	yes	characteristic
98	#638	C₂²×C₅	1	2	yes	—
99	C₂×C₃⋊Dic₃	C₂²×C₅	1	2	yes	—
100	C₄×C₃⋊D₃	C₂²×C₅	1	1	yes	characteristic
101	#634	C₂²×C₅	1	1	yes	characteristic
102	C₃×#67	C₂²×C₅	1	4	yes	—
103	C₂×C₃×Dic₃	C₂²×C₅	1	4	yes	—
104	C₃×C₄×D₃	C₂²×C₅	1	2	yes	—
105	C₃×Dic₆	C₂²×C₅	1	2	yes	—
106	#627	C₂²×C₅	1	4	yes	—
107	#627	C₂²×C₅	1	2	yes	—
108	#627	C₂²×C₅	1	1	yes	characteristic
109	#626	C₂²×C₅	1	2	yes	—
110	#625	C₂²×C₅	1	1	yes	characteristic
111	#624	C₂²×C₅	1	2	yes	—
112	D₃×Dic₃	C₂²×C₅	1	4	yes	—
113	C₂²×C₃×C₅	—	2	2	no	—
114	C₂²×C₃×C₅	—	3	2	no	—
115	C₂²×C₃×C₅	C₂²×D₃	1	4	yes	—
116	C₂×C₅×D₃	—	3	2	no	—
117	C₂×C₅×D₃	—	6	1	no	—
118	C₂×C₅×D₃	C₂²×D₃	1	2	yes	—
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	—	3	2	no	—
121	C₃×C₄×C₅	—	3	4	no	—
122	C₃×C₄×C₅	C₂²×D₃	1	2	yes	—
123	C₅×Dic₃	—	3	4	no	—
124	C₅×Dic₃	—	3	2	no	—
125	C₅×Dic₃	—	6	2	no	—
126	C₅×Dic₃	—	6	1	no	—
127	C₅×Dic₃	C₂²×D₃	1	2	yes	—
128	C₅×Dic₃	C₂²×D₃	1	4	yes	—
129	C₃×#41	—	3	2	no	—
130	D₃×Q₈	—	3	2	no	—
131	D₃×Q₈	—	3	2	no	—
132	#237	—	3	2	no	—
133	#237	—	3	4	no	—
134	#237	—	3	4	no	—
135	#237	—	6	1	no	—
136	#237	C₅×D₃	1	2	yes	complemented
137	#235	—	3	4	no	—
138	#235	—	3	2	no	—
139	C₂×Dic₆	—	3	4	no	—
140	C₂×Dic₆	—	3	2	no	—
141	C₃²×C₅	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Hall
142	C₅×Q₈	—	3	2	no	—
143	C₅×Q₈	—	9	1	no	—
144	C₅×Q₈	—	9	4	no	—
145	C₅×Q₈	—	9	2	no	—
146	C₅×Q₈	—	9	1	no	—
147	C₅×D₄	—	3	4	no	—
148	C₅×D₄	—	9	2	no	—
149	C₅×D₄	—	9	1	no	—
150	C₅×D₄	—	9	2	no	—
151	C₅×D₄	D₃²	1	1	yes	characteristic
152	C₂×C₄×C₅	—	3	4	no	—
153	C₂×C₄×C₅	—	3	2	no	—
154	C₂×C₄×C₅	—	9	2	no	—
155	C₂×C₄×C₅	—	9	4	no	—
156	C₂×C₄×C₅	—	9	2	no	—
157	C₂×C₄×C₅	—	9	1	no	—
158	C₂²×C₃²	C₂³×C₅	1	2	yes	—
159	C₂×C₃⋊D₃	C₂³×C₅	1	1	yes	characteristic
160	C₂×C₃×D₃	C₂³×C₅	1	2	yes	—
161	C₃²×C₄	C₂³×C₅	1	1	yes	characteristic
162	C₃⋊Dic₃	C₂³×C₅	1	2	yes	—
163	C₃⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
164	C₃×Dic₃	C₂³×C₅	1	4	yes	—
165	C₃×Dic₃	C₂³×C₅	1	2	yes	—
166	2₋¹⁺⁴	—	9	1	no	Sylow
167	C₂×C₃×C₅	—	2	4	no	—
168	C₂×C₃×C₅	—	2	1	no	—
169	C₂×C₃×C₅	—	4	2	no	—
170	C₂×C₃×C₅	—	6	2	no	—
171	C₂×C₃×C₅	C₂³×D₃	1	2	yes	—
172	C₅×D₃	—	2	2	no	—
173	C₅×D₃	—	6	2	no	—
174	C₅×D₃	—	12	1	no	—
175	C₃×Q₈	—	3	2	no	—
176	C₃×D₄	—	2	1	no	—
177	C₃×D₄	—	3	4	no	—
178	C₃×D₄	C₂×C₅×D₃	1	2	yes	—
179	C₂×C₃×C₄	—	3	4	no	—
180	C₂×C₃×C₄	—	3	2	no	—
181	#67	—	3	4	no	—
182	#67	—	3	2	no	—
183	#67	—	3	2	no	—
184	#67	—	6	2	no	—
185	#67	C₂×C₅×D₃	1	4	yes	—
186	C₂×Dic₃	—	3	4	no	—
187	C₂×Dic₃	—	3	4	no	—
188	C₂×Dic₃	—	6	2	no	—
189	C₂×Dic₃	C₂×C₅×D₃	1	4	yes	—
190	D₁₂	—	3	2	no	—
191	C₄×D₃	—	3	4	no	—
192	C₄×D₃	—	3	2	no	—
193	C₄×D₃	—	3	4	no	—
194	C₄×D₃	—	6	1	no	—
195	C₄×D₃	C₂×C₅×D₃	1	2	yes	—
196	Dic₆	—	3	2	no	—
197	Dic₆	—	3	2	no	—
198	Dic₆	—	3	4	no	—
199	Dic₆	—	3	4	no	—
200	Dic₆	—	3	4	no	—
201	Dic₆	—	6	1	no	—
202	Dic₆	C₂×C₅×D₃	1	2	yes	—
203	C₂²×C₅	—	3	2	no	—
204	C₂²×C₅	—	9	1	no	—
205	C₂²×C₅	C₂×D₃²	1	2	yes	—
206	C₄×C₅	—	3	2	no	—
207	C₄×C₅	—	3	4	no	—
208	C₄×C₅	—	9	2	no	—
209	C₄×C₅	—	9	1	no	—
210	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
211	C₂×C₃²	—	2	2	no	—
212	C₂×C₃²	C₂⁴×C₅	1	1	yes	characteristic
213	C₃⋊D₃	—	2	1	no	—
214	C₃×D₃	—	2	2	no	—
215	#41	—	3	2	no	—
216	#41	—	9	1	no	—
217	#41	—	9	4	no	—
218	#41	—	9	2	no	—
219	#41	—	9	1	no	—
220	C₂×Q₈	—	9	2	no	—
221	C₂×Q₈	—	9	2	no	—
222	C₂×Q₈	—	9	1	no	—
223	C₃×C₅	—	2	1	no	—
224	C₃×C₅	#1010	1	2	yes	complemented
225	C₂²×C₃	—	2	2	no	—
226	C₂²×C₃	—	3	2	no	—
227	C₂²×C₃	C₂²×C₅×D₃	1	4	yes	—
228	C₂×D₃	—	3	2	no	—
229	C₂×D₃	—	6	1	no	—
230	C₂×D₃	C₂²×C₅×D₃	1	2	yes	—
231	C₃×C₄	—	2	1	no	—
232	C₃×C₄	—	3	2	no	—
233	C₃×C₄	—	3	4	no	—
234	C₃×C₄	C₂²×C₅×D₃	1	2	yes	—
235	Dic₃	—	3	4	no	—
236	Dic₃	—	3	2	no	—
237	Dic₃	—	6	2	no	—
238	Dic₃	—	6	1	no	—
239	Dic₃	C₂²×C₅×D₃	1	2	yes	—
240	Dic₃	C₂²×C₅×D₃	1	4	yes	—
241	C₂×C₅	—	2	2	no	—
242	C₂×C₅	—	6	2	no	—
243	C₂×C₅	—	18	1	no	—
244	C₂×C₅	C₂²×D₃²	1	1	yes	characteristic
245	C₃²	C₅×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
246	Q₈	—	3	2	no	—
247	Q₈	—	9	1	no	—
248	Q₈	—	9	4	no	—
249	Q₈	—	9	2	no	—
250	Q₈	—	9	1	no	—
251	D₄	—	3	4	no	—
252	D₄	—	9	2	no	—
253	D₄	—	9	1	no	—
254	D₄	—	9	2	no	—
255	D₄	C₅×D₃²	1	1	yes	characteristic
256	C₂×C₄	—	3	4	no	—
257	C₂×C₄	—	3	2	no	—
258	C₂×C₄	—	9	2	no	—
259	C₂×C₄	—	9	4	no	—
260	C₂×C₄	—	9	2	no	—
261	C₂×C₄	—	9	1	no	—
262	C₂×C₃	—	2	4	no	—
263	C₂×C₃	—	2	1	no	—
264	C₂×C₃	—	4	2	no	—
265	C₂×C₃	—	6	2	no	—
266	C₂×C₃	C₂³×C₅×D₃	1	2	yes	—
267	D₃	—	2	2	no	—
268	D₃	—	6	2	no	—
269	D₃	—	12	1	no	—
270	C₅	#64309	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	3	2	no	—
272	C₂²	—	9	1	no	—
273	C₂²	C₂×C₅×D₃²	1	2	yes	—
274	C₄	—	3	2	no	—
275	C₄	—	3	4	no	—
276	C₄	—	9	2	no	—
277	C₄	—	9	1	no	—
278	C₄	C₂×C₅×D₃²	1	1	yes	characteristic
279	C₃	—	2	1	no	—
280	C₃	C₅×#1010	1	2	yes	complemented
281	C₂	—	2	2	no	—
282	C₂	—	6	2	no	—
283	C₂	—	18	1	no	—
284	C₂	C₂²×C₅×D₃²	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₅	C₂²×D₃²
commutator, cosocle	C₂×C₃²	C₂⁴×C₅
Frattini	C₂	C₂²×C₅×D₃²
Fitting	C₃²×C₅×D₄	C₂²
socle	C₂×C₃²×C₅	C₂⁴
hypercenter	C₅×D₄	D₃²
hypocenter	C₃²	C₅×2₋¹⁺⁴
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₅ ◁ C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₅×#3830	C₂	semidirect
2. C₃×C₅×#237	C₂	semidirect
3. C₂×C₅×#627	C₂	semidirect
4. C₅×#3805	C₂	semidirect
5. C₅×#3804	C₂	semidirect
6. C₅×#3798	C₂	semidirect
7. C₅×#3797	C₂	semidirect
8. C₅×D₃×Dic₆	C₂	semidirect
26. #64309	C₅	direct (N × H)
34. C₅×#237	D₃	semidirect
48. #3830	C₂×C₅	semidirect
49. C₃×#237	C₂×C₅	semidirect
50. C₂×#627	C₂×C₅	semidirect
51. #3805	C₂×C₅	semidirect
52. #3804	C₂×C₅	semidirect
53. #3798	C₂×C₅	semidirect
54. #3797	C₂×C₅	semidirect
55. D₃×Dic₆	C₂×C₅	semidirect
136. #237	C₅×D₃	semidirect
141. C₃²×C₅	2₋¹⁺⁴	semidirect
224. C₃×C₅	#1010	semidirect
245. C₃²	C₅×2₋¹⁺⁴	semidirect
270. C₅	#64309	direct (N × H)
280. C₃	C₅×#1010	semidirect

Maximal subgroups

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

subgroup	index
C₅×#3830	2
C₃×C₅×#237	2
C₂×C₅×#627	2
C₅×#3805	2
C₅×#3804	2
C₅×#3798	2
C₅×#3797	2
C₅×D₃×Dic₆	2
C₅×#1010	3
#64309	5

Subgroups — C5×#64309 #13089322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#64309 #13089322