Subgroups — C₅×#64312 #13089325

Order 1440.

710 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×D₄×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#237	C₂	1	2	yes	maximal, complemented
3	C₅×#3811	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#624	C₂	1	2	yes	maximal, complemented
5	C₅×#3804	C₂	1	4	yes	maximal, complemented
6	C₄×C₅×D₃²	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#3799	C₂	1	2	yes	maximal, complemented
8	C₅×#3797	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×D₃×#41	—	3	2	no	maximal
10	C₂²×C₅×C₃⋊D₃	C₂²	1	2	yes	—
11	C₂×C₅×D₃²	C₂²	1	1	yes	characteristic
12	C₃²×C₅×D₄	C₂²	1	1	yes	characteristic, complemented
13	C₅×#638	C₂²	1	2	yes	complemented
14	C₅×C₄⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic, complemented
15	C₄×C₅×C₃⋊D₃	C₂²	1	1	yes	characteristic
16	C₃×C₅×#67	C₂²	1	4	yes	complemented
17	C₂×C₃×C₅×Dic₃	C₂²	1	4	yes	complemented
18	C₃×C₄×C₅×D₃	C₂²	1	2	yes	complemented
19	C₃×C₅×Dic₆	C₂²	1	2	yes	complemented
20	C₅×#627	C₂²	1	2	yes	complemented
21	C₅×#626	C₂²	1	4	yes	complemented
22	C₅×#624	C₂²	1	2	yes	—
23	C₅×#624	C₂²	1	1	yes	characteristic
24	C₅×#624	C₂²	1	4	yes	complemented
25	C₅×D₃×Dic₃	C₂²	1	2	yes	complemented
26	#64312	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₃×C₅×#41	—	3	2	no	—
28	C₅×#239	—	3	2	no	—
29	C₅×D₃×Q₈	—	3	2	no	—
30	C₅×#237	—	3	4	no	—
31	C₅×#237	D₃	1	2	yes	complemented
32	C₅×D₃×D₄	—	3	2	no	—
33	C₅×D₃×D₄	—	3	4	no	—
34	C₅×D₃×D₄	—	6	1	no	—
35	C₅×#235	—	3	4	no	—
36	C₅×#235	—	3	2	no	—
37	C₂×C₄×C₅×D₃	—	3	4	no	—
38	C₂×C₄×C₅×D₃	—	3	2	no	—
39	C₂²×C₃²×C₅	C₂³	1	2	yes	—
40	C₂×C₅×C₃⋊D₃	—	2	4	no	—
41	C₂×C₅×C₃⋊D₃	C₂³	1	2	yes	—
42	C₂×C₅×C₃⋊D₃	C₂³	1	1	yes	characteristic
43	C₂×C₃×C₅×D₃	C₂³	1	2	yes	—
44	C₅×D₃²	—	2	2	no	—
45	C₃²×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₅×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
47	C₃×C₅×Dic₃	C₂³	1	2	yes	—
48	C₃×C₅×Dic₃	C₂³	1	4	yes	—
49	C₂×C₅×#41	—	9	1	no	Hall
50	D₄×C₃⋊D₃	C₂×C₅	1	1	yes	characteristic, complemented
51	C₃×#237	C₂×C₅	1	2	yes	complemented
52	#3811	C₂×C₅	1	2	yes	complemented
53	C₂×#624	C₂×C₅	1	2	yes	complemented
54	#3804	C₂×C₅	1	4	yes	complemented
55	C₄×D₃²	C₂×C₅	1	1	yes	characteristic, complemented
56	#3799	C₂×C₅	1	2	yes	complemented
57	#3797	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂²×C₅×D₃	—	3	4	no	—
59	C₂²×C₅×D₃	—	3	2	no	—
60	C₂²×C₅×D₃	—	6	2	no	—
61	C₃×C₅×Q₈	—	3	2	no	—
62	C₃×C₅×D₄	—	2	1	no	—
63	C₃×C₅×D₄	—	3	4	no	—
64	C₃×C₅×D₄	C₂×D₃	1	2	yes	complemented
65	C₂×C₃×C₄×C₅	—	3	4	no	—
66	C₂×C₃×C₄×C₅	—	3	2	no	—
67	C₅×#67	—	3	4	no	—
68	C₅×#67	—	3	4	no	—
69	C₅×#67	—	6	2	no	—
70	C₅×#67	C₂×D₃	1	4	yes	complemented
71	C₂×C₅×Dic₃	—	3	2	no	—
72	C₂×C₅×Dic₃	C₂×D₃	1	4	yes	complemented
73	C₅×D₁₂	—	3	4	no	—
74	C₅×D₁₂	—	3	2	no	—
75	C₅×D₁₂	—	6	1	no	—
76	C₄×C₅×D₃	—	3	2	no	—
77	C₄×C₅×D₃	—	3	2	no	—
78	C₄×C₅×D₃	—	3	2	no	—
79	C₄×C₅×D₃	—	3	4	no	—
80	C₄×C₅×D₃	—	3	4	no	—
81	C₄×C₅×D₃	—	3	4	no	—
82	C₄×C₅×D₃	—	6	1	no	—
83	C₄×C₅×D₃	C₂×D₃	1	2	yes	complemented
84	C₅×Dic₆	—	3	4	no	—
85	C₅×Dic₆	C₂×D₃	1	2	yes	complemented
86	D₃×#41	—	3	2	no	—
87	C₂×C₃²×C₅	—	2	2	no	—
88	C₂×C₃²×C₅	C₂⁴	1	1	yes	characteristic
89	C₅×C₃⋊D₃	—	2	2	no	—
90	C₅×C₃⋊D₃	#41	1	2	yes	complemented
91	C₃×C₅×D₃	—	2	2	no	—
92	C₅×#41	—	3	2	no	—
93	C₅×#41	—	9	4	no	—
94	C₅×#41	—	9	2	no	—
95	C₂×C₅×Q₈	—	9	1	no	—
96	C₂×C₅×D₄	—	9	2	no	—
97	C₂×C₅×D₄	—	9	1	no	—
98	C₂²×C₄×C₅	—	9	2	no	—
99	C₂²×C₄×C₅	—	9	1	no	—
100	C₂²×C₃⋊D₃	C₂²×C₅	1	2	yes	—
101	C₂×D₃²	C₂²×C₅	1	1	yes	characteristic
102	C₃²×D₄	C₂²×C₅	1	1	yes	characteristic, complemented
103	#638	C₂²×C₅	1	2	yes	complemented
104	C₄⋊(C₃⋊D₃)	C₂²×C₅	1	1	yes	characteristic, complemented
105	C₄×C₃⋊D₃	C₂²×C₅	1	1	yes	characteristic
106	C₃×#67	C₂²×C₅	1	4	yes	complemented
107	C₂×C₃×Dic₃	C₂²×C₅	1	4	yes	complemented
108	C₃×C₄×D₃	C₂²×C₅	1	2	yes	complemented
109	C₃×Dic₆	C₂²×C₅	1	2	yes	complemented
110	#627	C₂²×C₅	1	2	yes	complemented
111	#626	C₂²×C₅	1	4	yes	complemented
112	#624	C₂²×C₅	1	2	yes	—
113	#624	C₂²×C₅	1	1	yes	characteristic
114	#624	C₂²×C₅	1	4	yes	complemented
115	D₃×Dic₃	C₂²×C₅	1	2	yes	complemented
116	C₂²×C₃×C₅	—	2	2	no	—
117	C₂²×C₃×C₅	—	3	2	no	—
118	C₂²×C₃×C₅	C₂²×D₃	1	4	yes	—
119	C₂×C₅×D₃	—	3	2	no	—
120	C₂×C₅×D₃	—	3	4	no	—
121	C₂×C₅×D₃	—	6	1	no	—
122	C₂×C₅×D₃	—	6	2	no	—
123	C₂×C₅×D₃	—	6	4	no	—
124	C₂×C₅×D₃	—	6	8	no	—
125	C₂×C₅×D₃	—	12	4	no	—
126	C₂×C₅×D₃	C₂²×D₃	1	2	yes	—
127	C₃×C₄×C₅	—	2	1	no	—
128	C₃×C₄×C₅	—	3	4	no	—
129	C₃×C₄×C₅	—	3	2	no	—
130	C₃×C₄×C₅	C₂²×D₃	1	2	yes	—
131	C₅×Dic₃	—	3	2	no	—
132	C₅×Dic₃	—	6	1	no	—
133	C₅×Dic₃	C₂²×D₃	1	4	yes	—
134	C₅×Dic₃	C₂²×D₃	1	2	yes	—
135	C₃×#41	—	3	2	no	—
136	#239	—	3	2	no	—
137	D₃×Q₈	—	3	2	no	—
138	#237	—	3	4	no	—
139	#237	C₅×D₃	1	2	yes	complemented
140	D₃×D₄	—	3	2	no	—
141	D₃×D₄	—	3	4	no	—
142	D₃×D₄	—	6	1	no	—
143	#235	—	3	4	no	—
144	#235	—	3	2	no	—
145	C₂×C₄×D₃	—	3	4	no	—
146	C₂×C₄×D₃	—	3	2	no	—
147	C₃²×C₅	C₂×#41	1	1	yes	characteristic, complemented, Hall
148	C₂³×C₅	—	9	1	no	—
149	C₂³×C₅	—	9	2	no	—
150	C₅×Q₈	—	3	2	no	—
151	C₅×Q₈	—	9	2	no	—
152	C₅×D₄	—	3	4	no	—
153	C₅×D₄	—	9	4	no	—
154	C₅×D₄	—	9	2	no	—
155	C₅×D₄	—	9	1	no	—
156	C₅×D₄	D₃²	1	1	yes	characteristic, complemented
157	C₂×C₄×C₅	—	3	4	no	—
158	C₂×C₄×C₅	—	3	2	no	—
159	C₂×C₄×C₅	—	9	1	no	—
160	C₂×C₄×C₅	—	9	2	no	—
161	C₂×C₄×C₅	—	9	4	no	—
162	C₂×C₄×C₅	—	9	2	no	—
163	C₂×C₄×C₅	—	9	1	no	—
164	C₂²×C₃²	C₂³×C₅	1	2	yes	—
165	C₂×C₃⋊D₃	—	2	4	no	—
166	C₂×C₃⋊D₃	C₂³×C₅	1	2	yes	—
167	C₂×C₃⋊D₃	C₂³×C₅	1	1	yes	characteristic
168	C₂×C₃×D₃	C₂³×C₅	1	2	yes	—
169	D₃²	—	2	2	no	—
170	C₃²×C₄	C₂³×C₅	1	1	yes	characteristic
171	C₃⋊Dic₃	C₂³×C₅	1	1	yes	characteristic
172	C₃×Dic₃	C₂³×C₅	1	2	yes	—
173	C₃×Dic₃	C₂³×C₅	1	4	yes	—
174	C₂×#41	—	9	1	no	Sylow
175	C₂×C₃×C₅	—	2	4	no	—
176	C₂×C₃×C₅	—	2	1	no	—
177	C₂×C₃×C₅	—	4	2	no	—
178	C₂×C₃×C₅	—	6	2	no	—
179	C₂×C₃×C₅	C₂³×D₃	1	2	yes	—
180	C₅×D₃	—	2	2	no	—
181	C₅×D₃	—	3	4	no	—
182	C₅×D₃	—	6	2	no	—
183	C₅×D₃	—	6	4	no	—
184	C₅×D₃	—	12	2	no	—
185	C₂²×D₃	—	3	4	no	—
186	C₂²×D₃	—	3	2	no	—
187	C₂²×D₃	—	6	2	no	—
188	C₃×Q₈	—	3	2	no	—
189	C₃×D₄	—	2	1	no	—
190	C₃×D₄	—	3	4	no	—
191	C₃×D₄	C₂×C₅×D₃	1	2	yes	complemented
192	C₂×C₃×C₄	—	3	4	no	—
193	C₂×C₃×C₄	—	3	2	no	—
194	#67	—	3	4	no	—
195	#67	—	3	4	no	—
196	#67	—	6	2	no	—
197	#67	C₂×C₅×D₃	1	4	yes	complemented
198	C₂×Dic₃	—	3	2	no	—
199	C₂×Dic₃	C₂×C₅×D₃	1	4	yes	complemented
200	D₁₂	—	3	4	no	—
201	D₁₂	—	3	2	no	—
202	D₁₂	—	6	1	no	—
203	C₄×D₃	—	3	2	no	—
204	C₄×D₃	—	3	2	no	—
205	C₄×D₃	—	3	2	no	—
206	C₄×D₃	—	3	4	no	—
207	C₄×D₃	—	3	4	no	—
208	C₄×D₃	—	3	4	no	—
209	C₄×D₃	—	6	1	no	—
210	C₄×D₃	C₂×C₅×D₃	1	2	yes	complemented
211	Dic₆	—	3	4	no	—
212	Dic₆	C₂×C₅×D₃	1	2	yes	complemented
213	C₂²×C₅	—	3	2	no	—
214	C₂²×C₅	—	9	1	no	—
215	C₂²×C₅	—	9	2	no	—
216	C₂²×C₅	—	18	4	no	—
217	C₂²×C₅	—	18	2	no	—
218	C₂²×C₅	C₂×D₃²	1	2	yes	—
219	C₄×C₅	—	3	2	no	—
220	C₄×C₅	—	3	4	no	—
221	C₄×C₅	—	9	1	no	—
222	C₄×C₅	C₂×D₃²	1	1	yes	characteristic
223	C₂×C₃²	—	2	2	no	—
224	C₂×C₃²	C₂⁴×C₅	1	1	yes	characteristic
225	C₃⋊D₃	—	2	2	no	—
226	C₃⋊D₃	C₅×#41	1	2	yes	complemented
227	C₃×D₃	—	2	2	no	—
228	#41	—	3	2	no	—
229	#41	—	9	4	no	—
230	#41	—	9	2	no	—
231	C₂×Q₈	—	9	1	no	—
232	C₂×D₄	—	9	2	no	—
233	C₂×D₄	—	9	1	no	—
234	C₂²×C₄	—	9	2	no	—
235	C₂²×C₄	—	9	1	no	—
236	C₃×C₅	—	2	1	no	—
237	C₃×C₅	D₃×#41	1	2	yes	complemented
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	—	3	2	no	—
240	C₂²×C₃	C₂²×C₅×D₃	1	4	yes	—
241	C₂×D₃	—	3	2	no	—
242	C₂×D₃	—	3	4	no	—
243	C₂×D₃	—	6	1	no	—
244	C₂×D₃	—	6	2	no	—
245	C₂×D₃	—	6	4	no	—
246	C₂×D₃	—	6	8	no	—
247	C₂×D₃	—	12	4	no	—
248	C₂×D₃	C₂²×C₅×D₃	1	2	yes	—
249	C₃×C₄	—	2	1	no	—
250	C₃×C₄	—	3	4	no	—
251	C₃×C₄	—	3	2	no	—
252	C₃×C₄	C₂²×C₅×D₃	1	2	yes	—
253	Dic₃	—	3	2	no	—
254	Dic₃	—	6	1	no	—
255	Dic₃	C₂²×C₅×D₃	1	4	yes	—
256	Dic₃	C₂²×C₅×D₃	1	2	yes	—
257	C₂×C₅	—	2	2	no	—
258	C₂×C₅	—	6	2	no	—
259	C₂×C₅	—	9	2	no	—
260	C₂×C₅	—	18	2	no	—
261	C₂×C₅	C₂²×D₃²	1	1	yes	characteristic
262	C₃²	C₂×C₅×#41	1	1	yes	characteristic, complemented, Sylow
263	C₂³	—	9	1	no	—
264	C₂³	—	9	2	no	—
265	Q₈	—	3	2	no	—
266	Q₈	—	9	2	no	—
267	D₄	—	3	4	no	—
268	D₄	—	9	4	no	—
269	D₄	—	9	2	no	—
270	D₄	—	9	1	no	—
271	D₄	C₅×D₃²	1	1	yes	characteristic, complemented
272	C₂×C₄	—	3	4	no	—
273	C₂×C₄	—	3	2	no	—
274	C₂×C₄	—	9	1	no	—
275	C₂×C₄	—	9	2	no	—
276	C₂×C₄	—	9	4	no	—
277	C₂×C₄	—	9	2	no	—
278	C₂×C₄	—	9	1	no	—
279	C₂×C₃	—	2	4	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	—	4	2	no	—
282	C₂×C₃	—	6	2	no	—
283	C₂×C₃	C₂³×C₅×D₃	1	2	yes	—
284	D₃	—	2	2	no	—
285	D₃	—	3	4	no	—
286	D₃	—	6	2	no	—
287	D₃	—	6	4	no	—
288	D₃	—	12	2	no	—
289	C₅	#64312	1	1	yes	characteristic, direct product factor, Sylow
290	C₂²	—	3	2	no	—
291	C₂²	—	9	1	no	—
292	C₂²	—	9	2	no	—
293	C₂²	—	18	4	no	—
294	C₂²	—	18	2	no	—
295	C₂²	C₂×C₅×D₃²	1	2	yes	—
296	C₄	—	3	2	no	—
297	C₄	—	3	4	no	—
298	C₄	—	9	1	no	—
299	C₄	C₂×C₅×D₃²	1	1	yes	characteristic
300	C₃	—	2	1	no	—
301	C₃	C₅×D₃×#41	1	2	yes	complemented
302	C₂	—	2	2	no	—
303	C₂	—	6	2	no	—
304	C₂	—	9	2	no	—
305	C₂	—	18	2	no	—
306	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	C₂×C₅×#41
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×D₄×C₃⋊D₃	C₂	semidirect
2. C₃×C₅×#237	C₂	semidirect
3. C₅×#3811	C₂	semidirect
4. C₂×C₅×#624	C₂	semidirect
5. C₅×#3804	C₂	semidirect
6. C₄×C₅×D₃²	C₂	semidirect
7. C₅×#3799	C₂	semidirect
8. C₅×#3797	C₂	semidirect
12. C₃²×C₅×D₄	C₂²	semidirect
13. C₅×#638	C₂²	semidirect
14. C₅×C₄⋊(C₃⋊D₃)	C₂²	semidirect
16. C₃×C₅×#67	C₂²	semidirect
17. C₂×C₃×C₅×Dic₃	C₂²	semidirect
18. C₃×C₄×C₅×D₃	C₂²	semidirect
19. C₃×C₅×Dic₆	C₂²	semidirect
20. C₅×#627	C₂²	semidirect
21. C₅×#626	C₂²	semidirect
24. C₅×#624	C₂²	semidirect
25. C₅×D₃×Dic₃	C₂²	semidirect
26. #64312	C₅	direct (N × H)
31. C₅×#237	D₃	semidirect
50. D₄×C₃⋊D₃	C₂×C₅	semidirect
51. C₃×#237	C₂×C₅	semidirect
52. #3811	C₂×C₅	semidirect
53. C₂×#624	C₂×C₅	semidirect
54. #3804	C₂×C₅	semidirect
55. C₄×D₃²	C₂×C₅	semidirect
56. #3799	C₂×C₅	semidirect
57. #3797	C₂×C₅	semidirect
64. C₃×C₅×D₄	C₂×D₃	semidirect
70. C₅×#67	C₂×D₃	semidirect
72. C₂×C₅×Dic₃	C₂×D₃	semidirect
83. C₄×C₅×D₃	C₂×D₃	semidirect
85. C₅×Dic₆	C₂×D₃	semidirect
90. C₅×C₃⋊D₃	#41	semidirect
102. C₃²×D₄	C₂²×C₅	semidirect
103. #638	C₂²×C₅	semidirect
104. C₄⋊(C₃⋊D₃)	C₂²×C₅	semidirect
106. C₃×#67	C₂²×C₅	semidirect
107. C₂×C₃×Dic₃	C₂²×C₅	semidirect
108. C₃×C₄×D₃	C₂²×C₅	semidirect
109. C₃×Dic₆	C₂²×C₅	semidirect
110. #627	C₂²×C₅	semidirect
111. #626	C₂²×C₅	semidirect
114. #624	C₂²×C₅	semidirect
115. D₃×Dic₃	C₂²×C₅	semidirect
139. #237	C₅×D₃	semidirect
147. C₃²×C₅	C₂×#41	semidirect
156. C₅×D₄	D₃²	semidirect
191. C₃×D₄	C₂×C₅×D₃	semidirect
197. #67	C₂×C₅×D₃	semidirect
199. C₂×Dic₃	C₂×C₅×D₃	semidirect
210. C₄×D₃	C₂×C₅×D₃	semidirect
212. Dic₆	C₂×C₅×D₃	semidirect
226. C₃⋊D₃	C₅×#41	semidirect
237. C₃×C₅	D₃×#41	semidirect
262. C₃²	C₂×C₅×#41	semidirect
271. D₄	C₅×D₃²	semidirect
289. C₅	#64312	direct (N × H)
301. C₃	C₅×D₃×#41	semidirect

Maximal subgroups

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

subgroup	index
C₅×D₄×C₃⋊D₃	2
C₃×C₅×#237	2
C₅×#3811	2
C₂×C₅×#624	2
C₅×#3804	2
C₄×C₅×D₃²	2
C₅×#3799	2
C₅×#3797	2
C₅×D₃×#41	3
#64312	5

Subgroups — C5×#64312 #13089325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#64312 #13089325