Subgroups — C₃×C₅×#4144 #72231614

Order 2400.

772 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×C₂²⋊D₅	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#711	C₂	1	4	yes	maximal, complemented
4	C₃×C₅×D₄×D₅	C₂	1	4	yes	maximal, complemented
5	C₃×C₅×#709	C₂	1	2	yes	maximal, complemented
6	C₅×#4144	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×C₅²	C₂²	1	2	yes	complemented
8	C₂²×C₃×C₅×D₅	C₂²	1	4	yes	complemented
9	C₃×C₅²×D₄	C₂²	1	4	yes	complemented
10	C₂×C₃×C₄×C₅²	C₂²	1	1	yes	characteristic, complemented
11	C₃×C₅×C₂²⋊D₅	C₂²	1	4	yes	complemented
12	C₃×C₅×C₂²⋊D₅	C₂²	1	8	yes	complemented
13	C₂×C₃×C₅×Dic₅	C₂²	1	4	yes	complemented
14	C₃×C₅×D₂₀	C₂²	1	2	yes	complemented
15	C₃×C₄×C₅×D₅	C₂²	1	4	yes	complemented
16	C₃×C₅×Dic₁₀	C₂²	1	2	yes	complemented
17	C₃×C₅×H(5,𝔽₂)	—	5	1	no	maximal
18	C₃×#4144	C₅	1	1	yes	maximal, characteristic, direct product factor
19	C₂×C₅²×D₄	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×C₅×C₂²⋊D₅	C₂×C₃	1	4	yes	complemented
21	C₅×#711	C₂×C₃	1	4	yes	complemented
22	C₅×D₄×D₅	C₂×C₃	1	4	yes	complemented
23	C₅×#709	C₂×C₃	1	2	yes	complemented
24	C₂²×C₃×C₅²	—	4	2	no	—
25	C₂²×C₃×C₅²	C₂³	1	4	yes	—
26	C₂²×C₃×C₅²	C₂³	1	1	yes	characteristic
27	C₂×C₃×C₅×D₅	—	4	4	no	—
28	C₂×C₃×C₅×D₅	C₂³	1	4	yes	—
29	C₃×C₄×C₅²	C₂³	1	2	yes	—
30	C₃×C₅×Dic₅	C₂³	1	4	yes	—
31	C₃×C₅×#41	—	5	4	no	—
32	C₃×C₅×#41	—	5	2	no	—
33	C₂×C₃×C₅×D₄	—	2	2	no	—
34	C₂×C₃×C₅×D₄	—	5	4	no	—
35	C₂×C₃×C₅×D₄	—	5	4	no	—
36	C₂×C₃×C₅×D₄	D₅	1	1	yes	characteristic, complemented
37	C₂×C₃×C₅×D₄	C₂×C₅	1	1	yes	characteristic, complemented
38	C₂×C₃×C₂²⋊D₅	C₂×C₅	1	4	yes	complemented
39	C₃×#711	C₂×C₅	1	4	yes	complemented
40	C₃×D₄×D₅	C₂×C₅	1	4	yes	complemented
41	C₃×#709	C₂×C₅	1	2	yes	complemented
42	C₂³×C₅²	C₂²×C₃	1	2	yes	complemented
43	C₂²×C₅×D₅	C₂²×C₃	1	4	yes	complemented
44	C₅²×D₄	C₂²×C₃	1	4	yes	complemented
45	C₂×C₄×C₅²	C₂²×C₃	1	1	yes	characteristic, complemented
46	C₅×C₂²⋊D₅	C₂²×C₃	1	4	yes	complemented
47	C₅×C₂²⋊D₅	C₂²×C₃	1	8	yes	complemented
48	C₂×C₅×Dic₅	C₂²×C₃	1	4	yes	complemented
49	C₅×D₂₀	C₂²×C₃	1	2	yes	complemented
50	C₄×C₅×D₅	C₂²×C₃	1	4	yes	complemented
51	C₅×Dic₁₀	C₂²×C₃	1	2	yes	complemented
52	C₅×H(5,𝔽₂)	—	5	1	no	—
53	#4144	C₃×C₅	1	1	yes	characteristic, direct product factor
54	C₂×C₃×C₅²	—	2	1	no	—
55	C₂×C₃×C₅²	—	2	4	no	—
56	C₂×C₃×C₅²	C₂⁴	1	1	yes	characteristic
57	C₃×C₅×D₅	—	2	4	no	—
58	C₂³×C₃×C₅	—	2	4	no	—
59	C₂³×C₃×C₅	—	5	4	no	—
60	C₂³×C₃×C₅	C₂×D₅	1	2	yes	complemented
61	C₂³×C₃×C₅	C₂²×C₅	1	2	yes	complemented
62	C₂²×C₃×D₅	C₂²×C₅	1	4	yes	complemented
63	C₃×C₅×Q₈	—	5	2	no	—
64	C₃×C₅×D₄	—	2	8	no	—
65	C₃×C₅×D₄	—	5	8	no	—
66	C₃×C₅×D₄	—	5	4	no	—
67	C₃×C₅×D₄	—	5	2	no	—
68	C₃×C₅×D₄	C₂×D₅	1	4	yes	complemented
69	C₃×C₅×D₄	C₂²×C₅	1	4	yes	complemented
70	C₂×C₃×C₄×C₅	—	2	2	no	—
71	C₂×C₃×C₄×C₅	—	5	4	no	—
72	C₂×C₃×C₄×C₅	—	5	4	no	—
73	C₂×C₃×C₄×C₅	C₂×D₅	1	1	yes	characteristic, complemented
74	C₂×C₃×C₄×C₅	C₂²×C₅	1	1	yes	characteristic, complemented
75	C₃×C₂²⋊D₅	C₂²×C₅	1	4	yes	complemented
76	C₃×C₂²⋊D₅	C₂²×C₅	1	8	yes	complemented
77	C₂×C₃×Dic₅	C₂²×C₅	1	4	yes	complemented
78	C₃×D₂₀	C₂²×C₅	1	2	yes	complemented
79	C₃×C₄×D₅	C₂²×C₅	1	4	yes	complemented
80	C₃×Dic₁₀	C₂²×C₅	1	2	yes	complemented
81	C₂²×C₅²	—	4	2	no	—
82	C₂²×C₅²	C₂³×C₃	1	4	yes	—
83	C₂²×C₅²	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₅×D₅	—	4	4	no	—
85	C₂×C₅×D₅	C₂³×C₃	1	4	yes	—
86	C₄×C₅²	C₂³×C₃	1	2	yes	—
87	C₅×Dic₅	C₂³×C₃	1	4	yes	—
88	C₃×H(5,𝔽₂)	—	5	1	no	Hall
89	C₅×#41	—	5	4	no	—
90	C₅×#41	—	5	2	no	—
91	C₂×C₅×D₄	—	2	2	no	—
92	C₂×C₅×D₄	—	5	4	no	—
93	C₂×C₅×D₄	—	5	4	no	—
94	C₂×C₅×D₄	C₃×D₅	1	1	yes	characteristic, complemented
95	C₂×C₅×D₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
96	C₂×C₂²⋊D₅	C₂×C₃×C₅	1	4	yes	complemented
97	#711	C₂×C₃×C₅	1	4	yes	complemented
98	D₄×D₅	C₂×C₃×C₅	1	4	yes	complemented
99	#709	C₂×C₃×C₅	1	2	yes	complemented
100	C₃×C₅²	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Hall
101	C₂²×C₃×C₅	—	2	2	no	—
102	C₂²×C₃×C₅	—	2	8	no	—
103	C₂²×C₃×C₅	—	4	2	no	—
104	C₂²×C₃×C₅	—	4	2	no	—
105	C₂²×C₃×C₅	—	4	4	no	—
106	C₂²×C₃×C₅	—	4	4	no	—
107	C₂²×C₃×C₅	—	5	4	no	—
108	C₂²×C₃×C₅	—	20	4	no	—
109	C₂²×C₃×C₅	C₂²×D₅	1	4	yes	—
110	C₂²×C₃×C₅	C₂²×D₅	1	1	yes	characteristic
111	C₂²×C₃×C₅	C₂³×C₅	1	4	yes	—
112	C₂²×C₃×C₅	C₂³×C₅	1	1	yes	characteristic
113	C₂×C₃×D₅	—	4	4	no	—
114	C₂×C₃×D₅	C₂³×C₅	1	4	yes	—
115	C₃×C₄×C₅	—	2	4	no	—
116	C₃×C₄×C₅	—	5	4	no	—
117	C₃×C₄×C₅	C₂²×D₅	1	2	yes	—
118	C₃×C₄×C₅	C₂³×C₅	1	2	yes	—
119	C₃×Dic₅	C₂³×C₅	1	4	yes	—
120	C₂×C₅²	—	2	1	no	—
121	C₂×C₅²	—	2	4	no	—
122	C₂×C₅²	C₂⁴×C₃	1	1	yes	characteristic
123	C₅×D₅	—	2	4	no	—
124	C₃×#41	—	5	4	no	—
125	C₃×#41	—	5	2	no	—
126	C₂×C₃×D₄	—	5	4	no	—
127	C₂×C₃×D₄	—	5	4	no	—
128	C₂×C₃×D₄	C₅×D₅	1	1	yes	characteristic, complemented
129	C₂³×C₅	—	2	4	no	—
130	C₂³×C₅	—	5	4	no	—
131	C₂³×C₅	C₂×C₃×D₅	1	2	yes	complemented
132	C₂³×C₅	C₂²×C₃×C₅	1	2	yes	complemented
133	C₂²×D₅	C₂²×C₃×C₅	1	4	yes	complemented
134	C₅×Q₈	—	5	2	no	—
135	C₅×D₄	—	2	8	no	—
136	C₅×D₄	—	5	8	no	—
137	C₅×D₄	—	5	4	no	—
138	C₅×D₄	—	5	2	no	—
139	C₅×D₄	C₂×C₃×D₅	1	4	yes	complemented
140	C₅×D₄	C₂²×C₃×C₅	1	4	yes	complemented
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	5	4	no	—
143	C₂×C₄×C₅	—	5	4	no	—
144	C₂×C₄×C₅	C₂×C₃×D₅	1	1	yes	characteristic, complemented
145	C₂×C₄×C₅	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
146	C₂²⋊D₅	C₂²×C₃×C₅	1	4	yes	complemented
147	C₂²⋊D₅	C₂²×C₃×C₅	1	8	yes	complemented
148	C₂×Dic₅	C₂²×C₃×C₅	1	4	yes	complemented
149	D₂₀	C₂²×C₃×C₅	1	2	yes	complemented
150	C₄×D₅	C₂²×C₃×C₅	1	4	yes	complemented
151	Dic₁₀	C₂²×C₃×C₅	1	2	yes	complemented
152	H(5,𝔽₂)	—	5	1	no	Sylow
153	C₂×C₃×C₅	—	2	2	no	—
154	C₂×C₃×C₅	—	2	2	no	—
155	C₂×C₃×C₅	—	2	1	no	—
156	C₂×C₃×C₅	—	2	4	no	—
157	C₂×C₃×C₅	—	2	4	no	—
158	C₂×C₃×C₅	—	2	1	no	—
159	C₂×C₃×C₅	—	2	2	no	—
160	C₂×C₃×C₅	—	4	8	no	—
161	C₂×C₃×C₅	—	10	4	no	—
162	C₂×C₃×C₅	C₂³×D₅	1	1	yes	characteristic
163	C₂×C₃×C₅	C₂⁴×C₅	1	1	yes	characteristic
164	C₃×D₅	—	2	4	no	—
165	C₅²	C₃×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₃	—	5	4	no	—
167	C₂³×C₃	C₂×C₅×D₅	1	2	yes	complemented
168	C₃×Q₈	—	5	2	no	—
169	C₃×D₄	—	5	8	no	—
170	C₃×D₄	—	5	4	no	—
171	C₃×D₄	—	5	2	no	—
172	C₃×D₄	C₂×C₅×D₅	1	4	yes	complemented
173	C₂×C₃×C₄	—	5	4	no	—
174	C₂×C₃×C₄	—	5	4	no	—
175	C₂×C₃×C₄	C₂×C₅×D₅	1	1	yes	characteristic, complemented
176	C₂²×C₅	—	2	2	no	—
177	C₂²×C₅	—	2	8	no	—
178	C₂²×C₅	—	4	2	no	—
179	C₂²×C₅	—	4	2	no	—
180	C₂²×C₅	—	4	4	no	—
181	C₂²×C₅	—	4	4	no	—
182	C₂²×C₅	—	5	4	no	—
183	C₂²×C₅	—	20	4	no	—
184	C₂²×C₅	C₂²×C₃×D₅	1	4	yes	—
185	C₂²×C₅	C₂²×C₃×D₅	1	1	yes	characteristic
186	C₂²×C₅	C₂³×C₃×C₅	1	4	yes	—
187	C₂²×C₅	C₂³×C₃×C₅	1	1	yes	characteristic
188	C₂×D₅	—	4	4	no	—
189	C₂×D₅	C₂³×C₃×C₅	1	4	yes	—
190	C₄×C₅	—	2	4	no	—
191	C₄×C₅	—	5	4	no	—
192	C₄×C₅	C₂²×C₃×D₅	1	2	yes	—
193	C₄×C₅	C₂³×C₃×C₅	1	2	yes	—
194	Dic₅	C₂³×C₃×C₅	1	4	yes	—
195	#41	—	5	4	no	—
196	#41	—	5	2	no	—
197	C₂×D₄	—	5	4	no	—
198	C₂×D₄	—	5	4	no	—
199	C₂×D₄	C₃×C₅×D₅	1	1	yes	characteristic, complemented
200	C₃×C₅	—	2	2	no	—
201	C₃×C₅	#4144	1	1	yes	characteristic, direct product factor
202	C₃×C₅	C₅×H(5,𝔽₂)	1	1	yes	characteristic, complemented
203	C₂²×C₃	—	4	2	no	—
204	C₂²×C₃	—	5	4	no	—
205	C₂²×C₃	—	20	4	no	—
206	C₂²×C₃	C₂²×C₅×D₅	1	4	yes	—
207	C₂²×C₃	C₂²×C₅×D₅	1	1	yes	characteristic
208	C₃×C₄	—	5	4	no	—
209	C₃×C₄	C₂²×C₅×D₅	1	2	yes	—
210	C₂×C₅	—	2	2	no	—
211	C₂×C₅	—	2	2	no	—
212	C₂×C₅	—	2	1	no	—
213	C₂×C₅	—	2	4	no	—
214	C₂×C₅	—	2	4	no	—
215	C₂×C₅	—	2	1	no	—
216	C₂×C₅	—	2	2	no	—
217	C₂×C₅	—	4	8	no	—
218	C₂×C₅	—	10	4	no	—
219	C₂×C₅	C₂³×C₃×D₅	1	1	yes	characteristic
220	C₂×C₅	C₂⁴×C₃×C₅	1	1	yes	characteristic
221	D₅	—	2	4	no	—
222	C₂³	—	5	4	no	—
223	C₂³	C₂×C₃×C₅×D₅	1	2	yes	complemented
224	Q₈	—	5	2	no	—
225	D₄	—	5	8	no	—
226	D₄	—	5	4	no	—
227	D₄	—	5	2	no	—
228	D₄	C₂×C₃×C₅×D₅	1	4	yes	complemented
229	C₂×C₄	—	5	4	no	—
230	C₂×C₄	—	5	4	no	—
231	C₂×C₄	C₂×C₃×C₅×D₅	1	1	yes	characteristic, complemented
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	2	4	no	—
234	C₂×C₃	—	10	4	no	—
235	C₂×C₃	C₂³×C₅×D₅	1	1	yes	characteristic
236	C₅	—	2	2	no	—
237	C₅	C₃×#4144	1	1	yes	characteristic, direct product factor
238	C₅	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic, complemented
239	C₂²	—	4	2	no	—
240	C₂²	—	5	4	no	—
241	C₂²	—	20	4	no	—
242	C₂²	C₂²×C₃×C₅×D₅	1	4	yes	—
243	C₂²	C₂²×C₃×C₅×D₅	1	1	yes	characteristic
244	C₄	—	5	4	no	—
245	C₄	C₂²×C₃×C₅×D₅	1	2	yes	—
246	C₃	C₅×#4144	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	2	1	no	—
248	C₂	—	2	4	no	—
249	C₂	—	10	4	no	—
250	C₂	C₂³×C₃×C₅×D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅	C₂³×D₅
commutator, cosocle	C₂×C₅	C₂⁴×C₃×C₅
Frattini	C₂	C₂³×C₃×C₅×D₅
Fitting	C₂×C₃×C₅²×D₄	C₂
socle	C₂×C₃×C₅²	C₂⁴
hypercenter	C₂×C₃×C₅×D₄	D₅
hypocenter	C₅	C₃×C₅×H(5,𝔽₂)
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₅ ◁ C₂×C₃×C₅×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅²×D₄	C₂	semidirect
2. C₂×C₃×C₅×C₂²⋊D₅	C₂	semidirect
3. C₃×C₅×#711	C₂	semidirect
4. C₃×C₅×D₄×D₅	C₂	semidirect
5. C₃×C₅×#709	C₂	semidirect
6. C₅×#4144	C₃	direct (N × H)
7. C₂³×C₃×C₅²	C₂²	semidirect
8. C₂²×C₃×C₅×D₅	C₂²	semidirect
9. C₃×C₅²×D₄	C₂²	semidirect
10. C₂×C₃×C₄×C₅²	C₂²	semidirect
11. C₃×C₅×C₂²⋊D₅	C₂²	semidirect
12. C₃×C₅×C₂²⋊D₅	C₂²	semidirect
13. C₂×C₃×C₅×Dic₅	C₂²	semidirect
14. C₃×C₅×D₂₀	C₂²	semidirect
15. C₃×C₄×C₅×D₅	C₂²	semidirect
16. C₃×C₅×Dic₁₀	C₂²	semidirect
18. C₃×#4144	C₅	direct (N × H)
19. C₂×C₅²×D₄	C₂×C₃	semidirect
20. C₂×C₅×C₂²⋊D₅	C₂×C₃	semidirect
21. C₅×#711	C₂×C₃	semidirect
22. C₅×D₄×D₅	C₂×C₃	semidirect
23. C₅×#709	C₂×C₃	semidirect
36. C₂×C₃×C₅×D₄	D₅	semidirect
37. C₂×C₃×C₅×D₄	C₂×C₅	semidirect
38. C₂×C₃×C₂²⋊D₅	C₂×C₅	semidirect
39. C₃×#711	C₂×C₅	semidirect
40. C₃×D₄×D₅	C₂×C₅	semidirect
41. C₃×#709	C₂×C₅	semidirect
42. C₂³×C₅²	C₂²×C₃	semidirect
43. C₂²×C₅×D₅	C₂²×C₃	semidirect
44. C₅²×D₄	C₂²×C₃	semidirect
45. C₂×C₄×C₅²	C₂²×C₃	semidirect
46. C₅×C₂²⋊D₅	C₂²×C₃	semidirect
47. C₅×C₂²⋊D₅	C₂²×C₃	semidirect
48. C₂×C₅×Dic₅	C₂²×C₃	semidirect
49. C₅×D₂₀	C₂²×C₃	semidirect
50. C₄×C₅×D₅	C₂²×C₃	semidirect
51. C₅×Dic₁₀	C₂²×C₃	semidirect
53. #4144	C₃×C₅	direct (N × H)
60. C₂³×C₃×C₅	C₂×D₅	semidirect
61. C₂³×C₃×C₅	C₂²×C₅	semidirect
62. C₂²×C₃×D₅	C₂²×C₅	semidirect
68. C₃×C₅×D₄	C₂×D₅	semidirect
69. C₃×C₅×D₄	C₂²×C₅	semidirect
73. C₂×C₃×C₄×C₅	C₂×D₅	semidirect
74. C₂×C₃×C₄×C₅	C₂²×C₅	semidirect
75. C₃×C₂²⋊D₅	C₂²×C₅	semidirect
76. C₃×C₂²⋊D₅	C₂²×C₅	semidirect
77. C₂×C₃×Dic₅	C₂²×C₅	semidirect
78. C₃×D₂₀	C₂²×C₅	semidirect
79. C₃×C₄×D₅	C₂²×C₅	semidirect
80. C₃×Dic₁₀	C₂²×C₅	semidirect
94. C₂×C₅×D₄	C₃×D₅	semidirect
95. C₂×C₅×D₄	C₂×C₃×C₅	semidirect
96. C₂×C₂²⋊D₅	C₂×C₃×C₅	semidirect
97. #711	C₂×C₃×C₅	semidirect
98. D₄×D₅	C₂×C₃×C₅	semidirect
99. #709	C₂×C₃×C₅	semidirect
100. C₃×C₅²	H(5,𝔽₂)	semidirect
128. C₂×C₃×D₄	C₅×D₅	semidirect
131. C₂³×C₅	C₂×C₃×D₅	semidirect
132. C₂³×C₅	C₂²×C₃×C₅	semidirect
133. C₂²×D₅	C₂²×C₃×C₅	semidirect
139. C₅×D₄	C₂×C₃×D₅	semidirect
140. C₅×D₄	C₂²×C₃×C₅	semidirect
144. C₂×C₄×C₅	C₂×C₃×D₅	semidirect
145. C₂×C₄×C₅	C₂²×C₃×C₅	semidirect
146. C₂²⋊D₅	C₂²×C₃×C₅	semidirect
147. C₂²⋊D₅	C₂²×C₃×C₅	semidirect
148. C₂×Dic₅	C₂²×C₃×C₅	semidirect
149. D₂₀	C₂²×C₃×C₅	semidirect
150. C₄×D₅	C₂²×C₃×C₅	semidirect
151. Dic₁₀	C₂²×C₃×C₅	semidirect
165. C₅²	C₃×H(5,𝔽₂)	semidirect
167. C₂³×C₃	C₂×C₅×D₅	semidirect
172. C₃×D₄	C₂×C₅×D₅	semidirect
175. C₂×C₃×C₄	C₂×C₅×D₅	semidirect
199. C₂×D₄	C₃×C₅×D₅	semidirect
201. C₃×C₅	#4144	direct (N × H)
202. C₃×C₅	C₅×H(5,𝔽₂)	semidirect
223. C₂³	C₂×C₃×C₅×D₅	semidirect
228. D₄	C₂×C₃×C₅×D₅	semidirect
231. C₂×C₄	C₂×C₃×C₅×D₅	semidirect
237. C₅	C₃×#4144	direct (N × H)
238. C₅	C₃×C₅×H(5,𝔽₂)	semidirect
246. C₃	C₅×#4144	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅²×D₄	2
C₂×C₃×C₅×C₂²⋊D₅	2
C₃×C₅×#711	2
C₃×C₅×D₄×D₅	2
C₃×C₅×#709	2
C₅×#4144	3
C₃×C₅×H(5,𝔽₂)	5
C₃×#4144	5

Subgroups — C3×C5×#4144 #72231614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#4144 #72231614