Subgroups — C₃×C₅×#895 #13087907

Order 1440.

402 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×D₁₂	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₅×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₄×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#121	—	3	1	no	maximal
7	C₅×#895	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×C₅×D₃	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
11	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic
13	C₃×C₅×D₁₂	—	2	4	no	—
14	C₃×C₅×D₁₂	—	2	2	no	—
15	C₃×C₄×C₅×D₃	—	2	2	no	—
16	C₃×#895	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂×C₃×C₅×D₄	—	3	2	no	—
18	C₂×C₃×C₅×D₄	—	3	1	no	—
19	C₂²×C₃×C₄×C₅	—	3	1	no	—
20	C₂×C₅×D₁₂	C₂×C₃	1	2	yes	complemented
21	C₂×C₅×D₁₂	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₄×C₅×D₃	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×C₅×C₄⋊C₄	—	2	1	no	—
24	C₃×C₅×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
25	C₃×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
26	C₃×C₅×C₂²⋊C₄	—	3	2	no	—
27	C₅×(C₂×D₃)⋊C₄	C₂×C₃	1	2	yes	complemented
28	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
29	C₂×C₃×C₅×D₃	—	2	1	no	—
30	C₂×C₃×C₅×D₃	—	2	1	no	—
31	C₂×C₃×C₅×D₃	—	2	2	no	—
32	C₂×C₃×C₅×D₃	—	2	2	no	—
33	C₂×C₃×C₅×D₃	—	2	2	no	—
34	C₂×C₃×C₅×D₃	D₄	1	2	yes	complemented
35	C₃²×C₄×C₅	—	2	2	no	—
36	C₃²×C₄×C₅	D₄	1	2	yes	complemented
37	C₃×C₅×Dic₃	—	2	1	no	—
38	C₅×#121	—	3	1	no	Hall
39	C₂×C₃×D₁₂	C₂×C₅	1	2	yes	complemented
40	C₂×C₃×D₁₂	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂×C₃×C₄×D₃	C₂×C₅	1	1	yes	characteristic, complemented
42	C₃²×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×(C₂×D₃)⋊C₄	C₂×C₅	1	2	yes	complemented
44	C₂³×C₃×C₅	—	3	1	no	—
45	C₂³×C₃×C₅	—	3	2	no	—
46	C₂²×C₅×D₃	C₂²×C₃	1	2	yes	—
47	C₂²×C₅×D₃	C₂²×C₃	1	1	yes	characteristic
48	C₃×C₅×D₄	—	6	4	no	—
49	C₃×C₅×D₄	—	6	2	no	—
50	C₂×C₃×C₄×C₅	—	2	1	no	—
51	C₂×C₃×C₄×C₅	—	2	2	no	—
52	C₂×C₃×C₄×C₅	—	3	1	no	—
53	C₂×C₃×C₄×C₅	—	6	2	no	—
54	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	—
55	C₂×C₃×C₄×C₅	C₂×D₃	1	1	yes	characteristic
56	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	—
57	C₂×C₃×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
58	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic
59	C₅×D₁₂	—	2	4	no	—
60	C₅×D₁₂	—	2	2	no	—
61	C₄×C₅×D₃	—	2	2	no	—
62	C₃×#121	—	3	1	no	—
63	#895	C₃×C₅	1	1	yes	characteristic, direct product factor
64	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
66	C₂×C₃²×C₅	#41	1	1	yes	characteristic
67	C₃×C₅×D₃	—	2	2	no	—
68	C₃×C₅×D₃	—	4	2	no	—
69	C₂×C₅×D₄	—	3	2	no	—
70	C₂×C₅×D₄	—	3	1	no	—
71	C₂²×C₄×C₅	—	3	1	no	—
72	C₅×C₄⋊C₄	C₃×D₃	1	1	yes	characteristic, complemented
73	C₅×C₂²⋊C₄	—	3	2	no	—
74	C₂²×C₃×D₃	C₂²×C₅	1	2	yes	—
75	C₂²×C₃×D₃	C₂²×C₅	1	1	yes	characteristic
76	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
77	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
78	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic
79	C₃×D₁₂	—	2	4	no	—
80	C₃×D₁₂	—	2	2	no	—
81	C₃×C₄×D₃	—	2	2	no	—
82	C₂²×C₃×C₅	—	2	1	no	—
83	C₂²×C₃×C₅	—	3	2	no	—
84	C₂²×C₃×C₅	—	6	1	no	—
85	C₂²×C₃×C₅	—	6	1	no	—
86	C₂²×C₃×C₅	—	6	2	no	—
87	C₂²×C₃×C₅	—	6	2	no	—
88	C₂²×C₃×C₅	—	6	2	no	—
89	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
90	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
91	C₂×C₅×D₃	—	2	1	no	—
92	C₂×C₅×D₃	—	2	1	no	—
93	C₂×C₅×D₃	—	2	2	no	—
94	C₂×C₅×D₃	—	2	2	no	—
95	C₂×C₅×D₃	—	2	2	no	—
96	C₂×C₅×D₃	C₃×D₄	1	2	yes	complemented
97	C₃×C₄×C₅	—	2	2	no	—
98	C₃×C₄×C₅	—	2	2	no	—
99	C₃×C₄×C₅	—	2	2	no	—
100	C₃×C₄×C₅	—	4	2	no	—
101	C₃×C₄×C₅	—	6	1	no	—
102	C₃×C₄×C₅	D₁₂	1	2	yes	complemented
103	C₃×C₄×C₅	C₃×D₄	1	2	yes	complemented
104	C₅×Dic₃	—	2	1	no	—
105	C₂×C₃×D₄	—	3	2	no	—
106	C₂×C₃×D₄	—	3	1	no	—
107	C₂²×C₃×C₄	—	3	1	no	—
108	C₂×D₁₂	C₂×C₃×C₅	1	2	yes	complemented
109	C₂×D₁₂	C₂×C₃×C₅	1	1	yes	characteristic, complemented
110	C₂×C₄×D₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
111	C₃×C₄⋊C₄	—	2	1	no	—
112	C₃×C₄⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
113	C₃×C₄⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
114	C₃×C₂²⋊C₄	—	3	2	no	—
115	(C₂×D₃)⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
116	C₃²×C₅	#121	1	1	yes	characteristic, complemented, Hall
117	C₂³×C₅	—	3	1	no	—
118	C₂³×C₅	—	3	2	no	—
119	C₅×D₄	—	6	4	no	—
120	C₅×D₄	—	6	2	no	—
121	C₂×C₄×C₅	—	3	1	no	—
122	C₂×C₄×C₅	—	6	2	no	—
123	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	—
124	C₂×C₄×C₅	C₂×C₃×D₃	1	1	yes	characteristic
125	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
126	C₂×C₃×D₃	—	2	1	no	—
127	C₂×C₃×D₃	—	2	1	no	—
128	C₂×C₃×D₃	—	2	2	no	—
129	C₂×C₃×D₃	—	2	2	no	—
130	C₂×C₃×D₃	—	2	2	no	—
131	C₂×C₃×D₃	C₅×D₄	1	2	yes	complemented
132	C₃²×C₄	—	2	2	no	—
133	C₃²×C₄	C₅×D₄	1	2	yes	complemented
134	C₃×Dic₃	—	2	1	no	—
135	#121	—	3	1	no	Sylow
136	C₂×C₃×C₅	—	2	1	no	—
137	C₂×C₃×C₅	—	2	1	no	—
138	C₂×C₃×C₅	—	2	1	no	—
139	C₂×C₃×C₅	—	6	2	no	—
140	C₂×C₃×C₅	—	12	2	no	—
141	C₂×C₃×C₅	C₂×D₁₂	1	1	yes	characteristic
142	C₂×C₃×C₅	D₃×D₄	1	1	yes	characteristic
143	C₂×C₃×C₅	#239	1	1	yes	characteristic
144	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
146	C₂×C₃×C₅	C₃×#41	1	1	yes	characteristic
147	C₅×D₃	—	2	2	no	—
148	C₅×D₃	—	4	2	no	—
149	C₂³×C₃	—	3	1	no	—
150	C₂³×C₃	—	3	2	no	—
151	C₂²×D₃	C₂²×C₃×C₅	1	2	yes	—
152	C₂²×D₃	C₂²×C₃×C₅	1	1	yes	characteristic
153	C₃×D₄	—	6	4	no	—
154	C₃×D₄	—	6	2	no	—
155	C₂×C₃×C₄	—	2	1	no	—
156	C₂×C₃×C₄	—	2	2	no	—
157	C₂×C₃×C₄	—	3	1	no	—
158	C₂×C₃×C₄	—	6	2	no	—
159	C₂×C₃×C₄	C₂×C₅×D₃	1	2	yes	—
160	C₂×C₃×C₄	C₂×C₅×D₃	1	1	yes	characteristic
161	C₂×C₃×C₄	C₂²×C₃×C₅	1	2	yes	—
162	C₂×C₃×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
163	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic
164	D₁₂	—	2	4	no	—
165	D₁₂	—	2	2	no	—
166	C₄×D₃	—	2	2	no	—
167	C₂²×C₅	—	3	2	no	—
168	C₂²×C₅	—	6	1	no	—
169	C₂²×C₅	—	6	1	no	—
170	C₂²×C₅	—	6	2	no	—
171	C₂²×C₅	—	6	2	no	—
172	C₂²×C₅	—	6	2	no	—
173	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
174	C₄×C₅	—	2	2	no	—
175	C₄×C₅	—	6	1	no	—
176	C₄×C₅	C₃×D₁₂	1	2	yes	complemented
177	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
178	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
179	C₂×C₃²	C₅×#41	1	1	yes	characteristic
180	C₃×D₃	—	2	2	no	—
181	C₃×D₃	—	4	2	no	—
182	C₂×D₄	—	3	2	no	—
183	C₂×D₄	—	3	1	no	—
184	C₂²×C₄	—	3	1	no	—
185	C₄⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic, complemented
186	C₂²⋊C₄	—	3	2	no	—
187	C₃×C₅	—	2	1	no	—
188	C₃×C₅	#895	1	1	yes	characteristic, direct product factor
189	C₃×C₅	C₃×#121	1	1	yes	characteristic, complemented
190	C₂²×C₃	—	2	1	no	—
191	C₂²×C₃	—	3	2	no	—
192	C₂²×C₃	—	6	1	no	—
193	C₂²×C₃	—	6	1	no	—
194	C₂²×C₃	—	6	2	no	—
195	C₂²×C₃	—	6	2	no	—
196	C₂²×C₃	—	6	2	no	—
197	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
198	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
199	C₂×D₃	—	2	1	no	—
200	C₂×D₃	—	2	1	no	—
201	C₂×D₃	—	2	2	no	—
202	C₂×D₃	—	2	2	no	—
203	C₂×D₃	—	2	2	no	—
204	C₂×D₃	C₃×C₅×D₄	1	2	yes	complemented
205	C₃×C₄	—	2	2	no	—
206	C₃×C₄	—	2	2	no	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	—	4	2	no	—
209	C₃×C₄	—	6	1	no	—
210	C₃×C₄	C₅×D₁₂	1	2	yes	complemented
211	C₃×C₄	C₃×C₅×D₄	1	2	yes	complemented
212	Dic₃	—	2	1	no	—
213	C₂×C₅	—	6	2	no	—
214	C₂×C₅	—	12	2	no	—
215	C₂×C₅	C₂×C₃×D₁₂	1	1	yes	characteristic
216	C₂×C₅	C₃×D₃×D₄	1	1	yes	characteristic
217	C₂×C₅	C₃×#239	1	1	yes	characteristic
218	C₃²	C₅×#121	1	1	yes	characteristic, complemented, Sylow
219	C₂³	—	3	1	no	—
220	C₂³	—	3	2	no	—
221	D₄	—	6	4	no	—
222	D₄	—	6	2	no	—
223	C₂×C₄	—	3	1	no	—
224	C₂×C₄	—	6	2	no	—
225	C₂×C₄	C₂×C₃×C₅×D₃	1	2	yes	—
226	C₂×C₄	C₂×C₃×C₅×D₃	1	1	yes	characteristic
227	C₂×C₃	—	2	1	no	—
228	C₂×C₃	—	2	1	no	—
229	C₂×C₃	—	2	1	no	—
230	C₂×C₃	—	6	2	no	—
231	C₂×C₃	—	12	2	no	—
232	C₂×C₃	C₂×C₅×D₁₂	1	1	yes	characteristic
233	C₂×C₃	C₅×D₃×D₄	1	1	yes	characteristic
234	C₂×C₃	C₅×#239	1	1	yes	characteristic
235	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
236	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
237	C₂×C₃	C₃×C₅×#41	1	1	yes	characteristic
238	D₃	—	2	2	no	—
239	D₃	—	4	2	no	—
240	C₅	C₃×#895	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²	—	3	2	no	—
242	C₂²	—	6	1	no	—
243	C₂²	—	6	1	no	—
244	C₂²	—	6	2	no	—
245	C₂²	—	6	2	no	—
246	C₂²	—	6	2	no	—
247	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	6	1	no	—
250	C₄	C₃×C₅×D₁₂	1	2	yes	complemented
251	C₃	—	2	1	no	—
252	C₃	C₅×#895	1	1	yes	characteristic, direct product factor
253	C₃	C₃×C₅×#121	1	1	yes	characteristic, complemented
254	C₂	—	6	2	no	—
255	C₂	—	12	2	no	—
256	C₂	C₂×C₃×C₅×D₁₂	1	1	yes	characteristic
257	C₂	C₃×C₅×D₃×D₄	1	1	yes	characteristic
258	C₂	C₃×C₅×#239	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₅	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₃×C₅
Frattini	C₂²	C₂²×C₃×C₅×D₃
Fitting	C₃²×C₅×C₄⋊C₄	C₂
socle	C₂²×C₃²×C₅	C₂³
hypercenter	C₃×C₅×C₄⋊C₄	D₃
hypocenter	C₃	C₃×C₅×#121
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×D₁₂	C₂	semidirect
2. C₂×C₃×C₅×D₁₂	C₂	semidirect
3. C₂×C₃×C₄×C₅×D₃	C₂	semidirect
4. C₃²×C₅×C₄⋊C₄	C₂	semidirect
5. C₃×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
7. C₅×#895	C₃	direct (N × H)
16. C₃×#895	C₅	direct (N × H)
20. C₂×C₅×D₁₂	C₂×C₃	semidirect
21. C₂×C₅×D₁₂	C₂×C₃	semidirect
22. C₂×C₄×C₅×D₃	C₂×C₃	semidirect
24. C₃×C₅×C₄⋊C₄	D₃	semidirect
25. C₃×C₅×C₄⋊C₄	C₂×C₃	semidirect
27. C₅×(C₂×D₃)⋊C₄	C₂×C₃	semidirect
34. C₂×C₃×C₅×D₃	D₄	semidirect
36. C₃²×C₄×C₅	D₄	semidirect
39. C₂×C₃×D₁₂	C₂×C₅	semidirect
40. C₂×C₃×D₁₂	C₂×C₅	semidirect
41. C₂×C₃×C₄×D₃	C₂×C₅	semidirect
42. C₃²×C₄⋊C₄	C₂×C₅	semidirect
43. C₃×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
63. #895	C₃×C₅	direct (N × H)
72. C₅×C₄⋊C₄	C₃×D₃	semidirect
96. C₂×C₅×D₃	C₃×D₄	semidirect
102. C₃×C₄×C₅	D₁₂	semidirect
103. C₃×C₄×C₅	C₃×D₄	semidirect
108. C₂×D₁₂	C₂×C₃×C₅	semidirect
109. C₂×D₁₂	C₂×C₃×C₅	semidirect
110. C₂×C₄×D₃	C₂×C₃×C₅	semidirect
112. C₃×C₄⋊C₄	C₅×D₃	semidirect
113. C₃×C₄⋊C₄	C₂×C₃×C₅	semidirect
115. (C₂×D₃)⋊C₄	C₂×C₃×C₅	semidirect
116. C₃²×C₅	#121	semidirect
131. C₂×C₃×D₃	C₅×D₄	semidirect
133. C₃²×C₄	C₅×D₄	semidirect
176. C₄×C₅	C₃×D₁₂	semidirect
185. C₄⋊C₄	C₃×C₅×D₃	semidirect
188. C₃×C₅	#895	direct (N × H)
189. C₃×C₅	C₃×#121	semidirect
204. C₂×D₃	C₃×C₅×D₄	semidirect
210. C₃×C₄	C₅×D₁₂	semidirect
211. C₃×C₄	C₃×C₅×D₄	semidirect
218. C₃²	C₅×#121	semidirect
240. C₅	C₃×#895	direct (N × H)
250. C₄	C₃×C₅×D₁₂	semidirect
252. C₃	C₅×#895	direct (N × H)
253. C₃	C₃×C₅×#121	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×D₁₂	2
C₂×C₃×C₄×C₅×D₃	2
C₃²×C₅×C₄⋊C₄	2
C₃×C₅×(C₂×D₃)⋊C₄	2
C₃×C₅×#121	3
C₅×#895	3
C₃×#895	5

Subgroups — C3×C5×#895 #13087907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#895 #13087907