Subgroups — C₃×C₅×#882 #13087894

Order 1440.

554 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₅×D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₅×#67	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×C₅×D₁₂	C₂	1	2	yes	maximal, complemented
4	C₃²×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₃×C₅×#120	—	3	1	no	maximal
7	C₅×#882	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₃²×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₅×D₃	—	2	2	no	—
10	C₂²×C₃×C₅×D₃	—	2	4	no	—
11	C₂²×C₃×C₅×D₃	C₂²	1	2	yes	—
12	C₂²×C₃×C₅×D₃	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	complemented
14	C₃×C₅×#67	—	2	2	no	—
15	C₂×C₃×C₅×Dic₃	C₂²	1	1	yes	characteristic, complemented
16	C₃×C₅×D₁₂	—	2	4	no	—
17	C₃×#882	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂⁴×C₃×C₅	—	3	1	no	—
19	C₂³×C₅×D₃	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×C₃×C₅×D₄	—	3	1	no	—
21	C₂×C₃×C₅×D₄	—	3	2	no	—
22	C₂×C₅×#67	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₅×D₁₂	C₂×C₃	1	2	yes	complemented
24	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
25	C₃×C₅×C₂²⋊C₄	—	3	2	no	—
26	C₃×C₅×C₂²⋊C₄	D₃	1	1	yes	characteristic, complemented
27	C₃×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic, complemented
28	C₅×(C₂×D₃)⋊C₄	C₂×C₃	1	2	yes	complemented
29	C₂²×C₃²×C₅	—	2	2	no	—
30	C₂²×C₃²×C₅	D₄	1	2	yes	complemented
31	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₅×D₃	—	2	8	no	—
33	C₂×C₃×C₅×D₃	—	2	2	no	—
34	C₂×C₃×C₅×D₃	—	2	2	no	—
35	C₂×C₃×C₅×D₃	—	2	2	no	—
36	C₂×C₃×C₅×D₃	—	2	1	no	—
37	C₂×C₃×C₅×D₃	D₄	1	4	yes	complemented
38	C₃²×C₄×C₅	—	2	2	no	—
39	C₃×C₅×Dic₃	—	2	1	no	—
40	C₅×#120	—	3	1	no	Hall
41	C₂³×C₃×D₃	C₂×C₅	1	1	yes	characteristic, complemented
42	C₂×C₃×#67	C₂×C₅	1	1	yes	characteristic, complemented
43	C₂×C₃×D₁₂	C₂×C₅	1	2	yes	complemented
44	C₃²×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
45	C₃×(C₂×D₃)⋊C₄	C₂×C₅	1	2	yes	complemented
46	C₂³×C₃×C₅	—	2	1	no	—
47	C₂³×C₃×C₅	—	3	2	no	—
48	C₂³×C₃×C₅	—	3	1	no	—
49	C₂³×C₃×C₅	—	6	4	no	—
50	C₂³×C₃×C₅	—	6	2	no	—
51	C₂³×C₃×C₅	C₂×D₃	1	1	yes	characteristic
52	C₂³×C₃×C₅	C₂²×C₃	1	1	yes	characteristic
53	C₂²×C₅×D₃	—	2	2	no	—
54	C₂²×C₅×D₃	—	2	4	no	—
55	C₂²×C₅×D₃	C₂²×C₃	1	2	yes	—
56	C₂²×C₅×D₃	C₂²×C₃	1	1	yes	characteristic, complemented
57	C₃×C₅×D₄	—	6	4	no	—
58	C₃×C₅×D₄	—	6	2	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	3	1	no	—
61	C₂×C₃×C₄×C₅	C₂×D₃	1	2	yes	complemented
62	C₂×C₃×C₄×C₅	C₂²×C₃	1	2	yes	complemented
63	C₅×#67	—	2	2	no	—
64	C₂×C₅×Dic₃	C₂²×C₃	1	1	yes	characteristic, complemented
65	C₅×D₁₂	—	2	4	no	—
66	C₃×#120	—	3	1	no	—
67	#882	C₃×C₅	1	1	yes	characteristic, direct product factor
68	C₂×C₃²×C₅	—	2	2	no	—
69	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃²×C₅	C₂×D₄	1	2	yes	—
71	C₃×C₅×D₃	—	2	4	no	—
72	C₃×C₅×D₃	—	4	1	no	—
73	C₂⁴×C₅	—	3	1	no	—
74	C₂×C₅×D₄	—	3	1	no	—
75	C₂×C₅×D₄	—	3	2	no	—
76	C₅×C₂²⋊C₄	—	3	2	no	—
77	C₅×C₂²⋊C₄	C₃×D₃	1	1	yes	characteristic, complemented
78	C₂³×C₃²	C₂²×C₅	1	1	yes	characteristic
79	C₂²×C₃×D₃	—	2	2	no	—
80	C₂²×C₃×D₃	—	2	4	no	—
81	C₂²×C₃×D₃	C₂²×C₅	1	2	yes	—
82	C₂²×C₃×D₃	C₂²×C₅	1	1	yes	characteristic, complemented
83	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	complemented
84	C₃×#67	—	2	2	no	—
85	C₂×C₃×Dic₃	C₂²×C₅	1	1	yes	characteristic, complemented
86	C₃×D₁₂	—	2	4	no	—
87	C₂²×C₃×C₅	—	2	2	no	—
88	C₂²×C₃×C₅	—	2	2	no	—
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	1	no	—
91	C₂²×C₃×C₅	—	3	4	no	—
92	C₂²×C₃×C₅	—	4	2	no	—
93	C₂²×C₃×C₅	—	6	8	no	—
94	C₂²×C₃×C₅	—	6	2	no	—
95	C₂²×C₃×C₅	—	6	2	no	—
96	C₂²×C₃×C₅	—	6	1	no	—
97	C₂²×C₃×C₅	—	6	2	no	—
98	C₂²×C₃×C₅	D₁₂	1	2	yes	complemented
99	C₂²×C₃×C₅	C₃×D₄	1	2	yes	complemented
100	C₂²×C₃×C₅	C₂²×D₃	1	1	yes	characteristic
101	C₂²×C₃×C₅	C₂³×C₃	1	1	yes	characteristic
102	C₂×C₅×D₃	—	2	8	no	—
103	C₂×C₅×D₃	—	2	2	no	—
104	C₂×C₅×D₃	—	2	2	no	—
105	C₂×C₅×D₃	—	2	2	no	—
106	C₂×C₅×D₃	—	2	1	no	—
107	C₂×C₅×D₃	C₃×D₄	1	4	yes	complemented
108	C₃×C₄×C₅	—	2	2	no	—
109	C₃×C₄×C₅	—	2	2	no	—
110	C₃×C₄×C₅	—	4	2	no	—
111	C₃×C₄×C₅	—	6	1	no	—
112	C₅×Dic₃	—	2	1	no	—
113	C₂⁴×C₃	—	3	1	no	—
114	C₂³×D₃	C₂×C₃×C₅	1	1	yes	characteristic, complemented
115	C₂×C₃×D₄	—	3	1	no	—
116	C₂×C₃×D₄	—	3	2	no	—
117	C₂×#67	C₂×C₃×C₅	1	1	yes	characteristic, complemented
118	C₂×D₁₂	C₂×C₃×C₅	1	2	yes	complemented
119	C₃×C₂²⋊C₄	—	2	1	no	—
120	C₃×C₂²⋊C₄	—	3	2	no	—
121	C₃×C₂²⋊C₄	C₅×D₃	1	1	yes	characteristic, complemented
122	C₃×C₂²⋊C₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
123	(C₂×D₃)⋊C₄	C₂×C₃×C₅	1	2	yes	complemented
124	C₃²×C₅	#120	1	1	yes	characteristic, complemented, Hall
125	C₂³×C₅	—	3	2	no	—
126	C₂³×C₅	—	3	1	no	—
127	C₂³×C₅	—	6	4	no	—
128	C₂³×C₅	—	6	2	no	—
129	C₂³×C₅	C₂×C₃×D₃	1	1	yes	characteristic
130	C₅×D₄	—	6	4	no	—
131	C₅×D₄	—	6	2	no	—
132	C₂×C₄×C₅	—	3	1	no	—
133	C₂×C₄×C₅	C₂×C₃×D₃	1	2	yes	complemented
134	C₂²×C₃²	—	2	2	no	—
135	C₂²×C₃²	C₅×D₄	1	2	yes	complemented
136	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
137	C₂×C₃×D₃	—	2	8	no	—
138	C₂×C₃×D₃	—	2	2	no	—
139	C₂×C₃×D₃	—	2	2	no	—
140	C₂×C₃×D₃	—	2	2	no	—
141	C₂×C₃×D₃	—	2	1	no	—
142	C₂×C₃×D₃	C₅×D₄	1	4	yes	complemented
143	C₃²×C₄	—	2	2	no	—
144	C₃×Dic₃	—	2	1	no	—
145	#120	—	3	1	no	Sylow
146	C₂×C₃×C₅	—	2	2	no	—
147	C₂×C₃×C₅	—	2	2	no	—
148	C₂×C₃×C₅	—	2	1	no	—
149	C₂×C₃×C₅	—	2	2	no	—
150	C₂×C₃×C₅	—	4	2	no	—
151	C₂×C₃×C₅	—	6	4	no	—
152	C₂×C₃×C₅	—	12	1	no	—
153	C₂×C₃×C₅	C₂×D₁₂	1	1	yes	characteristic
154	C₂×C₃×C₅	D₃×D₄	1	2	yes	—
155	C₂×C₃×C₅	C₂×C₃×D₄	1	1	yes	characteristic
156	C₂×C₃×C₅	C₂×C₃×D₄	1	2	yes	—
157	C₅×D₃	—	2	4	no	—
158	C₅×D₃	—	4	1	no	—
159	C₂³×C₃	—	2	1	no	—
160	C₂³×C₃	—	3	2	no	—
161	C₂³×C₃	—	3	1	no	—
162	C₂³×C₃	—	6	4	no	—
163	C₂³×C₃	—	6	2	no	—
164	C₂³×C₃	C₂×C₅×D₃	1	1	yes	characteristic
165	C₂³×C₃	C₂²×C₃×C₅	1	1	yes	characteristic
166	C₂²×D₃	—	2	2	no	—
167	C₂²×D₃	—	2	4	no	—
168	C₂²×D₃	C₂²×C₃×C₅	1	2	yes	—
169	C₂²×D₃	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
170	C₃×D₄	—	6	4	no	—
171	C₃×D₄	—	6	2	no	—
172	C₂×C₃×C₄	—	2	2	no	—
173	C₂×C₃×C₄	—	3	1	no	—
174	C₂×C₃×C₄	C₂×C₅×D₃	1	2	yes	complemented
175	C₂×C₃×C₄	C₂²×C₃×C₅	1	2	yes	complemented
176	#67	—	2	2	no	—
177	C₂×Dic₃	C₂²×C₃×C₅	1	1	yes	characteristic, complemented
178	D₁₂	—	2	4	no	—
179	C₂²×C₅	—	2	2	no	—
180	C₂²×C₅	—	3	4	no	—
181	C₂²×C₅	—	6	8	no	—
182	C₂²×C₅	—	6	2	no	—
183	C₂²×C₅	—	6	2	no	—
184	C₂²×C₅	—	6	1	no	—
185	C₂²×C₅	—	6	2	no	—
186	C₂²×C₅	C₃×D₁₂	1	2	yes	complemented
187	C₂²×C₅	C₂²×C₃×D₃	1	1	yes	characteristic
188	C₄×C₅	—	2	2	no	—
189	C₄×C₅	—	6	1	no	—
190	C₂×C₃²	—	2	2	no	—
191	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
192	C₂×C₃²	C₂×C₅×D₄	1	2	yes	—
193	C₃×D₃	—	2	4	no	—
194	C₃×D₃	—	4	1	no	—
195	C₂⁴	—	3	1	no	—
196	C₂×D₄	—	3	1	no	—
197	C₂×D₄	—	3	2	no	—
198	C₂²⋊C₄	—	3	2	no	—
199	C₂²⋊C₄	C₃×C₅×D₃	1	1	yes	characteristic, complemented
200	C₃×C₅	—	2	1	no	—
201	C₃×C₅	#882	1	1	yes	characteristic, direct product factor
202	C₃×C₅	C₃×#120	1	1	yes	characteristic, complemented
203	C₂²×C₃	—	2	2	no	—
204	C₂²×C₃	—	2	2	no	—
205	C₂²×C₃	—	2	2	no	—
206	C₂²×C₃	—	2	1	no	—
207	C₂²×C₃	—	3	4	no	—
208	C₂²×C₃	—	4	2	no	—
209	C₂²×C₃	—	6	8	no	—
210	C₂²×C₃	—	6	2	no	—
211	C₂²×C₃	—	6	2	no	—
212	C₂²×C₃	—	6	1	no	—
213	C₂²×C₃	—	6	2	no	—
214	C₂²×C₃	C₅×D₁₂	1	2	yes	complemented
215	C₂²×C₃	C₃×C₅×D₄	1	2	yes	complemented
216	C₂²×C₃	C₂²×C₅×D₃	1	1	yes	characteristic
217	C₂²×C₃	C₂³×C₃×C₅	1	1	yes	characteristic
218	C₂×D₃	—	2	8	no	—
219	C₂×D₃	—	2	2	no	—
220	C₂×D₃	—	2	2	no	—
221	C₂×D₃	—	2	2	no	—
222	C₂×D₃	—	2	1	no	—
223	C₂×D₃	C₃×C₅×D₄	1	4	yes	complemented
224	C₃×C₄	—	2	2	no	—
225	C₃×C₄	—	2	2	no	—
226	C₃×C₄	—	4	2	no	—
227	C₃×C₄	—	6	1	no	—
228	Dic₃	—	2	1	no	—
229	C₂×C₅	—	2	2	no	—
230	C₂×C₅	—	6	4	no	—
231	C₂×C₅	—	12	1	no	—
232	C₂×C₅	C₂×C₃×D₁₂	1	1	yes	characteristic
233	C₂×C₅	C₃×D₃×D₄	1	2	yes	—
234	C₃²	C₅×#120	1	1	yes	characteristic, complemented, Sylow
235	C₂³	—	3	2	no	—
236	C₂³	—	3	1	no	—
237	C₂³	—	6	4	no	—
238	C₂³	—	6	2	no	—
239	C₂³	C₂×C₃×C₅×D₃	1	1	yes	characteristic
240	D₄	—	6	4	no	—
241	D₄	—	6	2	no	—
242	C₂×C₄	—	3	1	no	—
243	C₂×C₄	C₂×C₃×C₅×D₃	1	2	yes	complemented
244	C₂×C₃	—	2	2	no	—
245	C₂×C₃	—	2	2	no	—
246	C₂×C₃	—	2	1	no	—
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	4	2	no	—
249	C₂×C₃	—	6	4	no	—
250	C₂×C₃	—	12	1	no	—
251	C₂×C₃	C₂×C₅×D₁₂	1	1	yes	characteristic
252	C₂×C₃	C₅×D₃×D₄	1	2	yes	—
253	C₂×C₃	C₂×C₃×C₅×D₄	1	1	yes	characteristic
254	C₂×C₃	C₂×C₃×C₅×D₄	1	2	yes	—
255	D₃	—	2	4	no	—
256	D₃	—	4	1	no	—
257	C₅	C₃×#882	1	1	yes	characteristic, direct product factor, Sylow
258	C₂²	—	2	2	no	—
259	C₂²	—	3	4	no	—
260	C₂²	—	6	8	no	—
261	C₂²	—	6	2	no	—
262	C₂²	—	6	2	no	—
263	C₂²	—	6	1	no	—
264	C₂²	—	6	2	no	—
265	C₂²	C₃×C₅×D₁₂	1	2	yes	complemented
266	C₂²	C₂²×C₃×C₅×D₃	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	6	1	no	—
269	C₃	—	2	1	no	—
270	C₃	C₅×#882	1	1	yes	characteristic, direct product factor
271	C₃	C₃×C₅×#120	1	1	yes	characteristic, complemented
272	C₂	—	2	2	no	—
273	C₂	—	6	4	no	—
274	C₂	—	12	1	no	—
275	C₂	C₂×C₃×C₅×D₁₂	1	1	yes	characteristic
276	C₂	C₃×C₅×D₃×D₄	1	2	yes	—

	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×C₂²⋊C₄	C₂
socle	C₂²×C₃²×C₅	C₂³
hypercenter	C₃×C₅×C₂²⋊C₄	D₃
hypocenter	C₃	C₃×C₅×#120
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×C₅×D₃	C₂	semidirect
2. C₂×C₃×C₅×#67	C₂	semidirect
3. C₂×C₃×C₅×D₁₂	C₂	semidirect
4. C₃²×C₅×C₂²⋊C₄	C₂	semidirect
5. C₃×C₅×(C₂×D₃)⋊C₄	C₂	semidirect
7. C₅×#882	C₃	direct (N × H)
12. C₂²×C₃×C₅×D₃	C₂²	semidirect
13. C₂×C₃²×C₄×C₅	C₂²	semidirect
15. C₂×C₃×C₅×Dic₃	C₂²	semidirect
17. C₃×#882	C₅	direct (N × H)
19. C₂³×C₅×D₃	C₂×C₃	semidirect
22. C₂×C₅×#67	C₂×C₃	semidirect
23. C₂×C₅×D₁₂	C₂×C₃	semidirect
26. C₃×C₅×C₂²⋊C₄	D₃	semidirect
27. C₃×C₅×C₂²⋊C₄	C₂×C₃	semidirect
28. C₅×(C₂×D₃)⋊C₄	C₂×C₃	semidirect
30. C₂²×C₃²×C₅	D₄	semidirect
37. C₂×C₃×C₅×D₃	D₄	semidirect
41. C₂³×C₃×D₃	C₂×C₅	semidirect
42. C₂×C₃×#67	C₂×C₅	semidirect
43. C₂×C₃×D₁₂	C₂×C₅	semidirect
44. C₃²×C₂²⋊C₄	C₂×C₅	semidirect
45. C₃×(C₂×D₃)⋊C₄	C₂×C₅	semidirect
56. C₂²×C₅×D₃	C₂²×C₃	semidirect
61. C₂×C₃×C₄×C₅	C₂×D₃	semidirect
62. C₂×C₃×C₄×C₅	C₂²×C₃	semidirect
64. C₂×C₅×Dic₃	C₂²×C₃	semidirect
67. #882	C₃×C₅	direct (N × H)
77. C₅×C₂²⋊C₄	C₃×D₃	semidirect
82. C₂²×C₃×D₃	C₂²×C₅	semidirect
83. C₂×C₃²×C₄	C₂²×C₅	semidirect
85. C₂×C₃×Dic₃	C₂²×C₅	semidirect
98. C₂²×C₃×C₅	D₁₂	semidirect
99. C₂²×C₃×C₅	C₃×D₄	semidirect
107. C₂×C₅×D₃	C₃×D₄	semidirect
114. C₂³×D₃	C₂×C₃×C₅	semidirect
117. C₂×#67	C₂×C₃×C₅	semidirect
118. C₂×D₁₂	C₂×C₃×C₅	semidirect
121. C₃×C₂²⋊C₄	C₅×D₃	semidirect
122. C₃×C₂²⋊C₄	C₂×C₃×C₅	semidirect
123. (C₂×D₃)⋊C₄	C₂×C₃×C₅	semidirect
124. C₃²×C₅	#120	semidirect
133. C₂×C₄×C₅	C₂×C₃×D₃	semidirect
135. C₂²×C₃²	C₅×D₄	semidirect
142. C₂×C₃×D₃	C₅×D₄	semidirect
169. C₂²×D₃	C₂²×C₃×C₅	semidirect
174. C₂×C₃×C₄	C₂×C₅×D₃	semidirect
175. C₂×C₃×C₄	C₂²×C₃×C₅	semidirect
177. C₂×Dic₃	C₂²×C₃×C₅	semidirect
186. C₂²×C₅	C₃×D₁₂	semidirect
199. C₂²⋊C₄	C₃×C₅×D₃	semidirect
201. C₃×C₅	#882	direct (N × H)
202. C₃×C₅	C₃×#120	semidirect
214. C₂²×C₃	C₅×D₁₂	semidirect
215. C₂²×C₃	C₃×C₅×D₄	semidirect
223. C₂×D₃	C₃×C₅×D₄	semidirect
234. C₃²	C₅×#120	semidirect
243. C₂×C₄	C₂×C₃×C₅×D₃	semidirect
257. C₅	C₃×#882	direct (N × H)
265. C₂²	C₃×C₅×D₁₂	semidirect
270. C₃	C₅×#882	direct (N × H)
271. C₃	C₃×C₅×#120	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₅×D₃	2
C₂×C₃×C₅×#67	2
C₂×C₃×C₅×D₁₂	2
C₃²×C₅×C₂²⋊C₄	2
C₃×C₅×(C₂×D₃)⋊C₄	2
C₃×C₅×#120	3
C₅×#882	3
C₃×#882	5

Subgroups — C3×C5×#882 #13087894

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#882 #13087894