Subgroups — C₃×C₅×#3259 #14276293

Order 1920.

740 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#540	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×#478	C₂	1	4	yes	maximal, complemented
3	C₃×C₅×#469	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#466	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#459	C₂	1	4	yes	maximal, complemented
6	C₃×C₅×#424	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#3259	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
8	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
9	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
10	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×#127	C₂²	1	2	yes	—
13	C₃×C₅×#125	C₂²	1	2	yes	—
14	C₃×C₅×#123	—	2	4	no	—
15	C₃×C₅×#121	—	2	4	no	—
16	C₃×C₅×#121	C₂²	1	4	yes	—
17	C₃×C₄×C₅×D₄	C₂²	1	4	yes	—
18	C₃×C₅×#117	C₂²	1	1	yes	characteristic
19	C₃×C₅×#107	C₂²	1	2	yes	—
20	C₃×C₅×#106	C₂²	1	2	yes	—
21	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
22	C₃×C₅×#102	C₂²	1	4	yes	—
23	C₃×C₅×#102	C₂²	1	4	yes	—
24	C₃×#3259	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₅×#540	C₂×C₃	1	2	yes	complemented
26	C₅×#478	C₂×C₃	1	4	yes	complemented
27	C₅×#469	C₂×C₃	1	2	yes	complemented
28	C₅×#466	C₂×C₃	1	2	yes	complemented
29	C₅×#459	C₂×C₃	1	4	yes	complemented
30	C₅×#424	C₂×C₃	1	1	yes	characteristic, complemented
31	C₂×C₃×C₅×D₄	—	2	4	no	—
32	C₂×C₃×C₅×D₄	—	2	4	no	—
33	C₂×C₃×C₅×D₄	C₂³	1	4	yes	—
34	C₂²×C₃×C₄×C₅	—	2	4	no	—
35	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
36	C₃×C₅×D₈	—	4	2	no	—
37	C₃×C₅×D₈	—	4	2	no	—
38	C₃×C₅×M₁₆	—	4	1	no	—
39	C₂×C₃×C₅×C₈	—	4	1	no	—
40	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
41	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
42	C₃×C₅×C₄⋊C₄	—	2	4	no	—
43	C₃×C₅×C₄⋊C₄	D₄	1	2	yes	—
44	C₃×C₅×C₄⋊C₄	C₂³	1	4	yes	—
45	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
46	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
47	C₃×C₅×C₂²⋊C₄	D₄	1	2	yes	—
48	C₃×C₄²×C₅	C₂³	1	2	yes	—
49	C₃×#540	C₂×C₅	1	2	yes	complemented
50	C₃×#478	C₂×C₅	1	4	yes	complemented
51	C₃×#469	C₂×C₅	1	2	yes	complemented
52	C₃×#466	C₂×C₅	1	2	yes	complemented
53	C₃×#459	C₂×C₅	1	4	yes	complemented
54	C₃×#424	C₂×C₅	1	1	yes	characteristic, complemented
55	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
56	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
57	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
58	C₂²×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
59	C₅×#127	C₂²×C₃	1	2	yes	—
60	C₅×#125	C₂²×C₃	1	2	yes	—
61	C₅×#123	—	2	4	no	—
62	C₅×#121	—	2	4	no	—
63	C₅×#121	C₂²×C₃	1	4	yes	—
64	C₄×C₅×D₄	C₂²×C₃	1	4	yes	—
65	C₅×#117	C₂²×C₃	1	1	yes	characteristic
66	C₅×#107	C₂²×C₃	1	2	yes	—
67	C₅×#106	C₂²×C₃	1	2	yes	—
68	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
69	C₅×#102	C₂²×C₃	1	4	yes	—
70	C₅×#102	C₂²×C₃	1	4	yes	—
71	#3259	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
72	C₂³×C₃×C₅	—	2	4	no	—
73	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
74	C₃×C₅×D₄	—	2	4	no	—
75	C₃×C₅×D₄	—	2	4	no	—
76	C₃×C₅×D₄	—	8	4	no	—
77	C₃×C₅×D₄	—	8	4	no	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	2	4	no	—
81	C₂×C₃×C₄×C₅	—	8	4	no	—
82	C₂×C₃×C₄×C₅	C₂×D₄	1	4	yes	—
83	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₃×C₅×C₈	—	2	2	no	—
86	C₃×C₅×C₈	—	2	2	no	—
87	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
88	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
89	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
90	C₂²×C₃×C₈	C₂²×C₅	1	1	yes	characteristic
91	C₃×#127	C₂²×C₅	1	2	yes	—
92	C₃×#125	C₂²×C₅	1	2	yes	—
93	C₃×#123	—	2	4	no	—
94	C₃×#121	—	2	4	no	—
95	C₃×#121	C₂²×C₅	1	4	yes	—
96	C₃×C₄×D₄	C₂²×C₅	1	4	yes	—
97	C₃×#117	C₂²×C₅	1	1	yes	characteristic
98	C₃×#107	C₂²×C₅	1	2	yes	—
99	C₃×#106	C₂²×C₅	1	2	yes	—
100	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
101	C₃×#102	C₂²×C₅	1	4	yes	—
102	C₃×#102	C₂²×C₅	1	4	yes	—
103	C₂×C₅×D₄	—	2	4	no	—
104	C₂×C₅×D₄	—	2	4	no	—
105	C₂×C₅×D₄	C₂³×C₃	1	4	yes	—
106	C₂²×C₄×C₅	—	2	4	no	—
107	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
108	C₅×D₈	—	4	2	no	—
109	C₅×D₈	—	4	2	no	—
110	C₅×M₁₆	—	4	1	no	—
111	C₂×C₅×C₈	—	4	1	no	—
112	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
113	C₂×C₅×C₈	C₂³×C₃	1	2	yes	—
114	C₅×C₄⋊C₄	—	2	4	no	—
115	C₅×C₄⋊C₄	C₃×D₄	1	2	yes	—
116	C₅×C₄⋊C₄	C₂³×C₃	1	4	yes	—
117	C₅×C₂²⋊C₄	—	2	4	no	—
118	C₅×C₂²⋊C₄	—	2	4	no	—
119	C₅×C₂²⋊C₄	C₃×D₄	1	2	yes	—
120	C₄²×C₅	C₂³×C₃	1	2	yes	—
121	#540	C₂×C₃×C₅	1	2	yes	complemented
122	#478	C₂×C₃×C₅	1	4	yes	complemented
123	#469	C₂×C₃×C₅	1	2	yes	complemented
124	#466	C₂×C₃×C₅	1	2	yes	complemented
125	#459	C₂×C₃×C₅	1	4	yes	complemented
126	#424	C₂×C₃×C₅	1	1	yes	characteristic, complemented
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	2	no	—
129	C₂²×C₃×C₅	—	4	4	no	—
130	C₂²×C₃×C₅	—	4	4	no	—
131	C₂²×C₃×C₅	—	4	4	no	—
132	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	—	2	4	no	—
135	C₃×C₄×C₅	—	4	4	no	—
136	C₃×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
137	C₃×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
138	C₂×C₃×D₄	—	2	4	no	—
139	C₂×C₃×D₄	—	2	4	no	—
140	C₂×C₃×D₄	C₂³×C₅	1	4	yes	—
141	C₂²×C₃×C₄	—	2	4	no	—
142	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
143	C₃×D₈	—	4	2	no	—
144	C₃×D₈	—	4	2	no	—
145	C₃×M₁₆	—	4	1	no	—
146	C₂×C₃×C₈	—	4	1	no	—
147	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
148	C₂×C₃×C₈	C₂³×C₅	1	2	yes	—
149	C₃×C₄⋊C₄	—	2	4	no	—
150	C₃×C₄⋊C₄	C₅×D₄	1	2	yes	—
151	C₃×C₄⋊C₄	C₂³×C₅	1	4	yes	—
152	C₃×C₂²⋊C₄	—	2	4	no	—
153	C₃×C₂²⋊C₄	—	2	4	no	—
154	C₃×C₂²⋊C₄	C₅×D₄	1	2	yes	—
155	C₃×C₄²	C₂³×C₅	1	2	yes	—
156	C₂³×C₅	—	2	4	no	—
157	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
158	C₅×D₄	—	2	4	no	—
159	C₅×D₄	—	2	4	no	—
160	C₅×D₄	—	8	4	no	—
161	C₅×D₄	—	8	4	no	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	4	no	—
165	C₂×C₄×C₅	—	8	4	no	—
166	C₂×C₄×C₅	C₂×C₃×D₄	1	4	yes	—
167	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂⁴×C₃	1	1	yes	characteristic
169	C₅×C₈	—	2	2	no	—
170	C₅×C₈	—	2	2	no	—
171	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
172	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
173	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
174	C₂²×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
175	#127	C₂²×C₃×C₅	1	2	yes	—
176	#125	C₂²×C₃×C₅	1	2	yes	—
177	#123	—	2	4	no	—
178	#121	—	2	4	no	—
179	#121	C₂²×C₃×C₅	1	4	yes	—
180	C₄×D₄	C₂²×C₃×C₅	1	4	yes	—
181	#117	C₂²×C₃×C₅	1	1	yes	characteristic
182	#107	C₂²×C₃×C₅	1	2	yes	—
183	#106	C₂²×C₃×C₅	1	2	yes	—
184	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
185	#102	C₂²×C₃×C₅	1	4	yes	—
186	#102	C₂²×C₃×C₅	1	4	yes	—
187	C₂×C₃×C₅	—	4	1	no	—
188	C₂×C₃×C₅	—	8	4	no	—
189	C₂×C₃×C₅	#537	1	1	yes	characteristic
190	C₂×C₃×C₅	#576	1	2	yes	—
191	C₂³×C₃	—	2	4	no	—
192	C₂³×C₃	C₂×C₅×D₄	1	1	yes	characteristic
193	C₃×D₄	—	2	4	no	—
194	C₃×D₄	—	2	4	no	—
195	C₃×D₄	—	8	4	no	—
196	C₃×D₄	—	8	4	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	8	4	no	—
201	C₂×C₃×C₄	C₂×C₅×D₄	1	4	yes	—
202	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂⁴×C₅	1	1	yes	characteristic
204	C₃×C₈	—	2	2	no	—
205	C₃×C₈	—	2	2	no	—
206	C₂²×C₅	—	2	2	no	—
207	C₂²×C₅	—	2	1	no	—
208	C₂²×C₅	—	4	4	no	—
209	C₂²×C₅	—	4	4	no	—
210	C₂²×C₅	—	4	4	no	—
211	C₂²×C₅	C₂²×C₃×D₄	1	1	yes	characteristic
212	C₄×C₅	—	2	4	no	—
213	C₄×C₅	—	2	1	no	—
214	C₄×C₅	—	4	4	no	—
215	C₄×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
216	C₄×C₅	C₃×2₋¹⁺⁴	1	1	yes	characteristic
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂×D₄	C₂³×C₃×C₅	1	4	yes	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
222	D₈	—	4	2	no	—
223	D₈	—	4	2	no	—
224	M₁₆	—	4	1	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
227	C₂×C₈	C₂³×C₃×C₅	1	2	yes	—
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	C₃×C₅×D₄	1	2	yes	—
230	C₄⋊C₄	C₂³×C₃×C₅	1	4	yes	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	C₃×C₅×D₄	1	2	yes	—
234	C₄²	C₂³×C₃×C₅	1	2	yes	—
235	C₃×C₅	#3259	1	1	yes	characteristic, direct product factor, Hall
236	C₂²×C₃	—	2	1	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	4	4	no	—
239	C₂²×C₃	—	4	4	no	—
240	C₂²×C₃	—	4	4	no	—
241	C₂²×C₃	C₂²×C₅×D₄	1	1	yes	characteristic
242	C₃×C₄	—	2	1	no	—
243	C₃×C₄	—	2	4	no	—
244	C₃×C₄	—	4	4	no	—
245	C₃×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
246	C₃×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
247	C₂×C₅	—	4	1	no	—
248	C₂×C₅	—	8	4	no	—
249	C₂×C₅	C₃×#537	1	1	yes	characteristic
250	C₂×C₅	C₃×#576	1	2	yes	—
251	C₂³	—	2	4	no	—
252	C₂³	C₂×C₃×C₅×D₄	1	1	yes	characteristic
253	D₄	—	2	4	no	—
254	D₄	—	2	4	no	—
255	D₄	—	8	4	no	—
256	D₄	—	8	4	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₂×C₃×C₅×D₄	1	4	yes	—
262	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
263	C₂×C₄	C₂⁴×C₃×C₅	1	1	yes	characteristic
264	C₈	—	2	2	no	—
265	C₈	—	2	2	no	—
266	C₂×C₃	—	4	1	no	—
267	C₂×C₃	—	8	4	no	—
268	C₂×C₃	C₅×#537	1	1	yes	characteristic
269	C₂×C₃	C₅×#576	1	2	yes	—
270	C₅	C₃×#3259	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²	—	2	1	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	4	4	no	—
274	C₂²	—	4	4	no	—
275	C₂²	—	4	4	no	—
276	C₂²	C₂²×C₃×C₅×D₄	1	1	yes	characteristic
277	C₄	—	2	1	no	—
278	C₄	—	2	4	no	—
279	C₄	—	4	4	no	—
280	C₄	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic
281	C₄	C₃×C₅×2₋¹⁺⁴	1	1	yes	characteristic
282	C₃	C₅×#3259	1	1	yes	characteristic, direct product factor, Sylow
283	C₂	—	4	1	no	—
284	C₂	—	8	4	no	—
285	C₂	C₃×C₅×#537	1	1	yes	characteristic
286	C₂	C₃×C₅×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#540	C₂	semidirect
2. C₃×C₅×#478	C₂	semidirect
3. C₃×C₅×#469	C₂	semidirect
4. C₃×C₅×#466	C₂	semidirect
5. C₃×C₅×#459	C₂	semidirect
6. C₃×C₅×#424	C₂	semidirect
7. C₅×#3259	C₃	direct (N × H)
24. C₃×#3259	C₅	direct (N × H)
25. C₅×#540	C₂×C₃	semidirect
26. C₅×#478	C₂×C₃	semidirect
27. C₅×#469	C₂×C₃	semidirect
28. C₅×#466	C₂×C₃	semidirect
29. C₅×#459	C₂×C₃	semidirect
30. C₅×#424	C₂×C₃	semidirect
49. C₃×#540	C₂×C₅	semidirect
50. C₃×#478	C₂×C₅	semidirect
51. C₃×#469	C₂×C₅	semidirect
52. C₃×#466	C₂×C₅	semidirect
53. C₃×#459	C₂×C₅	semidirect
54. C₃×#424	C₂×C₅	semidirect
71. #3259	C₃×C₅	direct (N × H)
121. #540	C₂×C₃×C₅	semidirect
122. #478	C₂×C₃×C₅	semidirect
123. #469	C₂×C₃×C₅	semidirect
124. #466	C₂×C₃×C₅	semidirect
125. #459	C₂×C₃×C₅	semidirect
126. #424	C₂×C₃×C₅	semidirect
235. C₃×C₅	#3259	direct (N × H)
270. C₅	C₃×#3259	direct (N × H)
282. C₃	C₅×#3259	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×#540	2
C₃×C₅×#478	2
C₃×C₅×#469	2
C₃×C₅×#466	2
C₃×C₅×#459	2
C₃×C₅×#424	2
C₅×#3259	3
C₃×#3259	5

Subgroups — C3×C5×#3259 #14276293

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#3259 #14276293