Subgroups — C₃×C₅×#1687 #14274721

Order 1920.

396 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#459	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#436	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#331	C₂	1	2	yes	maximal, complemented
6	C₅×#1687	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×D₈	—	2	2	no	—
8	C₂×C₃×C₅×D₈	—	2	1	no	—
9	C₂×C₃×C₅×D₈	—	2	1	no	—
10	C₂×C₃×C₅×M₁₆	—	2	1	no	—
11	C₃×C₅×#127	C₂²	1	2	yes	—
12	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃×C₅×#102	—	2	2	no	—
16	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
17	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₃×#1687	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
19	C₅×#493	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×#459	C₂×C₃	1	2	yes	complemented
21	C₅×#436	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#334	C₂×C₃	1	1	yes	characteristic, complemented
23	C₅×#331	C₂×C₃	1	2	yes	complemented
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	—	2	2	no	—
26	C₂×C₃×C₅×D₄	—	2	2	no	—
27	C₂×C₃×C₅×D₄	D₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	—	2	1	no	—
29	C₃×C₅×D₈	—	4	2	no	—
30	C₃×C₅×D₈	—	4	4	no	—
31	C₃×C₅×D₈	—	4	2	no	—
32	C₃×C₅×M₁₆	—	4	2	no	—
33	C₂×C₃×C₅×C₈	—	2	1	no	—
34	C₂×C₃×C₅×C₈	—	2	2	no	—
35	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
37	C₃×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₃×#493	C₂×C₅	1	1	yes	characteristic, complemented
41	C₃×#459	C₂×C₅	1	2	yes	complemented
42	C₃×#436	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#334	C₂×C₅	1	1	yes	characteristic, complemented
44	C₃×#331	C₂×C₅	1	2	yes	complemented
45	C₂×C₅×D₈	—	2	2	no	—
46	C₂×C₅×D₈	—	2	1	no	—
47	C₂×C₅×D₈	—	2	1	no	—
48	C₂×C₅×M₁₆	—	2	1	no	—
49	C₅×#127	C₂²×C₃	1	2	yes	—
50	C₄×C₅×D₄	C₂²×C₃	1	1	yes	characteristic
51	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
52	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
53	C₅×#102	—	2	2	no	—
54	C₅×C₂²⋊C₈	—	2	1	no	—
55	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
56	#1687	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃×C₅	—	2	1	no	—
58	C₂³×C₃×C₅	—	4	2	no	—
59	C₃×C₅×D₄	—	2	2	no	—
60	C₃×C₅×D₄	—	4	2	no	—
61	C₃×C₅×D₄	—	4	2	no	—
62	C₃×C₅×D₄	—	4	2	no	—
63	C₃×C₅×D₄	—	4	2	no	—
64	C₃×C₅×D₄	—	8	2	no	—
65	C₂×C₃×C₄×C₅	—	2	1	no	—
66	C₂×C₃×C₄×C₅	—	4	1	no	—
67	C₂×C₃×C₄×C₅	—	4	1	no	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
71	C₃×C₅×C₈	—	2	1	no	—
72	C₃×C₅×C₈	—	4	1	no	—
73	C₃×C₅×C₈	—	4	2	no	—
74	C₃×C₅×C₈	D₈	1	2	yes	complemented
75	C₂×C₃×D₈	—	2	2	no	—
76	C₂×C₃×D₈	—	2	1	no	—
77	C₂×C₃×D₈	—	2	1	no	—
78	C₂×C₃×M₁₆	—	2	1	no	—
79	C₃×#127	C₂²×C₅	1	2	yes	—
80	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
81	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
82	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
83	C₃×#102	—	2	2	no	—
84	C₃×C₂²⋊C₈	—	2	1	no	—
85	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₂×C₅×D₄	—	2	2	no	—
87	C₂×C₅×D₄	—	2	2	no	—
88	C₂×C₅×D₄	—	2	2	no	—
89	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	—	2	1	no	—
91	C₅×D₈	—	4	2	no	—
92	C₅×D₈	—	4	4	no	—
93	C₅×D₈	—	4	2	no	—
94	C₅×M₁₆	—	4	2	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
98	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
99	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
100	C₅×C₂²⋊C₄	—	2	1	no	—
101	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
102	#493	C₂×C₃×C₅	1	1	yes	characteristic, complemented
103	#459	C₂×C₃×C₅	1	2	yes	complemented
104	#436	C₂×C₃×C₅	1	1	yes	characteristic, complemented
105	#334	C₂×C₃×C₅	1	1	yes	characteristic, complemented
106	#331	C₂×C₃×C₅	1	2	yes	complemented
107	C₂²×C₃×C₅	—	4	1	no	—
108	C₂²×C₃×C₅	—	4	1	no	—
109	C₂²×C₃×C₅	—	4	1	no	—
110	C₂²×C₃×C₅	—	8	2	no	—
111	C₂²×C₃×C₅	—	8	2	no	—
112	C₂²×C₃×C₅	—	8	2	no	—
113	C₂²×C₃×C₅	#121	1	1	yes	characteristic
114	C₃×C₄×C₅	—	2	1	no	—
115	C₃×C₄×C₅	—	4	1	no	—
116	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
117	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₃×D₄	—	2	2	no	—
120	C₂×C₃×D₄	—	2	2	no	—
121	C₂×C₃×D₄	—	2	2	no	—
122	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
123	C₂²×C₃×C₄	—	2	1	no	—
124	C₃×D₈	—	4	2	no	—
125	C₃×D₈	—	4	4	no	—
126	C₃×D₈	—	4	2	no	—
127	C₃×M₁₆	—	4	2	no	—
128	C₂×C₃×C₈	—	2	1	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
131	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
132	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
133	C₃×C₂²⋊C₄	—	2	1	no	—
134	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
135	C₂³×C₅	—	2	1	no	—
136	C₂³×C₅	—	4	2	no	—
137	C₅×D₄	—	2	2	no	—
138	C₅×D₄	—	4	2	no	—
139	C₅×D₄	—	4	2	no	—
140	C₅×D₄	—	4	2	no	—
141	C₅×D₄	—	4	2	no	—
142	C₅×D₄	—	8	2	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	4	1	no	—
145	C₂×C₄×C₅	—	4	1	no	—
146	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
149	C₅×C₈	—	2	1	no	—
150	C₅×C₈	—	4	1	no	—
151	C₅×C₈	—	4	2	no	—
152	C₅×C₈	C₃×D₈	1	2	yes	complemented
153	C₂×D₈	—	2	2	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×M₁₆	—	2	1	no	—
157	#127	C₂²×C₃×C₅	1	2	yes	—
158	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
159	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
160	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
161	#102	—	2	2	no	—
162	C₂²⋊C₈	—	2	1	no	—
163	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
164	C₂×C₃×C₅	—	8	1	no	—
165	C₂×C₃×C₅	—	16	2	no	—
166	C₂×C₃×C₅	#459	1	1	yes	characteristic
167	C₂×C₃×C₅	#469	1	1	yes	characteristic
168	C₂×C₃×C₅	#472	1	1	yes	characteristic
169	C₂³×C₃	—	2	1	no	—
170	C₂³×C₃	—	4	2	no	—
171	C₃×D₄	—	2	2	no	—
172	C₃×D₄	—	4	2	no	—
173	C₃×D₄	—	4	2	no	—
174	C₃×D₄	—	4	2	no	—
175	C₃×D₄	—	4	2	no	—
176	C₃×D₄	—	8	2	no	—
177	C₂×C₃×C₄	—	2	1	no	—
178	C₂×C₃×C₄	—	4	1	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
181	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
182	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
183	C₃×C₈	—	2	1	no	—
184	C₃×C₈	—	4	1	no	—
185	C₃×C₈	—	4	2	no	—
186	C₃×C₈	C₅×D₈	1	2	yes	complemented
187	C₂²×C₅	—	4	1	no	—
188	C₂²×C₅	—	4	1	no	—
189	C₂²×C₅	—	4	1	no	—
190	C₂²×C₅	—	8	2	no	—
191	C₂²×C₅	—	8	2	no	—
192	C₂²×C₅	—	8	2	no	—
193	C₂²×C₅	C₃×#121	1	1	yes	characteristic
194	C₄×C₅	—	2	1	no	—
195	C₄×C₅	—	4	1	no	—
196	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
197	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
198	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
199	C₂×D₄	—	2	2	no	—
200	C₂×D₄	—	2	2	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
203	C₂²×C₄	—	2	1	no	—
204	D₈	—	4	2	no	—
205	D₈	—	4	4	no	—
206	D₈	—	4	2	no	—
207	M₁₆	—	4	2	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
211	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
212	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
215	C₃×C₅	#1687	1	1	yes	characteristic, direct product factor, Hall
216	C₂²×C₃	—	4	1	no	—
217	C₂²×C₃	—	4	1	no	—
218	C₂²×C₃	—	4	1	no	—
219	C₂²×C₃	—	8	2	no	—
220	C₂²×C₃	—	8	2	no	—
221	C₂²×C₃	—	8	2	no	—
222	C₂²×C₃	C₅×#121	1	1	yes	characteristic
223	C₃×C₄	—	2	1	no	—
224	C₃×C₄	—	4	1	no	—
225	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
226	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
227	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
228	C₂×C₅	—	8	1	no	—
229	C₂×C₅	—	16	2	no	—
230	C₂×C₅	C₃×#459	1	1	yes	characteristic
231	C₂×C₅	C₃×#469	1	1	yes	characteristic
232	C₂×C₅	C₃×#472	1	1	yes	characteristic
233	C₂³	—	2	1	no	—
234	C₂³	—	4	2	no	—
235	D₄	—	2	2	no	—
236	D₄	—	4	2	no	—
237	D₄	—	4	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	8	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	4	1	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
245	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
246	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
247	C₈	—	2	1	no	—
248	C₈	—	4	1	no	—
249	C₈	—	4	2	no	—
250	C₈	C₃×C₅×D₈	1	2	yes	complemented
251	C₂×C₃	—	8	1	no	—
252	C₂×C₃	—	16	2	no	—
253	C₂×C₃	C₅×#459	1	1	yes	characteristic
254	C₂×C₃	C₅×#469	1	1	yes	characteristic
255	C₂×C₃	C₅×#472	1	1	yes	characteristic
256	C₅	C₃×#1687	1	1	yes	characteristic, direct product factor, Sylow
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	C₃×C₅×#121	1	1	yes	characteristic
264	C₄	—	2	1	no	—
265	C₄	—	4	1	no	—
266	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
267	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
269	C₃	C₅×#1687	1	1	yes	characteristic, direct product factor, Sylow
270	C₂	—	8	1	no	—
271	C₂	—	16	2	no	—
272	C₂	C₃×C₅×#459	1	1	yes	characteristic
273	C₂	C₃×C₅×#469	1	1	yes	characteristic
274	C₂	C₃×C₅×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₃×C₅
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#493	C₂	semidirect
2. C₃×C₅×#459	C₂	semidirect
3. C₃×C₅×#436	C₂	semidirect
4. C₃×C₅×#334	C₂	semidirect
5. C₃×C₅×#331	C₂	semidirect
6. C₅×#1687	C₃	direct (N × H)
18. C₃×#1687	C₅	direct (N × H)
19. C₅×#493	C₂×C₃	semidirect
20. C₅×#459	C₂×C₃	semidirect
21. C₅×#436	C₂×C₃	semidirect
22. C₅×#334	C₂×C₃	semidirect
23. C₅×#331	C₂×C₃	semidirect
40. C₃×#493	C₂×C₅	semidirect
41. C₃×#459	C₂×C₅	semidirect
42. C₃×#436	C₂×C₅	semidirect
43. C₃×#334	C₂×C₅	semidirect
44. C₃×#331	C₂×C₅	semidirect
56. #1687	C₃×C₅	direct (N × H)
74. C₃×C₅×C₈	D₈	semidirect
102. #493	C₂×C₃×C₅	semidirect
103. #459	C₂×C₃×C₅	semidirect
104. #436	C₂×C₃×C₅	semidirect
105. #334	C₂×C₃×C₅	semidirect
106. #331	C₂×C₃×C₅	semidirect
152. C₅×C₈	C₃×D₈	semidirect
186. C₃×C₈	C₅×D₈	semidirect
215. C₃×C₅	#1687	direct (N × H)
250. C₈	C₃×C₅×D₈	semidirect
256. C₅	C₃×#1687	direct (N × H)
269. C₃	C₅×#1687	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×#493	2
C₃×C₅×#459	2
C₃×C₅×#436	2
C₃×C₅×#334	2
C₃×C₅×#331	2
C₅×#1687	3
C₃×#1687	5

Subgroups — C3×C5×#1687 #14274721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1687 #14274721