Subgroups — C₃×C₅×#1667 #14274701

Order 1920.

408 conjugacy classes of subgroups.

The subgroup lattice has 284 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×C₈×D₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₅×#331	C₂	1	2	yes	maximal, complemented
6	C₅×#1667	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	C₂²×C₃×C₅×C₈	—	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×#1667	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×C₈×D₄	C₂×C₃	1	1	yes	characteristic, complemented
22	C₅×#335	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	C₂×C₃×C₅×C₈	—	2	1	no	—
33	C₂×C₃×C₅×C₈	—	2	2	no	—
34	C₂×C₃×C₅×C₈	—	4	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×C₈×D₄	C₂×C₅	1	1	yes	characteristic, complemented
43	C₃×#335	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	C₂²×C₅×C₈	—	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	#1667	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	1	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	C₃×C₅×D₄	D₈	1	2	yes	complemented
66	C₂×C₃×C₄×C₅	—	2	1	no	—
67	C₂×C₃×C₄×C₅	—	4	1	no	—
68	C₂×C₃×C₄×C₅	—	4	1	no	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
72	C₃×C₅×C₈	—	2	1	no	—
73	C₃×C₅×C₈	—	4	1	no	—
74	C₃×C₅×C₈	—	4	2	no	—
75	C₃×C₅×C₈	D₈	1	2	yes	complemented
76	C₂×C₃×D₈	—	2	2	no	—
77	C₂×C₃×D₈	—	2	1	no	—
78	C₂×C₃×D₈	—	2	1	no	—
79	C₂²×C₃×C₈	—	2	1	no	—
80	C₃×#127	C₂²×C₅	1	2	yes	—
81	C₃×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
82	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
83	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
84	C₃×#102	—	2	2	no	—
85	C₃×C₂²⋊C₈	—	2	1	no	—
86	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
87	C₂×C₅×D₄	—	2	2	no	—
88	C₂×C₅×D₄	—	2	2	no	—
89	C₂×C₅×D₄	—	2	2	no	—
90	C₂×C₅×D₄	C₃×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	—	2	1	no	—
92	C₅×D₈	—	4	2	no	—
93	C₅×D₈	—	4	4	no	—
94	C₅×D₈	—	4	2	no	—
95	C₂×C₅×C₈	—	2	1	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	4	2	no	—
98	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
99	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
100	C₅×C₄⋊C₄	C₃×D₄	1	1	yes	characteristic
101	C₅×C₂²⋊C₄	—	2	1	no	—
102	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
103	#493	C₂×C₃×C₅	1	1	yes	characteristic, complemented
104	#459	C₂×C₃×C₅	1	2	yes	complemented
105	C₈×D₄	C₂×C₃×C₅	1	1	yes	characteristic, complemented
106	#335	C₂×C₃×C₅	1	1	yes	characteristic, complemented
107	#331	C₂×C₃×C₅	1	2	yes	complemented
108	C₂²×C₃×C₅	—	2	2	no	—
109	C₂²×C₃×C₅	—	4	1	no	—
110	C₂²×C₃×C₅	—	4	1	no	—
111	C₂²×C₃×C₅	—	8	2	no	—
112	C₂²×C₃×C₅	—	8	2	no	—
113	C₂²×C₃×C₅	—	8	2	no	—
114	C₂²×C₃×C₅	#121	1	1	yes	characteristic
115	C₃×C₄×C₅	—	2	1	no	—
116	C₃×C₄×C₅	—	4	1	no	—
117	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
118	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
119	C₃×C₄×C₅	#135	1	1	yes	characteristic
120	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₃×D₄	—	2	2	no	—
122	C₂×C₃×D₄	—	2	2	no	—
123	C₂×C₃×D₄	—	2	2	no	—
124	C₂×C₃×D₄	C₅×D₄	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	2	1	no	—
126	C₃×D₈	—	4	2	no	—
127	C₃×D₈	—	4	4	no	—
128	C₃×D₈	—	4	2	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	4	2	no	—
132	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
134	C₃×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
135	C₃×C₂²⋊C₄	—	2	1	no	—
136	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
137	C₂³×C₅	—	2	1	no	—
138	C₂³×C₅	—	4	2	no	—
139	C₅×D₄	—	2	1	no	—
140	C₅×D₄	—	4	2	no	—
141	C₅×D₄	—	4	2	no	—
142	C₅×D₄	—	4	2	no	—
143	C₅×D₄	—	4	2	no	—
144	C₅×D₄	—	8	2	no	—
145	C₅×D₄	C₃×D₈	1	2	yes	complemented
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	4	1	no	—
148	C₂×C₄×C₅	—	4	1	no	—
149	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
152	C₅×C₈	—	2	1	no	—
153	C₅×C₈	—	4	1	no	—
154	C₅×C₈	—	4	2	no	—
155	C₅×C₈	C₃×D₈	1	2	yes	complemented
156	C₂×D₈	—	2	2	no	—
157	C₂×D₈	—	2	1	no	—
158	C₂×D₈	—	2	1	no	—
159	C₂²×C₈	—	2	1	no	—
160	#127	C₂²×C₃×C₅	1	2	yes	—
161	C₄×D₄	C₂²×C₃×C₅	1	1	yes	characteristic
162	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
163	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
164	#102	—	2	2	no	—
165	C₂²⋊C₈	—	2	1	no	—
166	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
167	C₂×C₃×C₅	—	4	2	no	—
168	C₂×C₃×C₅	—	16	2	no	—
169	C₂×C₃×C₅	#459	1	1	yes	characteristic
170	C₂×C₃×C₅	#466	1	1	yes	characteristic
171	C₂×C₃×C₅	#472	1	1	yes	characteristic
172	C₂³×C₃	—	2	1	no	—
173	C₂³×C₃	—	4	2	no	—
174	C₃×D₄	—	2	1	no	—
175	C₃×D₄	—	4	2	no	—
176	C₃×D₄	—	4	2	no	—
177	C₃×D₄	—	4	2	no	—
178	C₃×D₄	—	4	2	no	—
179	C₃×D₄	—	8	2	no	—
180	C₃×D₄	C₅×D₈	1	2	yes	complemented
181	C₂×C₃×C₄	—	2	1	no	—
182	C₂×C₃×C₄	—	4	1	no	—
183	C₂×C₃×C₄	—	4	1	no	—
184	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
187	C₃×C₈	—	2	1	no	—
188	C₃×C₈	—	4	1	no	—
189	C₃×C₈	—	4	2	no	—
190	C₃×C₈	C₅×D₈	1	2	yes	complemented
191	C₂²×C₅	—	2	2	no	—
192	C₂²×C₅	—	4	1	no	—
193	C₂²×C₅	—	4	1	no	—
194	C₂²×C₅	—	8	2	no	—
195	C₂²×C₅	—	8	2	no	—
196	C₂²×C₅	—	8	2	no	—
197	C₂²×C₅	C₃×#121	1	1	yes	characteristic
198	C₄×C₅	—	2	1	no	—
199	C₄×C₅	—	4	1	no	—
200	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
201	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
202	C₄×C₅	C₃×#135	1	1	yes	characteristic
203	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	2	2	no	—
207	C₂×D₄	C₃×C₅×D₄	1	1	yes	characteristic
208	C₂²×C₄	—	2	1	no	—
209	D₈	—	4	2	no	—
210	D₈	—	4	4	no	—
211	D₈	—	4	2	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
216	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
217	C₄⋊C₄	C₃×C₅×D₄	1	1	yes	characteristic
218	C₂²⋊C₄	—	2	1	no	—
219	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
220	C₃×C₅	#1667	1	1	yes	characteristic, direct product factor, Hall
221	C₂²×C₃	—	2	2	no	—
222	C₂²×C₃	—	4	1	no	—
223	C₂²×C₃	—	4	1	no	—
224	C₂²×C₃	—	8	2	no	—
225	C₂²×C₃	—	8	2	no	—
226	C₂²×C₃	—	8	2	no	—
227	C₂²×C₃	C₅×#121	1	1	yes	characteristic
228	C₃×C₄	—	2	1	no	—
229	C₃×C₄	—	4	1	no	—
230	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
231	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
232	C₃×C₄	C₅×#135	1	1	yes	characteristic
233	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
234	C₂×C₅	—	4	2	no	—
235	C₂×C₅	—	16	2	no	—
236	C₂×C₅	C₃×#459	1	1	yes	characteristic
237	C₂×C₅	C₃×#466	1	1	yes	characteristic
238	C₂×C₅	C₃×#472	1	1	yes	characteristic
239	C₂³	—	2	1	no	—
240	C₂³	—	4	2	no	—
241	D₄	—	2	1	no	—
242	D₄	—	4	2	no	—
243	D₄	—	4	2	no	—
244	D₄	—	4	2	no	—
245	D₄	—	4	2	no	—
246	D₄	—	8	2	no	—
247	D₄	C₃×C₅×D₈	1	2	yes	complemented
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	4	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
252	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
253	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
254	C₈	—	2	1	no	—
255	C₈	—	4	1	no	—
256	C₈	—	4	2	no	—
257	C₈	C₃×C₅×D₈	1	2	yes	complemented
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	—	16	2	no	—
260	C₂×C₃	C₅×#459	1	1	yes	characteristic
261	C₂×C₃	C₅×#466	1	1	yes	characteristic
262	C₂×C₃	C₅×#472	1	1	yes	characteristic
263	C₅	C₃×#1667	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	2	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	8	2	no	—
268	C₂²	—	8	2	no	—
269	C₂²	—	8	2	no	—
270	C₂²	C₃×C₅×#121	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	4	1	no	—
273	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
274	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
275	C₄	C₃×C₅×#135	1	1	yes	characteristic
276	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
277	C₃	C₅×#1667	1	1	yes	characteristic, direct product factor, Sylow
278	C₂	—	4	2	no	—
279	C₂	—	16	2	no	—
280	C₂	C₃×C₅×#459	1	1	yes	characteristic
281	C₂	C₃×C₅×#466	1	1	yes	characteristic
282	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×C₈×D₄	C₂	semidirect
4. C₃×C₅×#335	C₂	semidirect
5. C₃×C₅×#331	C₂	semidirect
6. C₅×#1667	C₃	direct (N × H)
18. C₃×#1667	C₅	direct (N × H)
19. C₅×#493	C₂×C₃	semidirect
20. C₅×#459	C₂×C₃	semidirect
21. C₅×C₈×D₄	C₂×C₃	semidirect
22. C₅×#335	C₂×C₃	semidirect
23. C₅×#331	C₂×C₃	semidirect
40. C₃×#493	C₂×C₅	semidirect
41. C₃×#459	C₂×C₅	semidirect
42. C₃×C₈×D₄	C₂×C₅	semidirect
43. C₃×#335	C₂×C₅	semidirect
44. C₃×#331	C₂×C₅	semidirect
56. #1667	C₃×C₅	direct (N × H)
65. C₃×C₅×D₄	D₈	semidirect
75. C₃×C₅×C₈	D₈	semidirect
103. #493	C₂×C₃×C₅	semidirect
104. #459	C₂×C₃×C₅	semidirect
105. C₈×D₄	C₂×C₃×C₅	semidirect
106. #335	C₂×C₃×C₅	semidirect
107. #331	C₂×C₃×C₅	semidirect
145. C₅×D₄	C₃×D₈	semidirect
155. C₅×C₈	C₃×D₈	semidirect
180. C₃×D₄	C₅×D₈	semidirect
190. C₃×C₈	C₅×D₈	semidirect
220. C₃×C₅	#1667	direct (N × H)
247. D₄	C₃×C₅×D₈	semidirect
257. C₈	C₃×C₅×D₈	semidirect
263. C₅	C₃×#1667	direct (N × H)
277. C₃	C₅×#1667	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×C₈×D₄	2
C₃×C₅×#335	2
C₃×C₅×#331	2
C₅×#1667	3
C₃×#1667	5

Subgroups — C3×C5×#1667 #14274701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1667 #14274701