Subgroups — C₃×C₅×#1565 #14274599

Order 1920.

360 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×#335	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#334	C₂	1	2	yes	maximal, complemented
6	C₅×#1565	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	—	2	1	no	—
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	—	2	1	no	—
12	C₃×C₅×C₄⋊C₈	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×C₂²⋊C₈	—	2	1	no	—
16	C₃×C₅×#97	C₄	1	2	yes	—
17	C₃×C₄×C₅×C₈	—	2	1	no	—
18	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₃×#1565	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₅×#431	C₂×C₃	1	1	yes	characteristic
21	C₂×C₅×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
23	C₅×#335	C₂×C₃	1	2	yes	complemented
24	C₅×#334	C₂×C₃	1	2	yes	complemented
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	Q₈	1	1	yes	characteristic
28	C₃×C₅×M₁₆	—	2	2	no	—
29	C₃×C₅×M₁₆	C₈	1	4	yes	complemented
30	C₂×C₃×C₅×C₈	—	2	2	no	—
31	C₂×C₃×C₅×C₈	—	2	1	no	—
32	C₂×C₃×C₅×C₈	—	2	1	no	—
33	C₂×C₃×C₅×C₈	—	4	2	no	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	C₃×#431	C₂×C₅	1	1	yes	characteristic
41	C₂×C₃×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
42	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
43	C₃×#335	C₂×C₅	1	2	yes	complemented
44	C₃×#334	C₂×C₅	1	2	yes	complemented
45	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
46	C₂×C₅×M₁₆	C₃×C₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	—	2	1	no	—
48	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	—	2	1	no	—
50	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
51	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
52	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
53	C₅×C₂²⋊C₈	—	2	1	no	—
54	C₅×#97	C₃×C₄	1	2	yes	—
55	C₄×C₅×C₈	—	2	1	no	—
56	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
57	#1565	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
58	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	—	2	1	no	—
60	C₂×C₃×C₄×C₅	—	2	1	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
66	C₂×C₃×C₄×C₅	M₁₆	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
70	C₃×C₅×C₈	—	2	2	no	—
71	C₃×C₅×C₈	—	4	1	no	—
72	C₃×C₅×C₈	—	4	2	no	—
73	C₃×C₅×C₈	—	4	1	no	—
74	C₃×C₅×C₈	C₂×C₈	1	4	yes	complemented
75	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic
76	C₂×C₃×M₁₆	C₄×C₅	1	1	yes	characteristic
77	C₂²×C₃×C₈	—	2	1	no	—
78	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
79	C₃×C₄⋊C₈	—	2	1	no	—
80	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
81	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
82	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
83	C₃×C₂²⋊C₈	—	2	1	no	—
84	C₃×#97	C₄×C₅	1	2	yes	—
85	C₃×C₄×C₈	—	2	1	no	—
86	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
87	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
90	C₅×M₁₆	—	2	2	no	—
91	C₅×M₁₆	C₃×C₈	1	4	yes	complemented
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	1	no	—
94	C₂×C₅×C₈	—	2	1	no	—
95	C₂×C₅×C₈	—	4	2	no	—
96	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
97	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
98	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
99	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
100	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
101	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
102	#431	C₂×C₃×C₅	1	1	yes	characteristic
103	C₂×C₄⋊C₈	C₂×C₃×C₅	1	1	yes	characteristic
104	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
105	#335	C₂×C₃×C₅	1	2	yes	complemented
106	#334	C₂×C₃×C₅	1	2	yes	complemented
107	C₂²×C₃×C₅	—	2	1	no	—
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	C₄⋊C₈	1	2	yes	—
110	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	—	2	1	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	C₄⋊C₈	1	2	yes	—
115	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
117	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₃×C₄×C₅	#137	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
122	C₃×M₁₆	—	2	2	no	—
123	C₃×M₁₆	C₅×C₈	1	4	yes	complemented
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	2	1	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	4	2	no	—
128	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
129	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
130	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
131	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
132	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
133	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
134	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
140	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
142	C₂×C₄×C₅	C₃×M₁₆	1	2	yes	—
143	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
146	C₅×C₈	—	2	2	no	—
147	C₅×C₈	—	4	1	no	—
148	C₅×C₈	—	4	2	no	—
149	C₅×C₈	—	4	1	no	—
150	C₅×C₈	C₂×C₃×C₈	1	4	yes	complemented
151	C₂×M₁₆	C₃×C₄×C₅	1	1	yes	characteristic
152	C₂×M₁₆	C₃×C₄×C₅	1	1	yes	characteristic
153	C₂²×C₈	—	2	1	no	—
154	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
155	C₄⋊C₈	—	2	1	no	—
156	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
157	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
158	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
159	C₂²⋊C₈	—	2	1	no	—
160	#97	C₃×C₄×C₅	1	2	yes	—
161	C₄×C₈	—	2	1	no	—
162	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
163	C₂×C₃×C₅	—	2	2	no	—
164	C₂×C₃×C₅	C₂×C₄⋊C₈	1	1	yes	characteristic
165	C₂×C₃×C₅	#428	1	1	yes	characteristic
166	C₂×C₃×C₅	#430	1	1	yes	characteristic
167	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	—	2	1	no	—
169	C₂×C₃×C₄	—	2	1	no	—
170	C₂×C₃×C₄	—	4	1	no	—
171	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
175	C₂×C₃×C₄	C₅×M₁₆	1	2	yes	—
176	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
179	C₃×C₈	—	2	2	no	—
180	C₃×C₈	—	4	1	no	—
181	C₃×C₈	—	4	2	no	—
182	C₃×C₈	—	4	1	no	—
183	C₃×C₈	C₂×C₅×C₈	1	4	yes	complemented
184	C₂²×C₅	—	2	1	no	—
185	C₂²×C₅	—	2	1	no	—
186	C₂²×C₅	C₃×C₄⋊C₈	1	2	yes	—
187	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
188	C₄×C₅	—	2	1	no	—
189	C₄×C₅	—	2	1	no	—
190	C₄×C₅	—	2	1	no	—
191	C₄×C₅	C₃×C₄⋊C₈	1	2	yes	—
192	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
193	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
194	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₄×C₅	C₃×#137	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
197	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
199	M₁₆	—	2	2	no	—
200	M₁₆	C₃×C₅×C₈	1	4	yes	complemented
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
206	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
207	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
208	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
209	C₄²	C₃×C₅×Q₈	1	1	yes	characteristic
210	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
211	C₃×C₅	#1565	1	1	yes	characteristic, direct product factor, Hall
212	C₂²×C₃	—	2	1	no	—
213	C₂²×C₃	—	2	1	no	—
214	C₂²×C₃	C₅×C₄⋊C₈	1	2	yes	—
215	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
216	C₃×C₄	—	2	1	no	—
217	C₃×C₄	—	2	1	no	—
218	C₃×C₄	—	2	1	no	—
219	C₃×C₄	C₅×C₄⋊C₈	1	2	yes	—
220	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
221	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
222	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₃×C₄	C₅×#137	1	1	yes	characteristic
224	C₂×C₅	—	2	2	no	—
225	C₂×C₅	C₂×C₃×C₄⋊C₈	1	1	yes	characteristic
226	C₂×C₅	C₃×#428	1	1	yes	characteristic
227	C₂×C₅	C₃×#430	1	1	yes	characteristic
228	C₂³	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
233	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
234	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
236	C₂×C₄	C₃×C₅×M₁₆	1	2	yes	—
237	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
238	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
239	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	4	2	no	—
242	C₈	—	4	1	no	—
243	C₈	—	4	1	no	—
244	C₈	C₂×C₃×C₅×C₈	1	4	yes	complemented
245	C₂×C₃	—	2	2	no	—
246	C₂×C₃	C₂×C₅×C₄⋊C₈	1	1	yes	characteristic
247	C₂×C₃	C₅×#428	1	1	yes	characteristic
248	C₂×C₃	C₅×#430	1	1	yes	characteristic
249	C₅	C₃×#1565	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₃×C₅×C₄⋊C₈	1	2	yes	—
253	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	C₃×C₅×C₄⋊C₈	1	2	yes	—
258	C₄	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
259	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
260	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₄	C₃×C₅×#137	1	1	yes	characteristic
262	C₃	C₅×#1565	1	1	yes	characteristic, direct product factor, Sylow
263	C₂	—	2	2	no	—
264	C₂	C₂×C₃×C₅×C₄⋊C₈	1	1	yes	characteristic
265	C₂	C₃×C₅×#428	1	1	yes	characteristic
266	C₂	C₃×C₅×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₄	C₂²×C₃×C₅×C₈
Frattini, cosocle	C₄²	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×#335	C₂	semidirect
5. C₃×C₅×#334	C₂	semidirect
6. C₅×#1565	C₃	direct (N × H)
19. C₃×#1565	C₅	direct (N × H)
23. C₅×#335	C₂×C₃	semidirect
24. C₅×#334	C₂×C₃	semidirect
29. C₃×C₅×M₁₆	C₈	semidirect
43. C₃×#335	C₂×C₅	semidirect
44. C₃×#334	C₂×C₅	semidirect
57. #1565	C₃×C₅	direct (N × H)
74. C₃×C₅×C₈	C₂×C₈	semidirect
91. C₅×M₁₆	C₃×C₈	semidirect
105. #335	C₂×C₃×C₅	semidirect
106. #334	C₂×C₃×C₅	semidirect
123. C₃×M₁₆	C₅×C₈	semidirect
150. C₅×C₈	C₂×C₃×C₈	semidirect
183. C₃×C₈	C₂×C₅×C₈	semidirect
200. M₁₆	C₃×C₅×C₈	semidirect
211. C₃×C₅	#1565	direct (N × H)
244. C₈	C₂×C₃×C₅×C₈	semidirect
249. C₅	C₃×#1565	direct (N × H)
262. C₃	C₅×#1565	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×#431	2
C₂×C₃×C₅×C₄⋊C₈	2
C₃×C₄×C₅×M₁₆	2
C₃×C₅×#335	2
C₃×C₅×#334	2
C₅×#1565	3
C₃×#1565	5

Subgroups — C3×C5×#1565 #14274599

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1565 #14274599