Subgroups — C₃×C₅×#1560 #14274594

Order 1920.

508 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#532	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#333	C₂	1	2	yes	maximal, complemented
4	C₃×C₅×#331	C₂	1	2	yes	maximal, complemented
5	C₅×#1560	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×#41	C₄	1	2	yes	—
7	C₂×C₃×C₅×M₁₆	—	2	2	no	—
8	C₃×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×#127	C₂²	1	1	yes	characteristic
10	C₃×C₅×#124	C₄	1	2	yes	—
11	C₃×C₅×#121	—	2	2	no	—
12	C₃×C₄×C₅×D₄	—	2	2	no	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₄⋊C₈	—	2	2	no	—
15	C₃×C₅×C₄⋊C₈	C₂²	1	4	yes	—
16	C₃×#1560	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₅×#532	C₂×C₃	1	1	yes	characteristic
18	C₅×#423	C₂×C₃	1	2	yes	complemented
19	C₅×#333	C₂×C₃	1	2	yes	complemented
20	C₅×#331	C₂×C₃	1	2	yes	complemented
21	C₃×C₅×#41	—	4	4	no	—
22	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅×D₄	—	2	2	no	—
24	C₂×C₃×C₅×D₄	—	2	2	no	—
25	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
29	C₃×C₅×M₁₆	—	4	4	no	—
30	C₂×C₃×C₅×C₈	—	2	4	no	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	D₄	1	2	yes	—
35	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₃×#532	C₂×C₅	1	1	yes	characteristic
37	C₃×#423	C₂×C₅	1	2	yes	complemented
38	C₃×#333	C₂×C₅	1	2	yes	complemented
39	C₃×#331	C₂×C₅	1	2	yes	complemented
40	C₂×C₅×#41	C₃×C₄	1	2	yes	—
41	C₂×C₅×M₁₆	—	2	2	no	—
42	C₅×C₄⋊Q₈	C₂²×C₃	1	1	yes	characteristic
43	C₅×#127	C₂²×C₃	1	1	yes	characteristic
44	C₅×#124	C₃×C₄	1	2	yes	—
45	C₅×#121	—	2	2	no	—
46	C₄×C₅×D₄	—	2	2	no	—
47	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
48	C₅×C₄⋊C₈	—	2	2	no	—
49	C₅×C₄⋊C₈	C₂²×C₃	1	4	yes	—
50	#1560	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³×C₃×C₅	—	2	2	no	—
52	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₃×C₅×Q₈	—	4	2	no	—
54	C₃×C₅×D₄	—	4	2	no	—
55	C₃×C₅×D₄	—	4	4	no	—
56	C₃×C₅×D₄	—	4	2	no	—
57	C₃×C₅×D₄	—	4	2	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	4	4	no	—
63	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₃×C₅×C₈	—	4	4	no	—
68	C₂×C₃×#41	C₄×C₅	1	2	yes	—
69	C₂×C₃×M₁₆	—	2	2	no	—
70	C₃×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
71	C₃×#127	C₂²×C₅	1	1	yes	characteristic
72	C₃×#124	C₄×C₅	1	2	yes	—
73	C₃×#121	—	2	2	no	—
74	C₃×C₄×D₄	—	2	2	no	—
75	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
76	C₃×C₄⋊C₈	—	2	2	no	—
77	C₃×C₄⋊C₈	C₂²×C₅	1	4	yes	—
78	C₅×#41	—	4	4	no	—
79	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
80	C₂×C₅×D₄	—	2	2	no	—
81	C₂×C₅×D₄	—	2	2	no	—
82	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
83	C₂²×C₄×C₅	—	2	2	no	—
84	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
86	C₅×M₁₆	—	4	4	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₂²⋊C₄	—	2	4	no	—
90	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
91	C₄²×C₅	C₃×D₄	1	2	yes	—
92	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
93	#532	C₂×C₃×C₅	1	1	yes	characteristic
94	#423	C₂×C₃×C₅	1	2	yes	complemented
95	#333	C₂×C₃×C₅	1	2	yes	complemented
96	#331	C₂×C₃×C₅	1	2	yes	complemented
97	C₂²×C₃×C₅	—	2	1	no	—
98	C₂²×C₃×C₅	—	2	2	no	—
99	C₂²×C₃×C₅	—	4	2	no	—
100	C₂²×C₃×C₅	—	4	4	no	—
101	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₃×C₄×C₅	—	2	1	no	—
103	C₃×C₄×C₅	—	2	2	no	—
104	C₃×C₄×C₅	—	4	2	no	—
105	C₃×C₄×C₅	#100	1	2	yes	—
106	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
107	C₃×C₄×C₅	#137	1	2	yes	—
108	C₃×#41	—	4	4	no	—
109	C₂×C₃×Q₈	C₂×C₄×C₅	1	2	yes	—
110	C₂×C₃×D₄	—	2	2	no	—
111	C₂×C₃×D₄	—	2	2	no	—
112	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
113	C₂²×C₃×C₄	—	2	2	no	—
114	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₅×D₄	1	2	yes	—
116	C₃×M₁₆	—	4	4	no	—
117	C₂×C₃×C₈	—	2	4	no	—
118	C₃×C₄⋊C₄	—	2	2	no	—
119	C₃×C₂²⋊C₄	—	2	4	no	—
120	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
121	C₃×C₄²	C₅×D₄	1	2	yes	—
122	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
123	C₂³×C₅	—	2	2	no	—
124	C₂³×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
125	C₅×Q₈	—	4	2	no	—
126	C₅×D₄	—	4	2	no	—
127	C₅×D₄	—	4	4	no	—
128	C₅×D₄	—	4	2	no	—
129	C₅×D₄	—	4	2	no	—
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
136	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
139	C₅×C₈	—	4	4	no	—
140	C₂×#41	C₃×C₄×C₅	1	2	yes	—
141	C₂×M₁₆	—	2	2	no	—
142	C₄⋊Q₈	C₂²×C₃×C₅	1	1	yes	characteristic
143	#127	C₂²×C₃×C₅	1	1	yes	characteristic
144	#124	C₃×C₄×C₅	1	2	yes	—
145	#121	—	2	2	no	—
146	C₄×D₄	—	2	2	no	—
147	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
148	C₄⋊C₈	—	2	2	no	—
149	C₄⋊C₈	C₂²×C₃×C₅	1	4	yes	—
150	C₂×C₃×C₅	—	4	1	no	—
151	C₂×C₃×C₅	—	8	2	no	—
152	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
153	C₂×C₃×C₅	#417	1	2	yes	—
154	C₂³×C₃	—	2	2	no	—
155	C₂³×C₃	C₅×C₂²⋊C₄	1	1	yes	characteristic
156	C₃×Q₈	—	4	2	no	—
157	C₃×D₄	—	4	2	no	—
158	C₃×D₄	—	4	4	no	—
159	C₃×D₄	—	4	2	no	—
160	C₃×D₄	—	4	2	no	—
161	C₂×C₃×C₄	—	2	2	no	—
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	2	no	—
164	C₂×C₃×C₄	—	2	2	no	—
165	C₂×C₃×C₄	—	4	4	no	—
166	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
167	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₅×D₄	1	2	yes	—
170	C₃×C₈	—	4	4	no	—
171	C₂²×C₅	—	2	1	no	—
172	C₂²×C₅	—	2	2	no	—
173	C₂²×C₅	—	4	2	no	—
174	C₂²×C₅	—	4	4	no	—
175	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
176	C₄×C₅	—	2	1	no	—
177	C₄×C₅	—	2	2	no	—
178	C₄×C₅	—	4	2	no	—
179	C₄×C₅	C₃×#100	1	2	yes	—
180	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
181	C₄×C₅	C₃×#137	1	2	yes	—
182	#41	—	4	4	no	—
183	C₂×Q₈	C₂×C₃×C₄×C₅	1	2	yes	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
189	C₂²×C₄	C₃×C₅×D₄	1	2	yes	—
190	M₁₆	—	4	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₂²⋊C₄	—	2	4	no	—
194	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
195	C₄²	C₃×C₅×D₄	1	2	yes	—
196	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
197	C₃×C₅	#1560	1	1	yes	characteristic, direct product factor, Hall
198	C₂²×C₃	—	2	1	no	—
199	C₂²×C₃	—	2	2	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	—	4	4	no	—
202	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
203	C₃×C₄	—	2	1	no	—
204	C₃×C₄	—	2	2	no	—
205	C₃×C₄	—	4	2	no	—
206	C₃×C₄	C₅×#100	1	2	yes	—
207	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
208	C₃×C₄	C₅×#137	1	2	yes	—
209	C₂×C₅	—	4	1	no	—
210	C₂×C₅	—	8	2	no	—
211	C₂×C₅	C₂×C₃×#100	1	1	yes	characteristic
212	C₂×C₅	C₃×#417	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
215	Q₈	—	4	2	no	—
216	D₄	—	4	2	no	—
217	D₄	—	4	4	no	—
218	D₄	—	4	2	no	—
219	D₄	—	4	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
226	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
228	C₂×C₄	C₂×C₃×C₅×D₄	1	2	yes	—
229	C₈	—	4	4	no	—
230	C₂×C₃	—	4	1	no	—
231	C₂×C₃	—	8	2	no	—
232	C₂×C₃	C₂×C₅×#100	1	1	yes	characteristic
233	C₂×C₃	C₅×#417	1	2	yes	—
234	C₅	C₃×#1560	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	—	2	1	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	4	no	—
239	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	C₃×C₅×#100	1	2	yes	—
244	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
245	C₄	C₃×C₅×#137	1	2	yes	—
246	C₃	C₅×#1560	1	1	yes	characteristic, direct product factor, Sylow
247	C₂	—	4	1	no	—
248	C₂	—	8	2	no	—
249	C₂	C₂×C₃×C₅×#100	1	1	yes	characteristic
250	C₂	C₃×C₅×#417	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
prime radical	C₃×C₅×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂×C₃×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#423	C₂	semidirect
3. C₃×C₅×#333	C₂	semidirect
4. C₃×C₅×#331	C₂	semidirect
5. C₅×#1560	C₃	direct (N × H)
16. C₃×#1560	C₅	direct (N × H)
18. C₅×#423	C₂×C₃	semidirect
19. C₅×#333	C₂×C₃	semidirect
20. C₅×#331	C₂×C₃	semidirect
37. C₃×#423	C₂×C₅	semidirect
38. C₃×#333	C₂×C₅	semidirect
39. C₃×#331	C₂×C₅	semidirect
50. #1560	C₃×C₅	direct (N × H)
94. #423	C₂×C₃×C₅	semidirect
95. #333	C₂×C₃×C₅	semidirect
96. #331	C₂×C₃×C₅	semidirect
197. C₃×C₅	#1560	direct (N × H)
234. C₅	C₃×#1560	direct (N × H)
246. C₃	C₅×#1560	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×#532	2
C₃×C₅×#423	2
C₃×C₅×#333	2
C₃×C₅×#331	2
C₅×#1560	3
C₃×#1560	5

Subgroups — C3×C5×#1560 #14274594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1560 #14274594