Subgroups — C₃×C₅×#1569 #14274603

Order 1920.

332 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#432	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#423	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×#1569	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	—	2	1	no	—
8	C₂×C₃×C₅×M₁₆	C₄	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₃×C₅×C₄⋊C₈	—	2	1	no	—
11	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₅×C₂²⋊C₈	—	2	1	no	—
15	C₃×C₅×#97	—	2	1	no	—
16	C₃×C₅×#97	C₄	1	1	yes	characteristic
17	C₃×C₅×#97	C₄	1	1	yes	characteristic
18	C₃×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₃×#1569	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
20	C₅×#432	C₂×C₃	1	1	yes	characteristic
21	C₅×#423	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	4	no	—
29	C₃×C₅×M₁₆	—	4	2	no	—
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	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	Q₈	1	1	yes	characteristic
38	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
39	C₃×#432	C₂×C₅	1	1	yes	characteristic
40	C₃×#423	C₂×C₅	1	1	yes	characteristic
41	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
42	C₃×#335	C₂×C₅	1	2	yes	complemented
43	C₃×#334	C₂×C₅	1	2	yes	complemented
44	C₂×C₅×M₁₆	—	2	1	no	—
45	C₂×C₅×M₁₆	C₃×C₄	1	2	yes	—
46	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
47	C₅×C₄⋊C₈	—	2	1	no	—
48	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
49	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
50	C₅×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
51	C₅×C₂²⋊C₈	—	2	1	no	—
52	C₅×#97	—	2	1	no	—
53	C₅×#97	C₃×C₄	1	1	yes	characteristic
54	C₅×#97	C₃×C₄	1	1	yes	characteristic
55	C₄×C₅×C₈	C₂²×C₃	1	2	yes	—
56	#1569	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
57	C₂³×C₃×C₅	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	—	2	2	no	—
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	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
68	C₃×C₅×C₈	—	2	2	no	—
69	C₃×C₅×C₈	—	4	1	no	—
70	C₃×C₅×C₈	—	4	2	no	—
71	C₃×C₅×C₈	—	4	1	no	—
72	C₃×C₅×C₈	M₁₆	1	4	yes	complemented
73	C₂×C₃×M₁₆	—	2	1	no	—
74	C₂×C₃×M₁₆	C₄×C₅	1	2	yes	—
75	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
76	C₃×C₄⋊C₈	—	2	1	no	—
77	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
78	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
79	C₃×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
80	C₃×C₂²⋊C₈	—	2	1	no	—
81	C₃×#97	—	2	1	no	—
82	C₃×#97	C₄×C₅	1	1	yes	characteristic
83	C₃×#97	C₄×C₅	1	1	yes	characteristic
84	C₃×C₄×C₈	C₂²×C₅	1	2	yes	—
85	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₃×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₃×Q₈	1	1	yes	characteristic
88	C₅×M₁₆	—	2	4	no	—
89	C₅×M₁₆	—	4	2	no	—
90	C₂×C₅×C₈	—	2	2	no	—
91	C₂×C₅×C₈	—	2	1	no	—
92	C₂×C₅×C₈	—	2	1	no	—
93	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
95	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
96	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
97	C₄²×C₅	C₃×Q₈	1	1	yes	characteristic
98	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
99	#432	C₂×C₃×C₅	1	1	yes	characteristic
100	#423	C₂×C₃×C₅	1	1	yes	characteristic
101	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
102	#335	C₂×C₃×C₅	1	2	yes	complemented
103	#334	C₂×C₃×C₅	1	2	yes	complemented
104	C₂²×C₃×C₅	—	2	1	no	—
105	C₂²×C₃×C₅	—	2	1	no	—
106	C₂²×C₃×C₅	—	2	1	no	—
107	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₃×C₄×C₅	—	2	1	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	1	no	—
111	C₃×C₄×C₅	—	2	1	no	—
112	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
113	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₃×C₄×C₅	#137	1	1	yes	characteristic
115	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₅×D₄	1	1	yes	characteristic
117	C₂²×C₃×C₄	C₅×Q₈	1	1	yes	characteristic
118	C₃×M₁₆	—	2	4	no	—
119	C₃×M₁₆	—	4	2	no	—
120	C₂×C₃×C₈	—	2	1	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₂×C₃×C₈	—	2	2	no	—
123	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
124	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
125	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
126	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
127	C₃×C₄²	C₅×Q₈	1	1	yes	characteristic
128	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
129	C₂³×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅	—	2	2	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	4	1	no	—
134	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	C₃×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
140	C₅×C₈	—	2	2	no	—
141	C₅×C₈	—	4	1	no	—
142	C₅×C₈	—	4	2	no	—
143	C₅×C₈	—	4	1	no	—
144	C₅×C₈	C₃×M₁₆	1	4	yes	complemented
145	C₂×M₁₆	—	2	1	no	—
146	C₂×M₁₆	C₃×C₄×C₅	1	2	yes	—
147	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
148	C₄⋊C₈	—	2	1	no	—
149	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
150	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
151	C₄⋊C₈	C₂²×C₃×C₅	1	1	yes	characteristic
152	C₂²⋊C₈	—	2	1	no	—
153	#97	—	2	1	no	—
154	#97	C₃×C₄×C₅	1	1	yes	characteristic
155	#97	C₃×C₄×C₅	1	1	yes	characteristic
156	C₄×C₈	C₂²×C₃×C₅	1	2	yes	—
157	C₂×C₃×C₅	—	4	1	no	—
158	C₂×C₃×C₅	#423	1	1	yes	characteristic
159	C₂×C₃×C₅	#428	1	1	yes	characteristic
160	C₂×C₃×C₅	#430	1	1	yes	characteristic
161	C₂³×C₃	C₅×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₃×C₄	—	2	2	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	4	1	no	—
166	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
167	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₃×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
172	C₃×C₈	—	2	2	no	—
173	C₃×C₈	—	4	1	no	—
174	C₃×C₈	—	4	2	no	—
175	C₃×C₈	—	4	1	no	—
176	C₃×C₈	C₅×M₁₆	1	4	yes	complemented
177	C₂²×C₅	—	2	1	no	—
178	C₂²×C₅	—	2	1	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
181	C₄×C₅	—	2	1	no	—
182	C₄×C₅	—	2	1	no	—
183	C₄×C₅	—	2	1	no	—
184	C₄×C₅	—	2	1	no	—
185	C₄×C₅	C₂×C₃×M₁₆	1	2	yes	—
186	C₄×C₅	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
187	C₄×C₅	C₃×#137	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
189	C₂²×C₄	C₃×C₅×D₄	1	1	yes	characteristic
190	C₂²×C₄	C₃×C₅×Q₈	1	1	yes	characteristic
191	M₁₆	—	2	4	no	—
192	M₁₆	—	4	2	no	—
193	C₂×C₈	—	2	1	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	2	2	no	—
196	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
197	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
198	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
199	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
200	C₄²	C₃×C₅×Q₈	1	1	yes	characteristic
201	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
202	C₃×C₅	#1569	1	1	yes	characteristic, direct product factor, Hall
203	C₂²×C₃	—	2	1	no	—
204	C₂²×C₃	—	2	1	no	—
205	C₂²×C₃	—	2	1	no	—
206	C₂²×C₃	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
207	C₃×C₄	—	2	1	no	—
208	C₃×C₄	—	2	1	no	—
209	C₃×C₄	—	2	1	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	C₂×C₅×M₁₆	1	2	yes	—
212	C₃×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
213	C₃×C₄	C₅×#137	1	1	yes	characteristic
214	C₂×C₅	—	4	1	no	—
215	C₂×C₅	C₃×#423	1	1	yes	characteristic
216	C₂×C₅	C₃×#428	1	1	yes	characteristic
217	C₂×C₅	C₃×#430	1	1	yes	characteristic
218	C₂³	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	4	1	no	—
223	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
224	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
225	C₂×C₄	C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
226	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
227	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
228	C₂×C₄	C₂×C₃×C₅×Q₈	1	1	yes	characteristic
229	C₈	—	2	2	no	—
230	C₈	—	4	2	no	—
231	C₈	—	4	1	no	—
232	C₈	—	4	1	no	—
233	C₈	C₃×C₅×M₁₆	1	4	yes	complemented
234	C₂×C₃	—	4	1	no	—
235	C₂×C₃	C₅×#423	1	1	yes	characteristic
236	C₂×C₃	C₅×#428	1	1	yes	characteristic
237	C₂×C₃	C₅×#430	1	1	yes	characteristic
238	C₅	C₃×#1569	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	C₂×C₃×C₅×C₄⋊C₄	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	C₂×C₃×C₅×M₁₆	1	2	yes	—
248	C₄	C₃×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
249	C₄	C₃×C₅×#137	1	1	yes	characteristic
250	C₃	C₅×#1569	1	1	yes	characteristic, direct product factor, Sylow
251	C₂	—	4	1	no	—
252	C₂	C₃×C₅×#423	1	1	yes	characteristic
253	C₂	C₃×C₅×#428	1	1	yes	characteristic
254	C₂	C₃×C₅×#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₄²	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂³×C₃×C₅	C₄⋊C₄
LCS	G ▷ C₂×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×#1569	C₃	direct (N × H)
19. C₃×#1569	C₅	direct (N × H)
23. C₅×#335	C₂×C₃	semidirect
24. C₅×#334	C₂×C₃	semidirect
42. C₃×#335	C₂×C₅	semidirect
43. C₃×#334	C₂×C₅	semidirect
56. #1569	C₃×C₅	direct (N × H)
72. C₃×C₅×C₈	M₁₆	semidirect
102. #335	C₂×C₃×C₅	semidirect
103. #334	C₂×C₃×C₅	semidirect
144. C₅×C₈	C₃×M₁₆	semidirect
176. C₃×C₈	C₅×M₁₆	semidirect
202. C₃×C₅	#1569	direct (N × H)
233. C₈	C₃×C₅×M₁₆	semidirect
238. C₅	C₃×#1569	direct (N × H)
250. C₃	C₅×#1569	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×#432	2
C₃×C₅×#423	2
C₃×C₄×C₅×M₁₆	2
C₃×C₅×#335	2
C₃×C₅×#334	2
C₅×#1569	3
C₃×#1569	5

Subgroups — C3×C5×#1569 #14274603

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1569 #14274603