Subgroups — C₃×C₅×#1589 #14274623

Order 1920.

360 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#436	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1589	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×D₈	C₄	1	1	yes	characteristic
8	C₂×C₃×C₅×M₁₆	—	2	2	no	—
9	C₃×C₄×C₅×D₄	C₂²	1	2	yes	—
10	C₃×C₅×#107	C₄	1	1	yes	characteristic
11	C₃×C₅×C₄⋊C₈	C₂²	1	2	yes	—
12	C₃×C₅×#102	C₄	1	2	yes	—
13	C₃×C₅×C₂²⋊C₈	—	2	2	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	C₃×#1589	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₄×C₅×D₈	C₂×C₃	1	1	yes	characteristic
19	C₅×#436	C₂×C₃	1	2	yes	complemented
20	C₅×#335	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×D₄⋊C₈	C₂×C₃	1	2	yes	complemented
22	C₅×C₈²	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×C₅×D₄	C₂×C₄	1	2	yes	—
24	C₂²×C₃×C₄×C₅	—	2	2	no	—
25	C₃×C₅×D₈	—	2	2	no	—
26	C₃×C₅×M₁₆	—	4	4	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	2	2	no	—
29	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
33	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
34	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
35	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
36	C₃×C₄×D₈	C₂×C₅	1	1	yes	characteristic
37	C₃×#436	C₂×C₅	1	2	yes	complemented
38	C₃×#335	C₂×C₅	1	1	yes	characteristic, complemented
39	C₃×D₄⋊C₈	C₂×C₅	1	2	yes	complemented
40	C₃×C₈²	C₂×C₅	1	1	yes	characteristic, complemented
41	C₂×C₅×D₈	C₃×C₄	1	1	yes	characteristic
42	C₂×C₅×M₁₆	—	2	2	no	—
43	C₄×C₅×D₄	C₂²×C₃	1	2	yes	—
44	C₅×#107	C₃×C₄	1	1	yes	characteristic
45	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
46	C₅×#102	C₃×C₄	1	2	yes	—
47	C₅×C₂²⋊C₈	—	2	2	no	—
48	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
50	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
51	#1589	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
52	C₂³×C₃×C₅	—	2	2	no	—
53	C₃×C₅×D₄	—	2	2	no	—
54	C₃×C₅×D₄	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	2	2	no	—
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	—	4	2	no	—
58	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
61	C₃×C₅×C₈	—	2	2	no	—
62	C₃×C₅×C₈	—	2	1	no	—
63	C₃×C₅×C₈	—	2	1	no	—
64	C₃×C₅×C₈	—	4	2	no	—
65	C₃×C₅×C₈	M₁₆	1	2	yes	complemented
66	C₃×C₅×C₈	D₈	1	2	yes	complemented
67	C₂×C₃×D₈	C₄×C₅	1	1	yes	characteristic
68	C₂×C₃×M₁₆	—	2	2	no	—
69	C₃×C₄×D₄	C₂²×C₅	1	2	yes	—
70	C₃×#107	C₄×C₅	1	1	yes	characteristic
71	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
72	C₃×#102	C₄×C₅	1	2	yes	—
73	C₃×C₂²⋊C₈	—	2	2	no	—
74	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
76	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
77	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
78	C₂²×C₄×C₅	—	2	2	no	—
79	C₅×D₈	—	2	2	no	—
80	C₅×M₁₆	—	4	4	no	—
81	C₂×C₅×C₈	—	2	1	no	—
82	C₂×C₅×C₈	—	2	2	no	—
83	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
84	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
86	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
87	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
88	C₅×C₂²⋊C₄	—	2	2	no	—
89	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
90	C₄×D₈	C₂×C₃×C₅	1	1	yes	characteristic
91	#436	C₂×C₃×C₅	1	2	yes	complemented
92	#335	C₂×C₃×C₅	1	1	yes	characteristic, complemented
93	D₄⋊C₈	C₂×C₃×C₅	1	2	yes	complemented
94	C₈²	C₂×C₃×C₅	1	1	yes	characteristic, complemented
95	C₂²×C₃×C₅	—	4	2	no	—
96	C₂²×C₃×C₅	—	4	2	no	—
97	C₂²×C₃×C₅	—	4	2	no	—
98	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
99	C₃×C₄×C₅	—	2	1	no	—
100	C₃×C₄×C₅	—	4	2	no	—
101	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
102	C₃×C₄×C₅	#131	1	1	yes	characteristic
103	C₃×C₄×C₅	C₂×D₈	1	1	yes	characteristic
104	C₃×C₄×C₅	#135	1	1	yes	characteristic
105	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
106	C₂²×C₃×C₄	—	2	2	no	—
107	C₃×D₈	—	2	2	no	—
108	C₃×M₁₆	—	4	4	no	—
109	C₂×C₃×C₈	—	2	1	no	—
110	C₂×C₃×C₈	—	2	2	no	—
111	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
113	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
114	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
115	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
116	C₃×C₂²⋊C₄	—	2	2	no	—
117	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
118	C₂³×C₅	—	2	2	no	—
119	C₅×D₄	—	2	2	no	—
120	C₅×D₄	—	2	2	no	—
121	C₂×C₄×C₅	—	2	2	no	—
122	C₂×C₄×C₅	—	4	2	no	—
123	C₂×C₄×C₅	—	4	2	no	—
124	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
125	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
127	C₅×C₈	—	2	2	no	—
128	C₅×C₈	—	2	1	no	—
129	C₅×C₈	—	2	1	no	—
130	C₅×C₈	—	4	2	no	—
131	C₅×C₈	C₃×M₁₆	1	2	yes	complemented
132	C₅×C₈	C₃×D₈	1	2	yes	complemented
133	C₂×D₈	C₃×C₄×C₅	1	1	yes	characteristic
134	C₂×M₁₆	—	2	2	no	—
135	C₄×D₄	C₂²×C₃×C₅	1	2	yes	—
136	#107	C₃×C₄×C₅	1	1	yes	characteristic
137	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
138	#102	C₃×C₄×C₅	1	2	yes	—
139	C₂²⋊C₈	—	2	2	no	—
140	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
141	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
142	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
143	C₂×C₃×C₅	—	8	2	no	—
144	C₂×C₃×C₅	#436	1	1	yes	characteristic
145	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
146	C₂×C₃×C₅	#443	1	1	yes	characteristic
147	C₂³×C₃	—	2	2	no	—
148	C₃×D₄	—	2	2	no	—
149	C₃×D₄	—	2	2	no	—
150	C₂×C₃×C₄	—	2	2	no	—
151	C₂×C₃×C₄	—	4	2	no	—
152	C₂×C₃×C₄	—	4	2	no	—
153	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
154	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
155	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
156	C₃×C₈	—	2	2	no	—
157	C₃×C₈	—	2	1	no	—
158	C₃×C₈	—	2	1	no	—
159	C₃×C₈	—	4	2	no	—
160	C₃×C₈	C₅×M₁₆	1	2	yes	complemented
161	C₃×C₈	C₅×D₈	1	2	yes	complemented
162	C₂²×C₅	—	4	2	no	—
163	C₂²×C₅	—	4	2	no	—
164	C₂²×C₅	—	4	2	no	—
165	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
166	C₄×C₅	—	2	1	no	—
167	C₄×C₅	—	4	2	no	—
168	C₄×C₅	C₂×C₃×M₁₆	1	1	yes	characteristic
169	C₄×C₅	C₃×#131	1	1	yes	characteristic
170	C₄×C₅	C₂×C₃×D₈	1	1	yes	characteristic
171	C₄×C₅	C₃×#135	1	1	yes	characteristic
172	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
173	C₂²×C₄	—	2	2	no	—
174	D₈	—	2	2	no	—
175	M₁₆	—	4	4	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
179	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
180	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
181	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
182	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
183	C₂²⋊C₄	—	2	2	no	—
184	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
185	C₃×C₅	#1589	1	1	yes	characteristic, direct product factor, Hall
186	C₂²×C₃	—	4	2	no	—
187	C₂²×C₃	—	4	2	no	—
188	C₂²×C₃	—	4	2	no	—
189	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
190	C₃×C₄	—	2	1	no	—
191	C₃×C₄	—	4	2	no	—
192	C₃×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
193	C₃×C₄	C₅×#131	1	1	yes	characteristic
194	C₃×C₄	C₂×C₅×D₈	1	1	yes	characteristic
195	C₃×C₄	C₅×#135	1	1	yes	characteristic
196	C₂×C₅	—	8	2	no	—
197	C₂×C₅	C₃×#436	1	1	yes	characteristic
198	C₂×C₅	C₃×C₄×D₈	1	1	yes	characteristic
199	C₂×C₅	C₃×#443	1	1	yes	characteristic
200	C₂³	—	2	2	no	—
201	D₄	—	2	2	no	—
202	D₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
207	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
208	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
209	C₈	—	2	2	no	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	4	2	no	—
213	C₈	C₃×C₅×M₁₆	1	2	yes	complemented
214	C₈	C₃×C₅×D₈	1	2	yes	complemented
215	C₂×C₃	—	8	2	no	—
216	C₂×C₃	C₅×#436	1	1	yes	characteristic
217	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
218	C₂×C₃	C₅×#443	1	1	yes	characteristic
219	C₅	C₃×#1589	1	1	yes	characteristic, direct product factor, Sylow
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	C₃×C₄×C₅×D₄	1	1	yes	characteristic
224	C₄	—	2	1	no	—
225	C₄	—	4	2	no	—
226	C₄	C₂×C₃×C₅×M₁₆	1	1	yes	characteristic
227	C₄	C₃×C₅×#131	1	1	yes	characteristic
228	C₄	C₂×C₃×C₅×D₈	1	1	yes	characteristic
229	C₄	C₃×C₅×#135	1	1	yes	characteristic
230	C₃	C₅×#1589	1	1	yes	characteristic, direct product factor, Sylow
231	C₂	—	8	2	no	—
232	C₂	C₃×C₅×#436	1	1	yes	characteristic
233	C₂	C₃×C₄×C₅×D₈	1	1	yes	characteristic
234	C₂	C₃×C₅×#443	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	C₄×D₄
prime radical	C₂×C₃×C₅×D₈	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₅×#436	C₂	semidirect
3. C₃×C₅×#335	C₂	semidirect
4. C₃×C₅×D₄⋊C₈	C₂	semidirect
5. C₃×C₅×C₈²	C₂	semidirect
6. C₅×#1589	C₃	direct (N × H)
17. C₃×#1589	C₅	direct (N × H)
19. C₅×#436	C₂×C₃	semidirect
20. C₅×#335	C₂×C₃	semidirect
21. C₅×D₄⋊C₈	C₂×C₃	semidirect
22. C₅×C₈²	C₂×C₃	semidirect
37. C₃×#436	C₂×C₅	semidirect
38. C₃×#335	C₂×C₅	semidirect
39. C₃×D₄⋊C₈	C₂×C₅	semidirect
40. C₃×C₈²	C₂×C₅	semidirect
51. #1589	C₃×C₅	direct (N × H)
65. C₃×C₅×C₈	M₁₆	semidirect
66. C₃×C₅×C₈	D₈	semidirect
91. #436	C₂×C₃×C₅	semidirect
92. #335	C₂×C₃×C₅	semidirect
93. D₄⋊C₈	C₂×C₃×C₅	semidirect
94. C₈²	C₂×C₃×C₅	semidirect
131. C₅×C₈	C₃×M₁₆	semidirect
132. C₅×C₈	C₃×D₈	semidirect
160. C₃×C₈	C₅×M₁₆	semidirect
161. C₃×C₈	C₅×D₈	semidirect
185. C₃×C₅	#1589	direct (N × H)
213. C₈	C₃×C₅×M₁₆	semidirect
214. C₈	C₃×C₅×D₈	semidirect
219. C₅	C₃×#1589	direct (N × H)
230. C₃	C₅×#1589	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×C₄×C₅×D₈	2
C₃×C₅×#436	2
C₃×C₅×#335	2
C₃×C₅×D₄⋊C₈	2
C₃×C₅×C₈²	2
C₅×#1589	3
C₃×#1589	5

Subgroups — C3×C5×#1589 #14274623

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1589 #14274623