Subgroups — C₃×C₅×#1580 #14274614

Order 1920.

384 conjugacy classes of subgroups.

The subgroup lattice has 224 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×C₈×D₄	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#334	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₅×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1580	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×D₈	C₄	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₈	—	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×#1580	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₄×C₅×D₈	C₂×C₃	1	1	yes	characteristic
19	C₅×C₈×D₄	C₂×C₃	1	2	yes	complemented
20	C₅×#334	C₂×C₃	1	1	yes	characteristic, complemented
21	C₅×D₄⋊C₈	C₂×C₃	1	2	yes	complemented
22	C₅×#322	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	C₈	1	4	yes	complemented
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₂×C₃×C₅×C₈	—	2	1	no	—
28	C₂×C₃×C₅×C₈	—	4	4	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	2	yes	—
32	C₃×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₃×C₄×D₈	C₂×C₅	1	1	yes	characteristic
36	C₃×C₈×D₄	C₂×C₅	1	2	yes	complemented
37	C₃×#334	C₂×C₅	1	1	yes	characteristic, complemented
38	C₃×D₄⋊C₈	C₂×C₅	1	2	yes	complemented
39	C₃×#322	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂×C₅×D₈	C₃×C₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	—	2	2	no	—
42	C₄×C₅×D₄	C₂²×C₃	1	2	yes	—
43	C₅×#107	C₃×C₄	1	1	yes	characteristic
44	C₅×C₄⋊C₈	C₂²×C₃	1	2	yes	—
45	C₅×#102	C₃×C₄	1	2	yes	—
46	C₅×C₂²⋊C₈	—	2	2	no	—
47	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
50	#1580	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
51	C₂³×C₃×C₅	—	2	2	no	—
52	C₃×C₅×D₄	—	2	2	no	—
53	C₃×C₅×D₄	C₂×C₈	1	4	yes	complemented
54	C₂×C₃×C₄×C₅	—	2	2	no	—
55	C₂×C₃×C₄×C₅	—	4	2	no	—
56	C₂×C₃×C₄×C₅	—	4	2	no	—
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
60	C₃×C₅×C₈	—	2	1	no	—
61	C₃×C₅×C₈	—	2	2	no	—
62	C₃×C₅×C₈	—	4	1	no	—
63	C₃×C₅×C₈	—	4	2	no	—
64	C₃×C₅×C₈	C₂×C₈	1	2	yes	complemented
65	C₂×C₃×D₈	C₄×C₅	1	1	yes	characteristic
66	C₂²×C₃×C₈	—	2	2	no	—
67	C₃×C₄×D₄	C₂²×C₅	1	2	yes	—
68	C₃×#107	C₄×C₅	1	1	yes	characteristic
69	C₃×C₄⋊C₈	C₂²×C₅	1	2	yes	—
70	C₃×#102	C₄×C₅	1	2	yes	—
71	C₃×C₂²⋊C₈	—	2	2	no	—
72	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
73	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
74	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
75	C₂×C₅×D₄	C₂×C₃×C₄	1	2	yes	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₅×D₈	C₃×C₈	1	4	yes	complemented
78	C₂×C₅×C₈	—	2	2	no	—
79	C₂×C₅×C₈	—	2	1	no	—
80	C₂×C₅×C₈	—	4	4	no	—
81	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
83	C₂×C₅×C₈	C₃×D₄	1	2	yes	—
84	C₅×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
85	C₅×C₂²⋊C₄	—	2	2	no	—
86	C₄²×C₅	C₂³×C₃	1	1	yes	characteristic
87	C₄×D₈	C₂×C₃×C₅	1	1	yes	characteristic
88	C₈×D₄	C₂×C₃×C₅	1	2	yes	complemented
89	#334	C₂×C₃×C₅	1	1	yes	characteristic, complemented
90	D₄⋊C₈	C₂×C₃×C₅	1	2	yes	complemented
91	#322	C₂×C₃×C₅	1	1	yes	characteristic, complemented
92	C₂²×C₃×C₅	—	2	4	no	—
93	C₂²×C₃×C₅	—	4	2	no	—
94	C₂²×C₃×C₅	—	4	2	no	—
95	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
96	C₃×C₄×C₅	—	2	1	no	—
97	C₃×C₄×C₅	—	4	2	no	—
98	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
99	C₃×C₄×C₅	#131	1	1	yes	characteristic
100	C₃×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₃×D₄	C₂×C₄×C₅	1	2	yes	—
102	C₂²×C₃×C₄	—	2	2	no	—
103	C₃×D₈	C₅×C₈	1	4	yes	complemented
104	C₂×C₃×C₈	—	2	2	no	—
105	C₂×C₃×C₈	—	2	1	no	—
106	C₂×C₃×C₈	—	4	4	no	—
107	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
108	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
109	C₂×C₃×C₈	C₅×D₄	1	2	yes	—
110	C₃×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
111	C₃×C₂²⋊C₄	—	2	2	no	—
112	C₃×C₄²	C₂³×C₅	1	1	yes	characteristic
113	C₂³×C₅	—	2	2	no	—
114	C₅×D₄	—	2	2	no	—
115	C₅×D₄	C₂×C₃×C₈	1	4	yes	complemented
116	C₂×C₄×C₅	—	2	2	no	—
117	C₂×C₄×C₅	—	4	2	no	—
118	C₂×C₄×C₅	—	4	2	no	—
119	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
122	C₅×C₈	—	2	1	no	—
123	C₅×C₈	—	2	2	no	—
124	C₅×C₈	—	4	1	no	—
125	C₅×C₈	—	4	2	no	—
126	C₅×C₈	C₂×C₃×C₈	1	2	yes	complemented
127	C₂×D₈	C₃×C₄×C₅	1	1	yes	characteristic
128	C₂²×C₈	—	2	2	no	—
129	C₄×D₄	C₂²×C₃×C₅	1	2	yes	—
130	#107	C₃×C₄×C₅	1	1	yes	characteristic
131	C₄⋊C₈	C₂²×C₃×C₅	1	2	yes	—
132	#102	C₃×C₄×C₅	1	2	yes	—
133	C₂²⋊C₈	—	2	2	no	—
134	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
135	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
136	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
137	C₂×C₃×C₅	—	4	4	no	—
138	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
139	C₂×C₃×C₅	#442	1	1	yes	characteristic
140	C₂×C₃×C₅	#444	1	1	yes	characteristic
141	C₂³×C₃	—	2	2	no	—
142	C₃×D₄	—	2	2	no	—
143	C₃×D₄	C₂×C₅×C₈	1	4	yes	complemented
144	C₂×C₃×C₄	—	2	2	no	—
145	C₂×C₃×C₄	—	4	2	no	—
146	C₂×C₃×C₄	—	4	2	no	—
147	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
148	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
149	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
150	C₃×C₈	—	2	1	no	—
151	C₃×C₈	—	2	2	no	—
152	C₃×C₈	—	4	1	no	—
153	C₃×C₈	—	4	2	no	—
154	C₃×C₈	C₂×C₅×C₈	1	2	yes	complemented
155	C₂²×C₅	—	2	4	no	—
156	C₂²×C₅	—	4	2	no	—
157	C₂²×C₅	—	4	2	no	—
158	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
159	C₄×C₅	—	2	1	no	—
160	C₄×C₅	—	4	2	no	—
161	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
162	C₄×C₅	C₃×#131	1	1	yes	characteristic
163	C₄×C₅	C₃×AGL(1,ℤ/8)	1	2	yes	—
164	C₂×D₄	C₂×C₃×C₄×C₅	1	2	yes	—
165	C₂²×C₄	—	2	2	no	—
166	D₈	C₃×C₅×C₈	1	4	yes	complemented
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	4	4	no	—
170	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
171	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
172	C₂×C₈	C₃×C₅×D₄	1	2	yes	—
173	C₄⋊C₄	C₂×C₃×C₄×C₅	1	2	yes	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₄²	C₂³×C₃×C₅	1	1	yes	characteristic
176	C₃×C₅	#1580	1	1	yes	characteristic, direct product factor, Hall
177	C₂²×C₃	—	2	4	no	—
178	C₂²×C₃	—	4	2	no	—
179	C₂²×C₃	—	4	2	no	—
180	C₂²×C₃	C₄×C₅×D₄	1	1	yes	characteristic
181	C₃×C₄	—	2	1	no	—
182	C₃×C₄	—	4	2	no	—
183	C₃×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
184	C₃×C₄	C₅×#131	1	1	yes	characteristic
185	C₃×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
186	C₂×C₅	—	4	4	no	—
187	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
188	C₂×C₅	C₃×#442	1	1	yes	characteristic
189	C₂×C₅	C₃×#444	1	1	yes	characteristic
190	C₂³	—	2	2	no	—
191	D₄	—	2	2	no	—
192	D₄	C₂×C₃×C₅×C₈	1	4	yes	complemented
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	2	no	—
196	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
197	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
198	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
199	C₈	—	2	1	no	—
200	C₈	—	2	2	no	—
201	C₈	—	4	1	no	—
202	C₈	—	4	2	no	—
203	C₈	C₂×C₃×C₅×C₈	1	2	yes	complemented
204	C₂×C₃	—	4	4	no	—
205	C₂×C₃	C₅×C₈×D₄	1	1	yes	characteristic
206	C₂×C₃	C₅×#442	1	1	yes	characteristic
207	C₂×C₃	C₅×#444	1	1	yes	characteristic
208	C₅	C₃×#1580	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²	—	2	4	no	—
210	C₂²	—	4	2	no	—
211	C₂²	—	4	2	no	—
212	C₂²	C₃×C₄×C₅×D₄	1	1	yes	characteristic
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
216	C₄	C₃×C₅×#131	1	1	yes	characteristic
217	C₄	C₃×C₅×AGL(1,ℤ/8)	1	2	yes	—
218	C₃	C₅×#1580	1	1	yes	characteristic, direct product factor, Sylow
219	C₂	—	4	4	no	—
220	C₂	C₃×C₅×C₈×D₄	1	1	yes	characteristic
221	C₂	C₃×C₅×#442	1	1	yes	characteristic
222	C₂	C₃×C₅×#444	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	C₄×D₄
prime radical	C₂×C₃×C₅×D₈	C₄
LCS	G ▷ 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×C₈×D₄	C₂	semidirect
3. C₃×C₅×#334	C₂	semidirect
4. C₃×C₅×D₄⋊C₈	C₂	semidirect
5. C₃×C₅×#322	C₂	semidirect
6. C₅×#1580	C₃	direct (N × H)
17. C₃×#1580	C₅	direct (N × H)
19. C₅×C₈×D₄	C₂×C₃	semidirect
20. C₅×#334	C₂×C₃	semidirect
21. C₅×D₄⋊C₈	C₂×C₃	semidirect
22. C₅×#322	C₂×C₃	semidirect
25. C₃×C₅×D₈	C₈	semidirect
36. C₃×C₈×D₄	C₂×C₅	semidirect
37. C₃×#334	C₂×C₅	semidirect
38. C₃×D₄⋊C₈	C₂×C₅	semidirect
39. C₃×#322	C₂×C₅	semidirect
50. #1580	C₃×C₅	direct (N × H)
53. C₃×C₅×D₄	C₂×C₈	semidirect
64. C₃×C₅×C₈	C₂×C₈	semidirect
77. C₅×D₈	C₃×C₈	semidirect
88. C₈×D₄	C₂×C₃×C₅	semidirect
89. #334	C₂×C₃×C₅	semidirect
90. D₄⋊C₈	C₂×C₃×C₅	semidirect
91. #322	C₂×C₃×C₅	semidirect
103. C₃×D₈	C₅×C₈	semidirect
115. C₅×D₄	C₂×C₃×C₈	semidirect
126. C₅×C₈	C₂×C₃×C₈	semidirect
143. C₃×D₄	C₂×C₅×C₈	semidirect
154. C₃×C₈	C₂×C₅×C₈	semidirect
166. D₈	C₃×C₅×C₈	semidirect
176. C₃×C₅	#1580	direct (N × H)
192. D₄	C₂×C₃×C₅×C₈	semidirect
203. C₈	C₂×C₃×C₅×C₈	semidirect
208. C₅	C₃×#1580	direct (N × H)
218. C₃	C₅×#1580	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×C₈×D₄	2
C₃×C₅×#334	2
C₃×C₅×D₄⋊C₈	2
C₃×C₅×#322	2
C₅×#1580	3
C₃×#1580	5

Subgroups — C3×C5×#1580 #14274614

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1580 #14274614