Subgroups — C₃×C₅×#2240 #14275274

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₅×#493	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×#357	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2240	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃×C₅×D₈	—	2	1	no	—
7	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
8	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₃×C₅×#127	—	2	1	no	—
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₃×C₅×#107	—	2	1	no	—
12	C₃×C₅×#107	C₂²	1	2	yes	—
13	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₃×#2240	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₅×#500	C₂×C₃	1	1	yes	characteristic, complemented
16	C₅×#493	C₂×C₃	1	1	yes	characteristic
17	C₅×#357	C₂×C₃	1	4	yes	complemented
18	C₄×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂×C₃×C₅×Q₈	—	4	1	no	—
20	C₂×C₃×C₅×D₄	—	2	2	no	—
21	C₂×C₃×C₅×D₄	—	4	1	no	—
22	C₂×C₃×C₅×D₄	—	4	1	no	—
23	C₃×C₅×D₈	—	2	4	no	—
24	C₃×C₅×D₈	—	4	2	no	—
25	C₂×C₃×C₅×C₈	D₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₃×C₅×C₄⋊C₄	—	2	2	no	—
28	C₃×C₅×C₄⋊C₄	—	4	1	no	—
29	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₃×C₅×C₁₆	—	2	2	no	—
31	C₃×#500	C₂×C₅	1	1	yes	characteristic, complemented
32	C₃×#493	C₂×C₅	1	1	yes	characteristic
33	C₃×#357	C₂×C₅	1	4	yes	complemented
34	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
35	C₂×C₅×D₈	—	2	1	no	—
36	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
37	C₅×C₄⋊Q₈	—	2	1	no	—
38	C₅×#127	—	2	1	no	—
39	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
40	C₅×#107	—	2	1	no	—
41	C₅×#107	C₂²×C₃	1	2	yes	—
42	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
43	#2240	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
44	C₂³×C₃×C₅	—	4	2	no	—
45	C₃×C₅×Q₈	—	8	2	no	—
46	C₃×C₅×D₄	—	4	2	no	—
47	C₃×C₅×D₄	—	4	2	no	—
48	C₃×C₅×D₄	—	8	2	no	—
49	C₃×C₅×D₄	—	8	2	no	—
50	C₂×C₃×C₄×C₅	—	4	2	no	—
51	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
52	C₂×C₃×C₄×C₅	D₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
54	C₃×C₅×C₈	SD₁₆	1	2	yes	—
55	C₃×C₅×C₈	#41	1	2	yes	—
56	C₂×C₃×D₈	—	2	1	no	—
57	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
58	C₃×C₄⋊Q₈	—	2	1	no	—
59	C₃×#127	—	2	1	no	—
60	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
61	C₃×#107	—	2	1	no	—
62	C₃×#107	C₂²×C₅	1	2	yes	—
63	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
64	C₂×C₅×Q₈	—	4	1	no	—
65	C₂×C₅×D₄	—	2	2	no	—
66	C₂×C₅×D₄	—	4	1	no	—
67	C₂×C₅×D₄	—	4	1	no	—
68	C₅×D₈	—	2	4	no	—
69	C₅×D₈	—	4	2	no	—
70	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
71	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
72	C₅×C₄⋊C₄	—	2	2	no	—
73	C₅×C₄⋊C₄	—	4	1	no	—
74	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
75	C₅×C₁₆	—	2	2	no	—
76	#500	C₂×C₃×C₅	1	1	yes	characteristic, complemented
77	#493	C₂×C₃×C₅	1	1	yes	characteristic
78	#357	C₂×C₃×C₅	1	4	yes	complemented
79	C₄×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
80	C₂²×C₃×C₅	—	8	2	no	—
81	C₂²×C₃×C₅	—	8	2	no	—
82	C₂²×C₃×C₅	—	8	2	no	—
83	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
84	C₃×C₄×C₅	—	8	2	no	—
85	C₃×C₄×C₅	D₁₆	1	2	yes	—
86	C₃×C₄×C₅	SD₃₂	1	2	yes	—
87	C₃×C₄×C₅	#124	1	1	yes	characteristic
88	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₃×Q₈	—	4	1	no	—
90	C₂×C₃×D₄	—	2	2	no	—
91	C₂×C₃×D₄	—	4	1	no	—
92	C₂×C₃×D₄	—	4	1	no	—
93	C₃×D₈	—	2	4	no	—
94	C₃×D₈	—	4	2	no	—
95	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
96	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
97	C₃×C₄⋊C₄	—	2	2	no	—
98	C₃×C₄⋊C₄	—	4	1	no	—
99	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
100	C₃×C₁₆	—	2	2	no	—
101	C₂³×C₅	—	4	2	no	—
102	C₅×Q₈	—	8	2	no	—
103	C₅×D₄	—	4	2	no	—
104	C₅×D₄	—	4	2	no	—
105	C₅×D₄	—	8	2	no	—
106	C₅×D₄	—	8	2	no	—
107	C₂×C₄×C₅	—	4	2	no	—
108	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
109	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
111	C₅×C₈	C₃×SD₁₆	1	2	yes	—
112	C₅×C₈	C₃×#41	1	2	yes	—
113	C₂×D₈	—	2	1	no	—
114	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
115	C₄⋊Q₈	—	2	1	no	—
116	#127	—	2	1	no	—
117	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
118	#107	—	2	1	no	—
119	#107	C₂²×C₃×C₅	1	2	yes	—
120	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
121	C₂×C₃×C₅	—	16	2	no	—
122	C₂×C₃×C₅	#486	1	1	yes	characteristic
123	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
124	C₂×C₃×C₅	C₂×SD₃₂	1	1	yes	characteristic
125	C₂³×C₃	—	4	2	no	—
126	C₃×Q₈	—	8	2	no	—
127	C₃×D₄	—	4	2	no	—
128	C₃×D₄	—	4	2	no	—
129	C₃×D₄	—	8	2	no	—
130	C₃×D₄	—	8	2	no	—
131	C₂×C₃×C₄	—	4	2	no	—
132	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
133	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
134	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
135	C₃×C₈	C₅×SD₁₆	1	2	yes	—
136	C₃×C₈	C₅×#41	1	2	yes	—
137	C₂²×C₅	—	8	2	no	—
138	C₂²×C₅	—	8	2	no	—
139	C₂²×C₅	—	8	2	no	—
140	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
141	C₄×C₅	—	8	2	no	—
142	C₄×C₅	C₃×D₁₆	1	2	yes	—
143	C₄×C₅	C₃×SD₃₂	1	2	yes	—
144	C₄×C₅	C₃×#124	1	1	yes	characteristic
145	C₄×C₅	C₂×C₃×SD₁₆	1	1	yes	characteristic
146	C₂×Q₈	—	4	1	no	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	4	1	no	—
149	C₂×D₄	—	4	1	no	—
150	D₈	—	2	4	no	—
151	D₈	—	4	2	no	—
152	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
153	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
154	C₄⋊C₄	—	2	2	no	—
155	C₄⋊C₄	—	4	1	no	—
156	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
157	C₁₆	—	2	2	no	—
158	C₃×C₅	#2240	1	1	yes	characteristic, direct product factor, Hall
159	C₂²×C₃	—	8	2	no	—
160	C₂²×C₃	—	8	2	no	—
161	C₂²×C₃	—	8	2	no	—
162	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
163	C₃×C₄	—	8	2	no	—
164	C₃×C₄	C₅×D₁₆	1	2	yes	—
165	C₃×C₄	C₅×SD₃₂	1	2	yes	—
166	C₃×C₄	C₅×#124	1	1	yes	characteristic
167	C₃×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
168	C₂×C₅	—	16	2	no	—
169	C₂×C₅	C₃×#486	1	1	yes	characteristic
170	C₂×C₅	C₂×C₃×D₁₆	1	1	yes	characteristic
171	C₂×C₅	C₂×C₃×SD₃₂	1	1	yes	characteristic
172	C₂³	—	4	2	no	—
173	Q₈	—	8	2	no	—
174	D₄	—	4	2	no	—
175	D₄	—	4	2	no	—
176	D₄	—	8	2	no	—
177	D₄	—	8	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
180	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
181	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
182	C₈	C₃×C₅×SD₁₆	1	2	yes	—
183	C₈	C₃×C₅×#41	1	2	yes	—
184	C₂×C₃	—	16	2	no	—
185	C₂×C₃	C₅×#486	1	1	yes	characteristic
186	C₂×C₃	C₂×C₅×D₁₆	1	1	yes	characteristic
187	C₂×C₃	C₂×C₅×SD₃₂	1	1	yes	characteristic
188	C₅	C₃×#2240	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²	—	8	2	no	—
190	C₂²	—	8	2	no	—
191	C₂²	—	8	2	no	—
192	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
193	C₄	—	8	2	no	—
194	C₄	C₃×C₅×D₁₆	1	2	yes	—
195	C₄	C₃×C₅×SD₃₂	1	2	yes	—
196	C₄	C₃×C₅×#124	1	1	yes	characteristic
197	C₄	C₂×C₃×C₅×SD₁₆	1	1	yes	characteristic
198	C₃	C₅×#2240	1	1	yes	characteristic, direct product factor, Sylow
199	C₂	—	16	2	no	—
200	C₂	C₃×C₅×#486	1	1	yes	characteristic
201	C₂	C₂×C₃×C₅×D₁₆	1	1	yes	characteristic
202	C₂	C₂×C₃×C₅×SD₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
prime radical	C₃×C₅×#493	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₃×C₄²×C₅ ◁ C₃×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₅×#500	C₂	semidirect
3. C₃×C₅×#357	C₂	semidirect
4. C₃×C₄×C₅×C₁₆	C₂	semidirect
5. C₅×#2240	C₃	direct (N × H)
14. C₃×#2240	C₅	direct (N × H)
15. C₅×#500	C₂×C₃	semidirect
17. C₅×#357	C₂×C₃	semidirect
18. C₄×C₅×C₁₆	C₂×C₃	semidirect
31. C₃×#500	C₂×C₅	semidirect
33. C₃×#357	C₂×C₅	semidirect
34. C₃×C₄×C₁₆	C₂×C₅	semidirect
43. #2240	C₃×C₅	direct (N × H)
76. #500	C₂×C₃×C₅	semidirect
78. #357	C₂×C₃×C₅	semidirect
79. C₄×C₁₆	C₂×C₃×C₅	semidirect
158. C₃×C₅	#2240	direct (N × H)
188. C₅	C₃×#2240	direct (N × H)
198. C₃	C₅×#2240	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×#500	2
C₃×C₅×#493	2
C₃×C₅×#357	2
C₃×C₄×C₅×C₁₆	2
C₅×#2240	3
C₃×#2240	5

Subgroups — C3×C5×#2240 #14275274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#2240 #14275274