Subgroups — C₃×C₅×#1820 #14274854

Order 1920.

600 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#381	C₂	1	2	yes	maximal
3	C₃×C₅×#342	C₂	1	4	yes	maximal
4	C₅×#1820	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
6	C₃×C₅×#117	—	2	4	no	—
7	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	complemented
8	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
9	C₂×C₃×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄²×C₅	C₄	1	2	yes	complemented
11	C₃×C₅×#95	—	2	4	no	—
12	C₃×#1820	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₅×#117	C₂×C₃	1	1	yes	characteristic
14	C₅×#381	C₂×C₃	1	2	yes	—
15	C₅×#342	C₂×C₃	1	4	yes	—
16	C₂⁴×C₃×C₅	C₂³	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	—	2	4	no	—
18	C₂²×C₃×C₄×C₅	—	2	4	no	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
22	C₂²×C₃×C₄×C₅	D₄	1	4	yes	—
23	C₃×C₅×C₄⋊C₄	—	2	4	no	—
24	C₃×C₅×C₂²⋊C₄	—	2	4	no	—
25	C₃×C₅×C₂²⋊C₄	—	2	8	no	—
26	C₃×C₄²×C₅	—	2	4	no	—
27	C₂×C₃×#117	C₂×C₅	1	1	yes	characteristic
28	C₃×#381	C₂×C₅	1	2	yes	—
29	C₃×#342	C₂×C₅	1	4	yes	—
30	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
31	C₅×#117	—	2	4	no	—
32	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	complemented
33	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
34	C₂×C₅×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
35	C₂×C₄²×C₅	C₃×C₄	1	2	yes	complemented
36	C₅×#95	—	2	4	no	—
37	#1820	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
38	C₂³×C₃×C₅	—	4	2	no	—
39	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂³×C₃×C₅	C₂×D₄	1	2	yes	—
41	C₂³×C₃×C₅	#41	1	4	yes	—
42	C₂×C₃×C₄×C₅	—	2	2	no	—
43	C₂×C₃×C₄×C₅	—	2	2	no	—
44	C₂×C₃×C₄×C₅	—	2	8	no	—
45	C₂×C₃×C₄×C₅	—	2	4	no	—
46	C₂×C₃×C₄×C₅	—	4	4	no	—
47	C₂×C₃×C₄×C₅	—	4	8	no	—
48	C₂×C₃×C₄×C₅	—	4	4	no	—
49	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
50	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
51	C₂³×C₃×C₄	C₂²×C₅	1	1	yes	characteristic
52	C₃×#117	—	2	4	no	—
53	C₂×C₃×C₄⋊C₄	C₄×C₅	1	2	yes	complemented
54	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
55	C₂×C₃×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
56	C₂×C₃×C₄²	C₄×C₅	1	2	yes	complemented
57	C₃×#95	—	2	4	no	—
58	C₂⁴×C₅	C₂³×C₃	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
64	C₂²×C₄×C₅	C₃×D₄	1	4	yes	—
65	C₅×C₄⋊C₄	—	2	4	no	—
66	C₅×C₂²⋊C₄	—	2	4	no	—
67	C₅×C₂²⋊C₄	—	2	8	no	—
68	C₄²×C₅	—	2	4	no	—
69	C₂×#117	C₂×C₃×C₅	1	1	yes	characteristic
70	#381	C₂×C₃×C₅	1	2	yes	—
71	#342	C₂×C₃×C₅	1	4	yes	—
72	C₂²×C₃×C₅	—	2	2	no	—
73	C₂²×C₃×C₅	—	2	4	no	—
74	C₂²×C₃×C₅	—	4	4	no	—
75	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₃×C₅	#117	1	2	yes	—
77	C₂²×C₃×C₅	#123	1	4	yes	—
78	C₃×C₄×C₅	—	2	2	no	—
79	C₃×C₄×C₅	—	2	2	no	—
80	C₃×C₄×C₅	—	4	4	no	—
81	C₃×C₄×C₅	—	8	4	no	—
82	C₂⁴×C₃	C₂³×C₅	1	1	yes	characteristic
83	C₂²×C₃×C₄	—	2	4	no	—
84	C₂²×C₃×C₄	—	2	4	no	—
85	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
86	C₂²×C₃×C₄	C₂×C₄×C₅	1	1	yes	characteristic
87	C₂²×C₃×C₄	C₂×C₄×C₅	1	4	yes	—
88	C₂²×C₃×C₄	C₅×D₄	1	4	yes	—
89	C₃×C₄⋊C₄	—	2	4	no	—
90	C₃×C₂²⋊C₄	—	2	4	no	—
91	C₃×C₂²⋊C₄	—	2	8	no	—
92	C₃×C₄²	—	2	4	no	—
93	C₂³×C₅	—	4	2	no	—
94	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
95	C₂³×C₅	C₂×C₃×D₄	1	2	yes	—
96	C₂³×C₅	C₃×#41	1	4	yes	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	8	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	4	4	no	—
102	C₂×C₄×C₅	—	4	8	no	—
103	C₂×C₄×C₅	—	4	4	no	—
104	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
105	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
106	C₂³×C₄	C₂²×C₃×C₅	1	1	yes	characteristic
107	#117	—	2	4	no	—
108	C₂×C₄⋊C₄	C₃×C₄×C₅	1	2	yes	complemented
109	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	4	yes	—
110	C₂×C₂²⋊C₄	C₂²×C₃×C₅	1	2	yes	—
111	C₂×C₄²	C₃×C₄×C₅	1	2	yes	complemented
112	#95	—	2	4	no	—
113	C₂×C₃×C₅	—	2	4	no	—
114	C₂×C₃×C₅	—	2	2	no	—
115	C₂×C₃×C₅	#381	1	1	yes	characteristic
116	C₂×C₃×C₅	#410	1	2	yes	—
117	C₂³×C₃	—	4	2	no	—
118	C₂³×C₃	C₂²×C₄×C₅	1	1	yes	characteristic
119	C₂³×C₃	C₂×C₅×D₄	1	2	yes	—
120	C₂³×C₃	C₅×#41	1	4	yes	—
121	C₂×C₃×C₄	—	2	2	no	—
122	C₂×C₃×C₄	—	2	2	no	—
123	C₂×C₃×C₄	—	2	8	no	—
124	C₂×C₃×C₄	—	2	4	no	—
125	C₂×C₃×C₄	—	4	4	no	—
126	C₂×C₃×C₄	—	4	8	no	—
127	C₂×C₃×C₄	—	4	4	no	—
128	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
129	C₂×C₃×C₄	C₅×C₂²⋊C₄	1	2	yes	—
130	C₂²×C₅	—	2	2	no	—
131	C₂²×C₅	—	2	4	no	—
132	C₂²×C₅	—	4	4	no	—
133	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₅	C₃×#117	1	2	yes	—
135	C₂²×C₅	C₃×#123	1	4	yes	—
136	C₄×C₅	—	2	2	no	—
137	C₄×C₅	—	2	2	no	—
138	C₄×C₅	—	4	4	no	—
139	C₄×C₅	—	8	4	no	—
140	C₂⁴	C₂³×C₃×C₅	1	1	yes	characteristic
141	C₂²×C₄	—	2	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
144	C₂²×C₄	C₂×C₃×C₄×C₅	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₃×C₄×C₅	1	4	yes	—
146	C₂²×C₄	C₃×C₅×D₄	1	4	yes	—
147	C₄⋊C₄	—	2	4	no	—
148	C₂²⋊C₄	—	2	4	no	—
149	C₂²⋊C₄	—	2	8	no	—
150	C₄²	—	2	4	no	—
151	C₃×C₅	#1820	1	1	yes	characteristic, direct product factor, Hall
152	C₂²×C₃	—	2	2	no	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	4	4	no	—
155	C₂²×C₃	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
156	C₂²×C₃	C₅×#117	1	2	yes	—
157	C₂²×C₃	C₅×#123	1	4	yes	—
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	2	2	no	—
160	C₃×C₄	—	4	4	no	—
161	C₃×C₄	—	8	4	no	—
162	C₂×C₅	—	2	4	no	—
163	C₂×C₅	—	2	2	no	—
164	C₂×C₅	C₃×#381	1	1	yes	characteristic
165	C₂×C₅	C₃×#410	1	2	yes	—
166	C₂³	—	4	2	no	—
167	C₂³	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
168	C₂³	C₂×C₃×C₅×D₄	1	2	yes	—
169	C₂³	C₃×C₅×#41	1	4	yes	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	—	4	8	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
178	C₂×C₄	C₃×C₅×C₂²⋊C₄	1	2	yes	—
179	C₂×C₃	—	2	4	no	—
180	C₂×C₃	—	2	2	no	—
181	C₂×C₃	C₅×#381	1	1	yes	characteristic
182	C₂×C₃	C₅×#410	1	2	yes	—
183	C₅	C₃×#1820	1	1	yes	characteristic, direct product factor, Sylow
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	C₂×C₃×C₅×C₂²⋊C₄	1	1	yes	characteristic
188	C₂²	C₃×C₅×#117	1	2	yes	—
189	C₂²	C₃×C₅×#123	1	4	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	4	4	no	—
193	C₄	—	8	4	no	—
194	C₃	C₅×#1820	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	—	2	4	no	—
196	C₂	—	2	2	no	—
197	C₂	C₃×C₅×#381	1	1	yes	characteristic
198	C₂	C₃×C₅×#410	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂⁴	C₂³×C₃×C₅
prime radical	C₂⁴×C₃×C₅	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ C₂³×C₃×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1820	C₃	direct (N × H)
7. C₂×C₃×C₅×C₄⋊C₄	C₄	semidirect
10. C₂×C₃×C₄²×C₅	C₄	semidirect
12. C₃×#1820	C₅	direct (N × H)
32. C₂×C₅×C₄⋊C₄	C₃×C₄	semidirect
35. C₂×C₄²×C₅	C₃×C₄	semidirect
37. #1820	C₃×C₅	direct (N × H)
53. C₂×C₃×C₄⋊C₄	C₄×C₅	semidirect
56. C₂×C₃×C₄²	C₄×C₅	semidirect
108. C₂×C₄⋊C₄	C₃×C₄×C₅	semidirect
111. C₂×C₄²	C₃×C₄×C₅	semidirect
151. C₃×C₅	#1820	direct (N × H)
183. C₅	C₃×#1820	direct (N × H)
194. C₃	C₅×#1820	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×C₅×#117	2
C₃×C₅×#381	2
C₃×C₅×#342	2
C₅×#1820	3
C₃×#1820	5

Subgroups — C3×C5×#1820 #14274854

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1820 #14274854