Subgroups — C₃×C₅×#1841 #14274875

Order 1920.

324 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₅×#344	C₂	1	2	yes	maximal
3	C₃×C₅×#339	C₂	1	2	yes	maximal
4	C₃×C₅×#338	C₂	1	1	yes	maximal, characteristic
5	C₃×C₅×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₅×#1841	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₃×C₅×#117	C₂²	1	2	yes	—
11	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₃×C₅×#97	C₄	1	2	yes	complemented
13	C₃×C₄×C₅×C₈	C₄	1	2	yes	complemented
14	C₃×#1841	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
15	C₄×C₅×M₁₆	C₂×C₃	1	1	yes	characteristic
16	C₅×#344	C₂×C₃	1	2	yes	—
17	C₅×#339	C₂×C₃	1	2	yes	—
18	C₅×#338	C₂×C₃	1	1	yes	characteristic
19	C₅×H(3,ℤ/4)	C₂×C₃	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₅	—	2	1	no	—
21	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
22	C₃×C₅×M₁₆	—	2	2	no	—
23	C₃×C₅×M₁₆	—	2	2	no	—
24	C₃×C₅×M₁₆	—	2	2	no	—
25	C₃×C₅×M₁₆	—	2	2	no	—
26	C₂×C₃×C₅×C₈	—	2	2	no	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
29	C₃×C₅×C₄⋊C₄	—	2	2	no	—
30	C₃×C₅×C₂²⋊C₄	—	2	2	no	—
31	C₃×C₄²×C₅	—	2	1	no	—
32	C₃×C₄²×C₅	—	2	2	no	—
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄×M₁₆	C₂×C₅	1	1	yes	characteristic
36	C₃×#344	C₂×C₅	1	2	yes	—
37	C₃×#339	C₂×C₅	1	2	yes	—
38	C₃×#338	C₂×C₅	1	1	yes	characteristic
39	C₃×H(3,ℤ/4)	C₂×C₅	1	1	yes	characteristic
40	C₂×C₅×M₁₆	C₂²×C₃	1	2	yes	—
41	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₅×M₁₆	C₂²×C₃	1	1	yes	characteristic
43	C₅×#117	C₂²×C₃	1	2	yes	—
44	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
45	C₅×#97	C₃×C₄	1	2	yes	complemented
46	C₄×C₅×C₈	C₃×C₄	1	2	yes	complemented
47	#1841	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
48	C₂³×C₃×C₅	#41	1	1	yes	characteristic
49	C₂×C₃×C₄×C₅	—	2	1	no	—
50	C₂×C₃×C₄×C₅	—	2	1	no	—
51	C₂×C₃×C₄×C₅	—	4	2	no	—
52	C₂×C₃×C₄×C₅	—	4	2	no	—
53	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄×C₅	#41	1	2	yes	—
55	C₂×C₃×C₄×C₅	#41	1	2	yes	—
56	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
57	C₃×C₅×C₈	—	2	2	no	—
58	C₃×C₅×C₈	—	2	2	no	—
59	C₃×C₅×C₈	—	4	2	no	—
60	C₂×C₃×M₁₆	C₂²×C₅	1	2	yes	—
61	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
62	C₂×C₃×M₁₆	C₂²×C₅	1	1	yes	characteristic
63	C₃×#117	C₂²×C₅	1	2	yes	—
64	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
65	C₃×#97	C₄×C₅	1	2	yes	complemented
66	C₃×C₄×C₈	C₄×C₅	1	2	yes	complemented
67	C₂²×C₄×C₅	—	2	1	no	—
68	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
69	C₅×M₁₆	—	2	2	no	—
70	C₅×M₁₆	—	2	2	no	—
71	C₅×M₁₆	—	2	2	no	—
72	C₅×M₁₆	—	2	2	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
75	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
76	C₅×C₄⋊C₄	—	2	2	no	—
77	C₅×C₂²⋊C₄	—	2	2	no	—
78	C₄²×C₅	—	2	1	no	—
79	C₄²×C₅	—	2	2	no	—
80	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
82	C₄×M₁₆	C₂×C₃×C₅	1	1	yes	characteristic
83	#344	C₂×C₃×C₅	1	2	yes	—
84	#339	C₂×C₃×C₅	1	2	yes	—
85	#338	C₂×C₃×C₅	1	1	yes	characteristic
86	H(3,ℤ/4)	C₂×C₃×C₅	1	1	yes	characteristic
87	C₂²×C₃×C₅	—	4	1	no	—
88	C₂²×C₃×C₅	#117	1	1	yes	characteristic
89	C₂²×C₃×C₅	#126	1	2	yes	—
90	C₃×C₄×C₅	—	2	2	no	—
91	C₃×C₄×C₅	—	4	1	no	—
92	C₃×C₄×C₅	—	8	2	no	—
93	C₃×C₄×C₅	#117	1	1	yes	characteristic
94	C₃×C₄×C₅	#117	1	1	yes	characteristic
95	C₃×C₄×C₅	#126	1	2	yes	—
96	C₂²×C₃×C₄	—	2	1	no	—
97	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
98	C₃×M₁₆	—	2	2	no	—
99	C₃×M₁₆	—	2	2	no	—
100	C₃×M₁₆	—	2	2	no	—
101	C₃×M₁₆	—	2	2	no	—
102	C₂×C₃×C₈	—	2	2	no	—
103	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
104	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
105	C₃×C₄⋊C₄	—	2	2	no	—
106	C₃×C₂²⋊C₄	—	2	2	no	—
107	C₃×C₄²	—	2	1	no	—
108	C₃×C₄²	—	2	2	no	—
109	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
110	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
111	C₂³×C₅	C₃×#41	1	1	yes	characteristic
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	4	2	no	—
115	C₂×C₄×C₅	—	4	2	no	—
116	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
117	C₂×C₄×C₅	C₃×#41	1	2	yes	—
118	C₂×C₄×C₅	C₃×#41	1	2	yes	—
119	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
120	C₅×C₈	—	2	2	no	—
121	C₅×C₈	—	2	2	no	—
122	C₅×C₈	—	4	2	no	—
123	C₂×M₁₆	C₂²×C₃×C₅	1	2	yes	—
124	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
125	C₂×M₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
126	#117	C₂²×C₃×C₅	1	2	yes	—
127	C₂×C₄²	C₂²×C₃×C₅	1	1	yes	characteristic
128	#97	C₃×C₄×C₅	1	2	yes	complemented
129	C₄×C₈	C₃×C₄×C₅	1	2	yes	complemented
130	C₂×C₃×C₅	—	2	1	no	—
131	C₂×C₃×C₅	—	2	2	no	—
132	C₂×C₃×C₅	#383	1	1	yes	characteristic
133	C₂³×C₃	C₅×#41	1	1	yes	characteristic
134	C₂×C₃×C₄	—	2	1	no	—
135	C₂×C₃×C₄	—	2	1	no	—
136	C₂×C₃×C₄	—	4	2	no	—
137	C₂×C₃×C₄	—	4	2	no	—
138	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
139	C₂×C₃×C₄	C₅×#41	1	2	yes	—
140	C₂×C₃×C₄	C₅×#41	1	2	yes	—
141	C₂×C₃×C₄	C₅×#41	1	1	yes	characteristic
142	C₃×C₈	—	2	2	no	—
143	C₃×C₈	—	2	2	no	—
144	C₃×C₈	—	4	2	no	—
145	C₂²×C₅	—	4	1	no	—
146	C₂²×C₅	C₃×#117	1	1	yes	characteristic
147	C₂²×C₅	C₃×#126	1	2	yes	—
148	C₄×C₅	—	2	2	no	—
149	C₄×C₅	—	4	1	no	—
150	C₄×C₅	—	8	2	no	—
151	C₄×C₅	C₃×#117	1	1	yes	characteristic
152	C₄×C₅	C₃×#117	1	1	yes	characteristic
153	C₄×C₅	C₃×#126	1	2	yes	—
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	C₂³×C₃×C₅	1	1	yes	characteristic
156	M₁₆	—	2	2	no	—
157	M₁₆	—	2	2	no	—
158	M₁₆	—	2	2	no	—
159	M₁₆	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
162	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
163	C₄⋊C₄	—	2	2	no	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₄²	—	2	1	no	—
166	C₄²	—	2	2	no	—
167	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
168	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
169	C₃×C₅	#1841	1	1	yes	characteristic, direct product factor, Hall
170	C₂²×C₃	—	4	1	no	—
171	C₂²×C₃	C₅×#117	1	1	yes	characteristic
172	C₂²×C₃	C₅×#126	1	2	yes	—
173	C₃×C₄	—	2	2	no	—
174	C₃×C₄	—	4	1	no	—
175	C₃×C₄	—	8	2	no	—
176	C₃×C₄	C₅×#117	1	1	yes	characteristic
177	C₃×C₄	C₅×#117	1	1	yes	characteristic
178	C₃×C₄	C₅×#126	1	2	yes	—
179	C₂×C₅	—	2	1	no	—
180	C₂×C₅	—	2	2	no	—
181	C₂×C₅	C₃×#383	1	1	yes	characteristic
182	C₂³	C₃×C₅×#41	1	1	yes	characteristic
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
188	C₂×C₄	C₃×C₅×#41	1	2	yes	—
189	C₂×C₄	C₃×C₅×#41	1	2	yes	—
190	C₂×C₄	C₃×C₅×#41	1	1	yes	characteristic
191	C₈	—	2	2	no	—
192	C₈	—	2	2	no	—
193	C₈	—	4	2	no	—
194	C₂×C₃	—	2	1	no	—
195	C₂×C₃	—	2	2	no	—
196	C₂×C₃	C₅×#383	1	1	yes	characteristic
197	C₅	C₃×#1841	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²	—	4	1	no	—
199	C₂²	C₃×C₅×#117	1	1	yes	characteristic
200	C₂²	C₃×C₅×#126	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	4	1	no	—
203	C₄	—	8	2	no	—
204	C₄	C₃×C₅×#117	1	1	yes	characteristic
205	C₄	C₃×C₅×#117	1	1	yes	characteristic
206	C₄	C₃×C₅×#126	1	2	yes	—
207	C₃	C₅×#1841	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	—	2	1	no	—
209	C₂	—	2	2	no	—
210	C₂	C₃×C₅×#383	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₅	#117
commutator	C₂×C₄	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂³×C₃×C₅
socle	C₂×C₃×C₅	#383
prime radical	C₂³×C₃×C₅	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄×C₅ ◁ C₂×C₃×C₅×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1841	C₃	direct (N × H)
12. C₃×C₅×#97	C₄	semidirect
13. C₃×C₄×C₅×C₈	C₄	semidirect
14. C₃×#1841	C₅	direct (N × H)
45. C₅×#97	C₃×C₄	semidirect
46. C₄×C₅×C₈	C₃×C₄	semidirect
47. #1841	C₃×C₅	direct (N × H)
65. C₃×#97	C₄×C₅	semidirect
66. C₃×C₄×C₈	C₄×C₅	semidirect
128. #97	C₃×C₄×C₅	semidirect
129. C₄×C₈	C₃×C₄×C₅	semidirect
169. C₃×C₅	#1841	direct (N × H)
197. C₅	C₃×#1841	direct (N × H)
207. C₃	C₅×#1841	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×M₁₆	2
C₃×C₅×#344	2
C₃×C₅×#339	2
C₃×C₅×#338	2
C₃×C₅×H(3,ℤ/4)	2
C₅×#1841	3
C₃×#1841	5

Subgroups — C3×C5×#1841 #14274875

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×#1841 #14274875