Subgroups — C₃×C₄×D₅×C₁₆ #14312504

Order 1920.

432 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	C₂×C₃×D₅×C₁₆	C₂	1	2	yes	maximal
2	C₃×C₄×C₈×D₅	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₁₆×Dic₅	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×#672	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₅×C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₈×D₅	C₄	1	1	yes	characteristic
8	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
9	C₃×C₄²×D₅	C₄	1	1	yes	characteristic
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic
13	C₃×C₈×Dic₅	C₂²	1	1	yes	characteristic
14	C₂×C₃×#672	C₂²	1	2	yes	—
15	C₃×C₄×#168	C₄	1	1	yes	characteristic
16	C₃×D₅×C₁₆	C₄	1	8	yes	direct product factor
17	C₂×C₃×C₄×C₁₆	—	5	1	no	maximal, Hall
18	C₂×D₅×C₁₆	C₂×C₃	1	2	yes	—
19	C₄×C₈×D₅	C₂×C₃	1	1	yes	characteristic
20	C₄×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
21	C₁₆×Dic₅	C₂×C₃	1	2	yes	complemented
22	C₄×#672	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×C₄×D₅	C₈	1	2	yes	—
24	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₃×C₄×Dic₅	C₈	1	2	yes	—
29	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
32	C₃×C₈×D₅	C₈	1	4	yes	—
33	C₃×C₈×D₅	C₂×C₄	1	4	yes	—
34	C₃×C₅×C₁₆	C₂×C₄	1	4	yes	complemented
35	C₃×#672	C₂×C₄	1	4	yes	complemented
36	C₂²×C₃×C₁₆	—	5	2	no	—
37	C₂×C₃×C₄×C₈	—	5	1	no	—
38	C₃×C₄×C₁₆	—	5	2	no	—
39	C₃×C₄×C₁₆	—	5	1	no	—
40	C₃×C₄×C₁₆	D₅	1	1	yes	characteristic, direct product factor
41	C₂×C₈×D₅	C₃×C₄	1	1	yes	characteristic
42	C₂×C₈×D₅	C₂²×C₃	1	1	yes	characteristic
43	C₄²×D₅	C₃×C₄	1	1	yes	characteristic
44	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
45	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
46	C₈×Dic₅	C₃×C₄	1	1	yes	characteristic
47	C₈×Dic₅	C₂²×C₃	1	1	yes	characteristic
48	C₂×#672	C₂²×C₃	1	2	yes	—
49	C₄×#168	C₃×C₄	1	1	yes	characteristic
50	D₅×C₁₆	C₃×C₄	1	8	yes	direct product factor
51	C₂×C₄×C₁₆	—	5	1	no	Sylow
52	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
53	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
54	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
56	C₂×C₃×Dic₅	C₂×C₈	1	2	yes	—
57	C₃×C₄×D₅	C₁₆	1	8	yes	direct product factor
58	C₃×C₄×D₅	C₄²	1	2	yes	—
59	C₃×C₄×D₅	C₂×C₈	1	2	yes	—
60	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
61	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
62	C₃×#168	C₄²	1	2	yes	—
63	C₃×#168	C₂×C₈	1	2	yes	—
64	C₂²×C₃×C₈	—	5	1	no	—
65	C₂²×C₃×C₈	—	5	1	no	—
66	C₂×C₃×C₄²	—	5	1	no	—
67	C₂×C₃×C₁₆	—	5	2	no	—
68	C₂×C₃×C₁₆	—	5	8	no	—
69	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	—
70	C₃×C₄×C₈	—	5	1	no	—
71	C₃×C₄×C₈	—	5	1	no	—
72	C₃×C₄×C₈	—	5	1	no	—
73	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
74	C₂×C₄×D₅	C₃×C₈	1	2	yes	—
75	C₂×C₄×D₅	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
78	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
79	C₄×Dic₅	C₃×C₈	1	2	yes	—
80	C₄×Dic₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
82	C₂×#168	C₂×C₃×C₄	1	1	yes	characteristic
83	C₈×D₅	C₃×C₈	1	4	yes	—
84	C₈×D₅	C₂×C₃×C₄	1	4	yes	—
85	C₅×C₁₆	C₂×C₃×C₄	1	4	yes	complemented
86	#672	C₂×C₃×C₄	1	4	yes	complemented
87	C₂²×C₁₆	—	5	2	no	—
88	C₂×C₄×C₈	—	5	1	no	—
89	C₄×C₁₆	—	5	2	no	—
90	C₄×C₁₆	—	5	1	no	—
91	C₄×C₁₆	C₃×D₅	1	1	yes	characteristic, direct product factor
92	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₃×D₅	C₄×C₈	1	2	yes	—
94	C₂×C₃×D₅	C₂×C₁₆	1	4	yes	—
95	C₃×C₄×C₅	C₂×C₁₆	1	4	yes	complemented
96	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
97	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
98	C₃×Dic₅	C₄×C₈	1	2	yes	—
99	C₃×Dic₅	C₂×C₁₆	1	4	yes	complemented
100	C₂²×C₃×C₄	—	5	2	no	—
101	C₂²×C₃×C₄	—	5	1	no	—
102	C₂×C₃×C₈	—	5	1	no	—
103	C₂×C₃×C₈	—	5	1	no	—
104	C₂×C₃×C₈	—	5	4	no	—
105	C₂×C₃×C₈	—	5	4	no	—
106	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
107	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
108	C₃×C₄²	—	5	2	no	—
109	C₃×C₄²	—	5	1	no	—
110	C₃×C₄²	C₄×D₅	1	1	yes	characteristic
111	C₃×C₁₆	—	5	4	no	—
112	C₃×C₁₆	C₄×D₅	1	4	yes	direct product factor
113	C₂²×D₅	C₂×C₃×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
115	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
116	C₂×Dic₅	C₂×C₃×C₈	1	1	yes	characteristic
117	C₂×Dic₅	C₂×C₃×C₈	1	2	yes	—
118	C₄×D₅	C₃×C₁₆	1	8	yes	direct product factor
119	C₄×D₅	C₃×C₄²	1	2	yes	—
120	C₄×D₅	C₂×C₃×C₈	1	2	yes	—
121	C₅×C₈	C₂×C₃×C₈	1	2	yes	—
122	C₅×C₈	C₂²×C₃×C₄	1	2	yes	—
123	#168	C₃×C₄²	1	2	yes	—
124	#168	C₂×C₃×C₈	1	2	yes	—
125	C₂²×C₈	—	5	1	no	—
126	C₂²×C₈	—	5	1	no	—
127	C₂×C₄²	—	5	1	no	—
128	C₂×C₁₆	—	5	2	no	—
129	C₂×C₁₆	—	5	8	no	—
130	C₂×C₁₆	C₂×C₃×D₅	1	2	yes	—
131	C₄×C₈	—	5	1	no	—
132	C₄×C₈	—	5	1	no	—
133	C₄×C₈	—	5	1	no	—
134	C₄×C₈	C₂×C₃×D₅	1	1	yes	characteristic
135	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
136	C₂×C₃×C₅	C₂²×C₁₆	1	2	yes	—
137	C₃×D₅	C₄×C₁₆	1	4	yes	direct product factor
138	C₂³×C₃	—	5	1	no	—
139	C₂×C₃×C₄	—	5	2	no	—
140	C₂×C₃×C₄	—	5	1	no	—
141	C₂×C₃×C₄	—	5	8	no	—
142	C₂×C₃×C₄	—	5	2	no	—
143	C₂×C₃×C₄	—	5	2	no	—
144	C₂×C₃×C₄	C₈×D₅	1	2	yes	—
145	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
146	C₃×C₈	—	5	2	no	—
147	C₃×C₈	—	5	2	no	—
148	C₃×C₈	C₈×D₅	1	2	yes	—
149	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
150	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
151	C₂×D₅	C₃×C₄×C₈	1	2	yes	—
152	C₂×D₅	C₂×C₃×C₁₆	1	4	yes	—
153	C₄×C₅	C₂×C₃×C₁₆	1	4	yes	complemented
154	C₄×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
155	C₄×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
156	Dic₅	C₃×C₄×C₈	1	2	yes	—
157	Dic₅	C₂×C₃×C₁₆	1	4	yes	complemented
158	C₂²×C₄	—	5	2	no	—
159	C₂²×C₄	—	5	1	no	—
160	C₂×C₈	—	5	1	no	—
161	C₂×C₈	—	5	1	no	—
162	C₂×C₈	—	5	4	no	—
163	C₂×C₈	—	5	4	no	—
164	C₂×C₈	C₃×C₄×D₅	1	1	yes	characteristic
165	C₂×C₈	C₂²×C₃×D₅	1	1	yes	characteristic
166	C₄²	—	5	2	no	—
167	C₄²	—	5	1	no	—
168	C₄²	C₃×C₄×D₅	1	1	yes	characteristic
169	C₁₆	—	5	4	no	—
170	C₁₆	C₃×C₄×D₅	1	4	yes	direct product factor
171	C₃×C₅	C₂×C₄×C₁₆	1	1	yes	characteristic, complemented, Hall
172	C₂²×C₃	—	5	4	no	—
173	C₂²×C₃	—	5	2	no	—
174	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
175	C₃×C₄	—	5	4	no	—
176	C₃×C₄	—	5	2	no	—
177	C₃×C₄	D₅×C₁₆	1	4	yes	direct product factor
178	C₃×C₄	C₄²×D₅	1	1	yes	characteristic
179	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
180	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
181	C₂×C₅	C₂²×C₃×C₁₆	1	2	yes	—
182	D₅	C₃×C₄×C₁₆	1	4	yes	direct product factor
183	C₂³	—	5	1	no	—
184	C₂×C₄	—	5	2	no	—
185	C₂×C₄	—	5	1	no	—
186	C₂×C₄	—	5	8	no	—
187	C₂×C₄	—	5	2	no	—
188	C₂×C₄	—	5	2	no	—
189	C₂×C₄	C₃×C₈×D₅	1	2	yes	—
190	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
191	C₈	—	5	2	no	—
192	C₈	—	5	2	no	—
193	C₈	C₃×C₈×D₅	1	2	yes	—
194	C₈	C₂×C₃×C₄×D₅	1	2	yes	—
195	C₂×C₃	—	5	4	no	—
196	C₂×C₃	C₄×C₈×D₅	1	1	yes	characteristic
197	C₂×C₃	C₂×D₅×C₁₆	1	2	yes	—
198	C₅	C₂×C₃×C₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
199	C₂²	—	5	4	no	—
200	C₂²	—	5	2	no	—
201	C₂²	C₂×C₃×C₈×D₅	1	1	yes	characteristic
202	C₄	—	5	4	no	—
203	C₄	—	5	2	no	—
204	C₄	C₃×D₅×C₁₆	1	4	yes	direct product factor
205	C₄	C₃×C₄²×D₅	1	1	yes	characteristic
206	C₄	C₂×C₃×C₈×D₅	1	1	yes	characteristic
207	C₃	C₄×D₅×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
208	C₂	—	5	4	no	—
209	C₂	C₃×C₄×C₈×D₅	1	1	yes	characteristic
210	C₂	C₂×C₃×D₅×C₁₆	1	2	yes	—

	subgroup	G/H
center, hypercenter	C₃×C₄×C₁₆	D₅
commutator, hypocenter	C₅	C₂×C₃×C₄×C₁₆
Frattini	C₂×C₈	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×C₁₆	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₂×C₅×C₈	C₂³×C₃
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. C₃×C₄×C₅×C₁₆	C₂	semidirect
4. C₃×C₁₆×Dic₅	C₂	semidirect
5. C₃×C₄×#672	C₂	semidirect
6. C₄×D₅×C₁₆	C₃	direct (N × H)
16. C₃×D₅×C₁₆	C₄	direct (N × H)
20. C₄×C₅×C₁₆	C₂×C₃	semidirect
21. C₁₆×Dic₅	C₂×C₃	semidirect
22. C₄×#672	C₂×C₃	semidirect
34. C₃×C₅×C₁₆	C₂×C₄	semidirect
35. C₃×#672	C₂×C₄	semidirect
40. C₃×C₄×C₁₆	D₅	direct (N × H)
50. D₅×C₁₆	C₃×C₄	direct (N × H)
57. C₃×C₄×D₅	C₁₆	direct (N × H)
85. C₅×C₁₆	C₂×C₃×C₄	semidirect
86. #672	C₂×C₃×C₄	semidirect
91. C₄×C₁₆	C₃×D₅	direct (N × H)
95. C₃×C₄×C₅	C₂×C₁₆	semidirect
99. C₃×Dic₅	C₂×C₁₆	semidirect
112. C₃×C₁₆	C₄×D₅	direct (N × H)
118. C₄×D₅	C₃×C₁₆	direct (N × H)
137. C₃×D₅	C₄×C₁₆	direct (N × H)
153. C₄×C₅	C₂×C₃×C₁₆	semidirect
157. Dic₅	C₂×C₃×C₁₆	semidirect
170. C₁₆	C₃×C₄×D₅	direct (N × H)
171. C₃×C₅	C₂×C₄×C₁₆	semidirect
177. C₃×C₄	D₅×C₁₆	direct (N × H)
182. D₅	C₃×C₄×C₁₆	direct (N × H)
198. C₅	C₂×C₃×C₄×C₁₆	semidirect
204. C₄	C₃×D₅×C₁₆	direct (N × H)
207. C₃	C₄×D₅×C₁₆	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×D₅×C₁₆	2
C₃×C₄×C₈×D₅	2
C₃×C₄×C₅×C₁₆	2
C₃×C₁₆×Dic₅	2
C₃×C₄×#672	2
C₄×D₅×C₁₆	3
C₂×C₃×C₄×C₁₆	5

Subgroups — C3×C4×D5×C16 #14312504

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×D₅×C₁₆ #14312504