Subgroups — C₃×C₅×D₄×C₁₆ #14275201

Order 1920.

348 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×C₁₆	C₂	1	2	yes	maximal, complemented
2	C₃×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
3	C₃×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×C₂²⋊C₁₆	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×D₄×C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×C₈	C₂²	1	2	yes	—
8	C₃×C₄×C₅×D₄	C₄	1	1	yes	characteristic
9	C₂×C₃×C₅×C₁₆	—	2	4	no	—
10	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₃×C₅×C₂²⋊C₈	C₄	1	2	yes	—
15	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₃×D₄×C₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
17	C₂²×C₅×C₁₆	C₂×C₃	1	2	yes	complemented
18	C₅×C₈×D₄	C₂×C₃	1	1	yes	characteristic
19	C₅×C₄⋊C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
20	C₅×C₂²⋊C₁₆	C₂×C₃	1	2	yes	complemented
21	C₄×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×C₅×D₄	C₈	1	1	yes	characteristic
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
24	C₂×C₃×C₅×C₈	—	2	2	no	—
25	C₂×C₃×C₅×C₈	—	2	2	no	—
26	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
29	C₃×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
30	C₃×C₅×C₂²⋊C₄	C₈	1	2	yes	—
31	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₃×C₅×C₁₆	—	2	2	no	—
33	C₃×C₅×C₁₆	—	2	1	no	—
34	C₃×C₅×C₁₆	D₄	1	2	yes	direct product factor
35	C₂²×C₃×C₁₆	C₂×C₅	1	2	yes	complemented
36	C₃×C₈×D₄	C₂×C₅	1	1	yes	characteristic
37	C₃×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
38	C₃×C₂²⋊C₁₆	C₂×C₅	1	2	yes	complemented
39	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂²×C₅×C₈	C₂²×C₃	1	2	yes	—
41	C₄×C₅×D₄	C₃×C₄	1	1	yes	characteristic
42	C₂×C₅×C₁₆	—	2	4	no	—
43	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₅×C₁₆	C₂²×C₃	1	2	yes	—
45	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
46	C₅×C₄⋊C₈	C₃×C₄	1	1	yes	characteristic
47	C₅×C₂²⋊C₈	C₃×C₄	1	2	yes	—
48	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
49	D₄×C₁₆	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
50	C₂³×C₃×C₅	C₂×C₈	1	2	yes	—
51	C₃×C₅×D₄	C₁₆	1	4	yes	direct product factor
52	C₂×C₃×C₄×C₅	—	2	2	no	—
53	C₂×C₃×C₄×C₅	—	2	2	no	—
54	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
56	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
57	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₅×C₈	—	2	2	no	—
59	C₃×C₅×C₈	—	2	1	no	—
60	C₃×C₅×C₈	C₂×D₄	1	1	yes	characteristic
61	C₃×C₅×C₈	#41	1	1	yes	characteristic
62	C₂²×C₃×C₈	C₂²×C₅	1	2	yes	—
63	C₃×C₄×D₄	C₄×C₅	1	1	yes	characteristic
64	C₂×C₃×C₁₆	—	2	4	no	—
65	C₂×C₃×C₁₆	C₂²×C₅	1	2	yes	—
66	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
67	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
68	C₃×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
69	C₃×C₂²⋊C₈	C₄×C₅	1	2	yes	—
70	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
71	C₂×C₅×D₄	C₃×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	C₂×C₃×C₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×C₃×C₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
78	C₅×C₄⋊C₄	C₃×C₈	1	1	yes	characteristic
79	C₅×C₂²⋊C₄	C₃×C₈	1	2	yes	—
80	C₄²×C₅	C₂×C₃×C₄	1	1	yes	characteristic
81	C₅×C₁₆	—	2	2	no	—
82	C₅×C₁₆	—	2	1	no	—
83	C₅×C₁₆	C₃×D₄	1	2	yes	direct product factor
84	C₂²×C₁₆	C₂×C₃×C₅	1	2	yes	complemented
85	C₈×D₄	C₂×C₃×C₅	1	1	yes	characteristic
86	C₄⋊C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
87	C₂²⋊C₁₆	C₂×C₃×C₅	1	2	yes	complemented
88	C₄×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
89	C₂²×C₃×C₅	—	2	2	no	—
90	C₂²×C₃×C₅	—	2	2	no	—
91	C₂²×C₃×C₅	C₂×C₁₆	1	4	yes	complemented
92	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
93	C₃×C₄×C₅	—	2	2	no	—
94	C₃×C₄×C₅	—	2	1	no	—
95	C₃×C₄×C₅	C₂×C₁₆	1	2	yes	complemented
96	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
97	C₃×C₄×C₅	#131	1	1	yes	characteristic
98	C₂×C₃×D₄	C₅×C₈	1	1	yes	characteristic
99	C₂²×C₃×C₄	C₂×C₄×C₅	1	2	yes	—
100	C₂×C₃×C₈	—	2	2	no	—
101	C₂×C₃×C₈	—	2	2	no	—
102	C₂×C₃×C₈	C₂×C₄×C₅	1	1	yes	characteristic
103	C₂×C₃×C₈	C₂×C₄×C₅	1	2	yes	—
104	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
105	C₃×C₄⋊C₄	C₅×C₈	1	1	yes	characteristic
106	C₃×C₂²⋊C₄	C₅×C₈	1	2	yes	—
107	C₃×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
108	C₃×C₁₆	—	2	2	no	—
109	C₃×C₁₆	—	2	1	no	—
110	C₃×C₁₆	C₅×D₄	1	2	yes	direct product factor
111	C₂³×C₅	C₂×C₃×C₈	1	2	yes	—
112	C₅×D₄	C₃×C₁₆	1	4	yes	direct product factor
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅	C₂×C₃×C₈	1	2	yes	—
117	C₂×C₄×C₅	C₂×C₃×C₈	1	1	yes	characteristic
118	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
119	C₅×C₈	—	2	2	no	—
120	C₅×C₈	—	2	1	no	—
121	C₅×C₈	C₂×C₃×D₄	1	1	yes	characteristic
122	C₅×C₈	C₃×#41	1	1	yes	characteristic
123	C₂²×C₈	C₂²×C₃×C₅	1	2	yes	—
124	C₄×D₄	C₃×C₄×C₅	1	1	yes	characteristic
125	C₂×C₁₆	—	2	4	no	—
126	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
127	C₂×C₁₆	C₂²×C₃×C₅	1	2	yes	—
128	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
129	C₄⋊C₈	C₃×C₄×C₅	1	1	yes	characteristic
130	C₂²⋊C₈	C₃×C₄×C₅	1	2	yes	—
131	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
132	C₂×C₃×C₅	—	2	4	no	—
133	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
134	C₂×C₃×C₅	C₂²×C₁₆	1	1	yes	characteristic
135	C₂×C₃×C₅	#504	1	1	yes	characteristic
136	C₂³×C₃	C₂×C₅×C₈	1	2	yes	—
137	C₃×D₄	C₅×C₁₆	1	4	yes	direct product factor
138	C₂×C₃×C₄	—	2	2	no	—
139	C₂×C₃×C₄	—	2	2	no	—
140	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
141	C₂×C₃×C₄	C₂×C₅×C₈	1	2	yes	—
142	C₂×C₃×C₄	C₂×C₅×C₈	1	1	yes	characteristic
143	C₂×C₃×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
144	C₃×C₈	—	2	2	no	—
145	C₃×C₈	—	2	1	no	—
146	C₃×C₈	C₂×C₅×D₄	1	1	yes	characteristic
147	C₃×C₈	C₅×#41	1	1	yes	characteristic
148	C₂²×C₅	—	2	2	no	—
149	C₂²×C₅	—	2	2	no	—
150	C₂²×C₅	C₂×C₃×C₁₆	1	4	yes	complemented
151	C₂²×C₅	C₂²×C₃×C₈	1	1	yes	characteristic
152	C₄×C₅	—	2	1	no	—
153	C₄×C₅	—	2	2	no	—
154	C₄×C₅	C₂×C₃×C₁₆	1	2	yes	complemented
155	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
156	C₄×C₅	C₃×#131	1	1	yes	characteristic
157	C₂×D₄	C₃×C₅×C₈	1	1	yes	characteristic
158	C₂²×C₄	C₂×C₃×C₄×C₅	1	2	yes	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	C₂×C₃×C₄×C₅	1	1	yes	characteristic
162	C₂×C₈	C₂×C₃×C₄×C₅	1	2	yes	—
163	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
164	C₄⋊C₄	C₃×C₅×C₈	1	1	yes	characteristic
165	C₂²⋊C₄	C₃×C₅×C₈	1	2	yes	—
166	C₄²	C₂×C₃×C₄×C₅	1	1	yes	characteristic
167	C₁₆	—	2	2	no	—
168	C₁₆	—	2	1	no	—
169	C₁₆	C₃×C₅×D₄	1	2	yes	direct product factor
170	C₃×C₅	D₄×C₁₆	1	1	yes	characteristic, direct product factor, Hall
171	C₂²×C₃	—	2	2	no	—
172	C₂²×C₃	—	2	2	no	—
173	C₂²×C₃	C₂×C₅×C₁₆	1	4	yes	complemented
174	C₂²×C₃	C₂²×C₅×C₈	1	1	yes	characteristic
175	C₃×C₄	—	2	2	no	—
176	C₃×C₄	—	2	1	no	—
177	C₃×C₄	C₂×C₅×C₁₆	1	2	yes	complemented
178	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
179	C₃×C₄	C₅×#131	1	1	yes	characteristic
180	C₂×C₅	—	2	4	no	—
181	C₂×C₅	C₃×C₈×D₄	1	1	yes	characteristic
182	C₂×C₅	C₂²×C₃×C₁₆	1	1	yes	characteristic
183	C₂×C₅	C₃×#504	1	1	yes	characteristic
184	C₂³	C₂×C₃×C₅×C₈	1	2	yes	—
185	D₄	C₃×C₅×C₁₆	1	4	yes	direct product factor
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
189	C₂×C₄	C₂×C₃×C₅×C₈	1	2	yes	—
190	C₂×C₄	C₂×C₃×C₅×C₈	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
192	C₈	—	2	2	no	—
193	C₈	—	2	1	no	—
194	C₈	C₂×C₃×C₅×D₄	1	1	yes	characteristic
195	C₈	C₃×C₅×#41	1	1	yes	characteristic
196	C₂×C₃	—	2	4	no	—
197	C₂×C₃	C₅×C₈×D₄	1	1	yes	characteristic
198	C₂×C₃	C₂²×C₅×C₁₆	1	1	yes	characteristic
199	C₂×C₃	C₅×#504	1	1	yes	characteristic
200	C₅	C₃×D₄×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
201	C₂²	—	2	2	no	—
202	C₂²	—	2	2	no	—
203	C₂²	C₂×C₃×C₅×C₁₆	1	4	yes	complemented
204	C₂²	C₂²×C₃×C₅×C₈	1	1	yes	characteristic
205	C₄	—	2	1	no	—
206	C₄	—	2	2	no	—
207	C₄	C₂×C₃×C₅×C₁₆	1	2	yes	complemented
208	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
209	C₄	C₃×C₅×#131	1	1	yes	characteristic
210	C₃	C₅×D₄×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
211	C₂	—	2	4	no	—
212	C₂	C₃×C₅×C₈×D₄	1	1	yes	characteristic
213	C₂	C₂²×C₃×C₅×C₁₆	1	1	yes	characteristic
214	C₂	C₃×C₅×#504	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₅×C₁₆	C₂²
commutator	C₂	C₂²×C₃×C₅×C₁₆
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×C₃×C₅×D₄	C₈
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₅×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×C₁₆	C₂	semidirect
3. C₃×C₅×C₄⋊C₁₆	C₂	semidirect
4. C₃×C₅×C₂²⋊C₁₆	C₂	semidirect
5. C₃×C₄×C₅×C₁₆	C₂	semidirect
6. C₅×D₄×C₁₆	C₃	direct (N × H)
16. C₃×D₄×C₁₆	C₅	direct (N × H)
17. C₂²×C₅×C₁₆	C₂×C₃	semidirect
19. C₅×C₄⋊C₁₆	C₂×C₃	semidirect
20. C₅×C₂²⋊C₁₆	C₂×C₃	semidirect
21. C₄×C₅×C₁₆	C₂×C₃	semidirect
34. C₃×C₅×C₁₆	D₄	direct (N × H)
35. C₂²×C₃×C₁₆	C₂×C₅	semidirect
37. C₃×C₄⋊C₁₆	C₂×C₅	semidirect
38. C₃×C₂²⋊C₁₆	C₂×C₅	semidirect
39. C₃×C₄×C₁₆	C₂×C₅	semidirect
49. D₄×C₁₆	C₃×C₅	direct (N × H)
51. C₃×C₅×D₄	C₁₆	direct (N × H)
83. C₅×C₁₆	C₃×D₄	direct (N × H)
84. C₂²×C₁₆	C₂×C₃×C₅	semidirect
86. C₄⋊C₁₆	C₂×C₃×C₅	semidirect
87. C₂²⋊C₁₆	C₂×C₃×C₅	semidirect
88. C₄×C₁₆	C₂×C₃×C₅	semidirect
91. C₂²×C₃×C₅	C₂×C₁₆	semidirect
95. C₃×C₄×C₅	C₂×C₁₆	semidirect
110. C₃×C₁₆	C₅×D₄	direct (N × H)
112. C₅×D₄	C₃×C₁₆	direct (N × H)
137. C₃×D₄	C₅×C₁₆	direct (N × H)
150. C₂²×C₅	C₂×C₃×C₁₆	semidirect
154. C₄×C₅	C₂×C₃×C₁₆	semidirect
169. C₁₆	C₃×C₅×D₄	direct (N × H)
170. C₃×C₅	D₄×C₁₆	direct (N × H)
173. C₂²×C₃	C₂×C₅×C₁₆	semidirect
177. C₃×C₄	C₂×C₅×C₁₆	semidirect
185. D₄	C₃×C₅×C₁₆	direct (N × H)
200. C₅	C₃×D₄×C₁₆	direct (N × H)
203. C₂²	C₂×C₃×C₅×C₁₆	semidirect
207. C₄	C₂×C₃×C₅×C₁₆	semidirect
210. C₃	C₅×D₄×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
C₂²×C₃×C₅×C₁₆	2
C₃×C₅×C₈×D₄	2
C₃×C₅×C₄⋊C₁₆	2
C₃×C₅×C₂²⋊C₁₆	2
C₃×C₄×C₅×C₁₆	2
C₅×D₄×C₁₆	3
C₃×D₄×C₁₆	5

Subgroups — C3×C5×D4×C16 #14275201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×D₄×C₁₆ #14275201