Subgroups — C₃×C₄×C₅×D₁₆ #14275206

Order 1920.

348 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×D₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×D₈	C₂	1	2	yes	maximal, complemented
3	C₃×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×#357	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₅×D₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×D₈	C₂²	1	2	yes	—
8	C₃×C₄×C₅×D₄	—	2	2	no	—
9	C₃×C₅×D₁₆	C₄	1	4	yes	direct product factor
10	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₅×C₁₆	C₂²	1	1	yes	characteristic
12	C₃×C₅×#107	C₂²	1	2	yes	—
13	C₃×C₅×#102	—	2	2	no	—
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₃×C₄×D₁₆	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₅×D₁₆	C₂×C₃	1	1	yes	characteristic
17	C₄×C₅×D₈	C₂×C₃	1	2	yes	complemented
18	C₅×#366	C₂×C₃	1	1	yes	characteristic, complemented
19	C₅×#357	C₂×C₃	1	2	yes	complemented
20	C₄×C₅×C₁₆	C₂×C₃	1	1	yes	characteristic, complemented
21	C₂×C₃×C₅×D₄	—	2	2	no	—
22	C₂²×C₃×C₄×C₅	—	4	2	no	—
23	C₃×C₅×D₈	—	2	2	no	—
24	C₃×C₅×D₈	C₂×C₄	1	4	yes	complemented
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×C₂²⋊C₄	—	4	2	no	—
29	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
30	C₃×C₅×C₁₆	—	2	1	no	—
31	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
32	C₂×C₃×D₁₆	C₂×C₅	1	1	yes	characteristic
33	C₃×C₄×D₈	C₂×C₅	1	2	yes	complemented
34	C₃×#366	C₂×C₅	1	1	yes	characteristic, complemented
35	C₃×#357	C₂×C₅	1	2	yes	complemented
36	C₃×C₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
37	C₂×C₅×D₈	C₂²×C₃	1	2	yes	—
38	C₄×C₅×D₄	—	2	2	no	—
39	C₅×D₁₆	C₃×C₄	1	4	yes	direct product factor
40	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₅×C₁₆	C₂²×C₃	1	1	yes	characteristic
42	C₅×#107	C₂²×C₃	1	2	yes	—
43	C₅×#102	—	2	2	no	—
44	C₄×C₅×C₈	C₂²×C₃	1	1	yes	characteristic
45	C₄×D₁₆	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
46	C₂³×C₃×C₅	—	4	2	no	—
47	C₃×C₅×D₄	—	2	4	no	—
48	C₃×C₅×D₄	—	4	2	no	—
49	C₂×C₃×C₄×C₅	—	4	2	no	—
50	C₂×C₃×C₄×C₅	—	8	4	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	—	2	1	no	—
55	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₅×C₈	#41	1	1	yes	characteristic
57	C₂×C₃×D₈	C₂²×C₅	1	2	yes	—
58	C₃×C₄×D₄	—	2	2	no	—
59	C₃×D₁₆	C₄×C₅	1	4	yes	direct product factor
60	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
61	C₂×C₃×C₁₆	C₂²×C₅	1	1	yes	characteristic
62	C₃×#107	C₂²×C₅	1	2	yes	—
63	C₃×#102	—	2	2	no	—
64	C₃×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
65	C₂×C₅×D₄	—	2	2	no	—
66	C₂²×C₄×C₅	—	4	2	no	—
67	C₅×D₈	—	2	2	no	—
68	C₅×D₈	C₂×C₃×C₄	1	4	yes	complemented
69	C₂×C₅×C₈	C₃×D₄	1	1	yes	characteristic
70	C₂×C₅×C₈	C₂³×C₃	1	1	yes	characteristic
71	C₅×C₄⋊C₄	—	2	2	no	—
72	C₅×C₂²⋊C₄	—	4	2	no	—
73	C₄²×C₅	C₃×D₄	1	1	yes	characteristic
74	C₅×C₁₆	—	2	1	no	—
75	C₅×C₁₆	C₂×C₃×C₄	1	2	yes	complemented
76	C₂×D₁₆	C₂×C₃×C₅	1	1	yes	characteristic
77	C₄×D₈	C₂×C₃×C₅	1	2	yes	complemented
78	#366	C₂×C₃×C₅	1	1	yes	characteristic, complemented
79	#357	C₂×C₃×C₅	1	2	yes	complemented
80	C₄×C₁₆	C₂×C₃×C₅	1	1	yes	characteristic, complemented
81	C₂²×C₃×C₅	—	4	4	no	—
82	C₂²×C₃×C₅	—	8	2	no	—
83	C₂²×C₃×C₅	—	8	2	no	—
84	C₂²×C₃×C₅	C₂×D₈	1	1	yes	characteristic
85	C₃×C₄×C₅	—	2	1	no	—
86	C₃×C₄×C₅	—	8	2	no	—
87	C₃×C₄×C₅	D₁₆	1	2	yes	direct product factor
88	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
89	C₃×C₄×C₅	#135	1	1	yes	characteristic
90	C₂×C₃×D₄	—	2	2	no	—
91	C₂²×C₃×C₄	—	4	2	no	—
92	C₃×D₈	—	2	2	no	—
93	C₃×D₈	C₂×C₄×C₅	1	4	yes	complemented
94	C₂×C₃×C₈	C₅×D₄	1	1	yes	characteristic
95	C₂×C₃×C₈	C₂³×C₅	1	1	yes	characteristic
96	C₃×C₄⋊C₄	—	2	2	no	—
97	C₃×C₂²⋊C₄	—	4	2	no	—
98	C₃×C₄²	C₅×D₄	1	1	yes	characteristic
99	C₃×C₁₆	—	2	1	no	—
100	C₃×C₁₆	C₂×C₄×C₅	1	2	yes	complemented
101	C₂³×C₅	—	4	2	no	—
102	C₅×D₄	—	2	4	no	—
103	C₅×D₄	—	4	2	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	—	8	4	no	—
106	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
107	C₂×C₄×C₅	C₃×D₈	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
109	C₅×C₈	—	2	1	no	—
110	C₅×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
111	C₅×C₈	C₃×#41	1	1	yes	characteristic
112	C₂×D₈	C₂²×C₃×C₅	1	2	yes	—
113	C₄×D₄	—	2	2	no	—
114	D₁₆	C₃×C₄×C₅	1	4	yes	direct product factor
115	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
116	C₂×C₁₆	C₂²×C₃×C₅	1	1	yes	characteristic
117	#107	C₂²×C₃×C₅	1	2	yes	—
118	#102	—	2	2	no	—
119	C₄×C₈	C₂²×C₃×C₅	1	1	yes	characteristic
120	C₂×C₃×C₅	—	8	4	no	—
121	C₂×C₃×C₅	C₄×D₈	1	1	yes	characteristic
122	C₂×C₃×C₅	C₂×D₁₆	1	1	yes	characteristic
123	C₂×C₃×C₅	#508	1	1	yes	characteristic
124	C₂³×C₃	—	4	2	no	—
125	C₃×D₄	—	2	4	no	—
126	C₃×D₄	—	4	2	no	—
127	C₂×C₃×C₄	—	4	2	no	—
128	C₂×C₃×C₄	—	8	4	no	—
129	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
130	C₂×C₃×C₄	C₅×D₈	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂×C₅×D₄	1	1	yes	characteristic
132	C₃×C₈	—	2	1	no	—
133	C₃×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
134	C₃×C₈	C₅×#41	1	1	yes	characteristic
135	C₂²×C₅	—	4	4	no	—
136	C₂²×C₅	—	8	2	no	—
137	C₂²×C₅	—	8	2	no	—
138	C₂²×C₅	C₂×C₃×D₈	1	1	yes	characteristic
139	C₄×C₅	—	2	1	no	—
140	C₄×C₅	—	8	2	no	—
141	C₄×C₅	C₃×D₁₆	1	2	yes	direct product factor
142	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
143	C₄×C₅	C₃×#135	1	1	yes	characteristic
144	C₂×D₄	—	2	2	no	—
145	C₂²×C₄	—	4	2	no	—
146	D₈	—	2	2	no	—
147	D₈	C₂×C₃×C₄×C₅	1	4	yes	complemented
148	C₂×C₈	C₃×C₅×D₄	1	1	yes	characteristic
149	C₂×C₈	C₂³×C₃×C₅	1	1	yes	characteristic
150	C₄⋊C₄	—	2	2	no	—
151	C₂²⋊C₄	—	4	2	no	—
152	C₄²	C₃×C₅×D₄	1	1	yes	characteristic
153	C₁₆	—	2	1	no	—
154	C₁₆	C₂×C₃×C₄×C₅	1	2	yes	complemented
155	C₃×C₅	C₄×D₁₆	1	1	yes	characteristic, direct product factor, Hall
156	C₂²×C₃	—	4	4	no	—
157	C₂²×C₃	—	8	2	no	—
158	C₂²×C₃	—	8	2	no	—
159	C₂²×C₃	C₂×C₅×D₈	1	1	yes	characteristic
160	C₃×C₄	—	2	1	no	—
161	C₃×C₄	—	8	2	no	—
162	C₃×C₄	C₅×D₁₆	1	2	yes	direct product factor
163	C₃×C₄	C₄×C₅×D₄	1	1	yes	characteristic
164	C₃×C₄	C₅×#135	1	1	yes	characteristic
165	C₂×C₅	—	8	4	no	—
166	C₂×C₅	C₃×C₄×D₈	1	1	yes	characteristic
167	C₂×C₅	C₂×C₃×D₁₆	1	1	yes	characteristic
168	C₂×C₅	C₃×#508	1	1	yes	characteristic
169	C₂³	—	4	2	no	—
170	D₄	—	2	4	no	—
171	D₄	—	4	2	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	8	4	no	—
174	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
175	C₂×C₄	C₃×C₅×D₈	1	1	yes	characteristic
176	C₂×C₄	C₂×C₃×C₅×D₄	1	1	yes	characteristic
177	C₈	—	2	1	no	—
178	C₈	C₂²×C₃×C₄×C₅	1	1	yes	characteristic
179	C₈	C₃×C₅×#41	1	1	yes	characteristic
180	C₂×C₃	—	8	4	no	—
181	C₂×C₃	C₄×C₅×D₈	1	1	yes	characteristic
182	C₂×C₃	C₂×C₅×D₁₆	1	1	yes	characteristic
183	C₂×C₃	C₅×#508	1	1	yes	characteristic
184	C₅	C₃×C₄×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
185	C₂²	—	4	4	no	—
186	C₂²	—	8	2	no	—
187	C₂²	—	8	2	no	—
188	C₂²	C₂×C₃×C₅×D₈	1	1	yes	characteristic
189	C₄	—	2	1	no	—
190	C₄	—	8	2	no	—
191	C₄	C₃×C₅×D₁₆	1	2	yes	direct product factor
192	C₄	C₃×C₄×C₅×D₄	1	1	yes	characteristic
193	C₄	C₃×C₅×#135	1	1	yes	characteristic
194	C₃	C₄×C₅×D₁₆	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	—	8	4	no	—
196	C₂	C₃×C₄×C₅×D₈	1	1	yes	characteristic
197	C₂	C₂×C₃×C₅×D₁₆	1	1	yes	characteristic
198	C₂	C₃×C₅×#508	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄×C₅	D₈
commutator	C₈	C₂²×C₃×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂³×C₃×C₅
socle	C₂²×C₃×C₅	C₂×D₈
prime radical	C₂×C₃×C₅×D₁₆	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×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
2. C₃×C₄×C₅×D₈	C₂	semidirect
3. C₃×C₅×#366	C₂	semidirect
4. C₃×C₅×#357	C₂	semidirect
5. C₃×C₄×C₅×C₁₆	C₂	semidirect
6. C₄×C₅×D₁₆	C₃	direct (N × H)
9. C₃×C₅×D₁₆	C₄	direct (N × H)
15. C₃×C₄×D₁₆	C₅	direct (N × H)
17. C₄×C₅×D₈	C₂×C₃	semidirect
18. C₅×#366	C₂×C₃	semidirect
19. C₅×#357	C₂×C₃	semidirect
20. C₄×C₅×C₁₆	C₂×C₃	semidirect
24. C₃×C₅×D₈	C₂×C₄	semidirect
31. C₃×C₅×C₁₆	C₂×C₄	semidirect
33. C₃×C₄×D₈	C₂×C₅	semidirect
34. C₃×#366	C₂×C₅	semidirect
35. C₃×#357	C₂×C₅	semidirect
36. C₃×C₄×C₁₆	C₂×C₅	semidirect
39. C₅×D₁₆	C₃×C₄	direct (N × H)
45. C₄×D₁₆	C₃×C₅	direct (N × H)
59. C₃×D₁₆	C₄×C₅	direct (N × H)
68. C₅×D₈	C₂×C₃×C₄	semidirect
75. C₅×C₁₆	C₂×C₃×C₄	semidirect
77. C₄×D₈	C₂×C₃×C₅	semidirect
78. #366	C₂×C₃×C₅	semidirect
79. #357	C₂×C₃×C₅	semidirect
80. C₄×C₁₆	C₂×C₃×C₅	semidirect
87. C₃×C₄×C₅	D₁₆	direct (N × H)
93. C₃×D₈	C₂×C₄×C₅	semidirect
100. C₃×C₁₆	C₂×C₄×C₅	semidirect
114. D₁₆	C₃×C₄×C₅	direct (N × H)
141. C₄×C₅	C₃×D₁₆	direct (N × H)
147. D₈	C₂×C₃×C₄×C₅	semidirect
154. C₁₆	C₂×C₃×C₄×C₅	semidirect
155. C₃×C₅	C₄×D₁₆	direct (N × H)
162. C₃×C₄	C₅×D₁₆	direct (N × H)
184. C₅	C₃×C₄×D₁₆	direct (N × H)
191. C₄	C₃×C₅×D₁₆	direct (N × H)
194. C₃	C₄×C₅×D₁₆	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×D₁₆	2
C₃×C₄×C₅×D₈	2
C₃×C₅×#366	2
C₃×C₅×#357	2
C₃×C₄×C₅×C₁₆	2
C₄×C₅×D₁₆	3
C₃×C₄×D₁₆	5

Subgroups — C3×C4×C5×D16 #14275206

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×C₅×D₁₆ #14275206