Subgroups — C₂×C₃×C₄²×C₁₆ #72181981

Order 1536.

2820 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₄×C₁₆	C₂	1	6	yes	maximal
2	C₂×C₃×C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄²×C₁₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄²×C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂³×C₃×C₁₆	C₂²	1	4	yes	—
6	C₂²×C₃×C₄×C₈	C₄	1	3	yes	—
7	C₂²×C₃×C₄×C₈	C₂²	1	3	yes	—
8	C₂×C₃×C₄³	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄×C₁₆	C₄	1	48	yes	direct product factor
10	C₂×C₃×C₄×C₁₆	C₂²	1	24	yes	—
11	C₃×C₄²×C₈	C₄	1	4	yes	—
12	C₃×C₄²×C₈	C₂²	1	4	yes	—
13	C₂²×C₄×C₁₆	C₂×C₃	1	6	yes	—
14	C₂×C₄²×C₈	C₂×C₃	1	1	yes	characteristic
15	C₄²×C₁₆	C₂×C₃	1	8	yes	direct product factor
16	C₂³×C₃×C₈	C₂×C₄	1	3	yes	—
17	C₂³×C₃×C₈	C₂³	1	1	yes	characteristic
18	C₂²×C₃×C₄²	C₈	1	4	yes	—
19	C₂²×C₃×C₄²	C₂×C₄	1	3	yes	—
20	C₂²×C₃×C₁₆	C₂×C₄	1	48	yes	—
21	C₂²×C₃×C₁₆	C₂³	1	8	yes	—
22	C₂×C₃×C₄×C₈	C₈	1	24	yes	—
23	C₂×C₃×C₄×C₈	C₂×C₄	1	6	yes	—
24	C₂×C₃×C₄×C₈	C₂×C₄	1	12	yes	—
25	C₂×C₃×C₄×C₈	C₂×C₄	1	24	yes	—
26	C₂×C₃×C₄×C₈	C₂³	1	6	yes	—
27	C₃×C₄³	C₈	1	4	yes	—
28	C₃×C₄³	C₂×C₄	1	4	yes	—
29	C₃×C₄×C₁₆	C₂×C₄	1	96	yes	direct product factor
30	C₂³×C₁₆	C₂²×C₃	1	4	yes	—
31	C₂²×C₄×C₈	C₃×C₄	1	3	yes	—
32	C₂²×C₄×C₈	C₂²×C₃	1	3	yes	—
33	C₂×C₄³	C₃×C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₆	C₃×C₄	1	48	yes	direct product factor
35	C₂×C₄×C₁₆	C₂²×C₃	1	24	yes	—
36	C₄²×C₈	C₃×C₄	1	4	yes	—
37	C₄²×C₈	C₂²×C₃	1	4	yes	—
38	C₂³×C₃×C₄	C₂×C₈	1	6	yes	—
39	C₂³×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₃×C₈	C₄²	1	12	yes	—
41	C₂²×C₃×C₈	C₂×C₈	1	24	yes	—
42	C₂²×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂²×C₄	1	6	yes	—
44	C₂²×C₃×C₈	C₂²×C₄	1	12	yes	—
45	C₂²×C₃×C₈	C₂⁴	1	1	yes	characteristic
46	C₂×C₃×C₄²	C₁₆	1	32	yes	direct product factor
47	C₂×C₃×C₄²	C₄²	1	12	yes	—
48	C₂×C₃×C₄²	C₂×C₈	1	6	yes	—
49	C₂×C₃×C₄²	C₂×C₈	1	12	yes	—
50	C₂×C₃×C₄²	C₂×C₈	1	16	yes	—
51	C₂×C₃×C₄²	C₂²×C₄	1	6	yes	—
52	C₂×C₃×C₁₆	C₄²	1	64	yes	direct product factor
53	C₂×C₃×C₁₆	C₂²×C₄	1	48	yes	—
54	C₃×C₄×C₈	C₄²	1	24	yes	—
55	C₃×C₄×C₈	C₂×C₈	1	48	yes	—
56	C₃×C₄×C₈	C₂²×C₄	1	24	yes	—
57	C₂³×C₈	C₂×C₃×C₄	1	3	yes	—
58	C₂³×C₈	C₂³×C₃	1	1	yes	characteristic
59	C₂²×C₄²	C₃×C₈	1	4	yes	—
60	C₂²×C₄²	C₂×C₃×C₄	1	3	yes	—
61	C₂²×C₁₆	C₂×C₃×C₄	1	48	yes	—
62	C₂²×C₁₆	C₂³×C₃	1	8	yes	—
63	C₂×C₄×C₈	C₃×C₈	1	24	yes	—
64	C₂×C₄×C₈	C₂×C₃×C₄	1	6	yes	—
65	C₂×C₄×C₈	C₂×C₃×C₄	1	12	yes	—
66	C₂×C₄×C₈	C₂×C₃×C₄	1	24	yes	—
67	C₂×C₄×C₈	C₂³×C₃	1	6	yes	—
68	C₄³	C₃×C₈	1	4	yes	—
69	C₄³	C₂×C₃×C₄	1	4	yes	—
70	C₄×C₁₆	C₂×C₃×C₄	1	96	yes	direct product factor
71	C₂⁴×C₃	C₂²×C₈	1	1	yes	characteristic
72	C₂²×C₃×C₄	C₄×C₈	1	24	yes	—
73	C₂²×C₃×C₄	C₂×C₁₆	1	48	yes	—
74	C₂²×C₃×C₄	C₂×C₄²	1	6	yes	—
75	C₂²×C₃×C₄	C₂²×C₈	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂²×C₈	1	6	yes	—
77	C₂²×C₃×C₄	C₂²×C₈	1	12	yes	—
78	C₂²×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
79	C₂×C₃×C₈	C₄×C₈	1	48	yes	—
80	C₂×C₃×C₈	C₂×C₄²	1	6	yes	—
81	C₂×C₃×C₈	C₂×C₄²	1	12	yes	—
82	C₂×C₃×C₈	C₂×C₄²	1	16	yes	—
83	C₂×C₃×C₈	C₂²×C₈	1	24	yes	—
84	C₂×C₃×C₈	C₂³×C₄	1	6	yes	—
85	C₃×C₄²	C₄×C₈	1	24	yes	—
86	C₃×C₄²	C₂×C₁₆	1	64	yes	direct product factor
87	C₃×C₄²	C₂×C₄²	1	12	yes	—
88	C₃×C₄²	C₂²×C₈	1	12	yes	—
89	C₃×C₁₆	C₂×C₄²	1	32	yes	direct product factor
90	C₂³×C₄	C₂×C₃×C₈	1	6	yes	—
91	C₂³×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
92	C₂²×C₈	C₃×C₄²	1	12	yes	—
93	C₂²×C₈	C₂×C₃×C₈	1	24	yes	—
94	C₂²×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
95	C₂²×C₈	C₂²×C₃×C₄	1	6	yes	—
96	C₂²×C₈	C₂²×C₃×C₄	1	12	yes	—
97	C₂²×C₈	C₂⁴×C₃	1	1	yes	characteristic
98	C₂×C₄²	C₃×C₁₆	1	32	yes	direct product factor
99	C₂×C₄²	C₃×C₄²	1	12	yes	—
100	C₂×C₄²	C₂×C₃×C₈	1	6	yes	—
101	C₂×C₄²	C₂×C₃×C₈	1	12	yes	—
102	C₂×C₄²	C₂×C₃×C₈	1	16	yes	—
103	C₂×C₄²	C₂²×C₃×C₄	1	6	yes	—
104	C₂×C₁₆	C₃×C₄²	1	64	yes	direct product factor
105	C₂×C₁₆	C₂²×C₃×C₄	1	48	yes	—
106	C₄×C₈	C₃×C₄²	1	24	yes	—
107	C₄×C₈	C₂×C₃×C₈	1	48	yes	—
108	C₄×C₈	C₂²×C₃×C₄	1	24	yes	—
109	C₂³×C₃	C₂×C₄×C₈	1	6	yes	—
110	C₂³×C₃	C₂²×C₁₆	1	8	yes	—
111	C₂³×C₃	C₂³×C₈	1	1	yes	characteristic
112	C₂×C₃×C₄	C₄×C₁₆	1	96	yes	direct product factor
113	C₂×C₃×C₄	C₄³	1	4	yes	—
114	C₂×C₃×C₄	C₂×C₄×C₈	1	6	yes	—
115	C₂×C₃×C₄	C₂×C₄×C₈	1	12	yes	—
116	C₂×C₃×C₄	C₂×C₄×C₈	1	24	yes	—
117	C₂×C₃×C₄	C₂²×C₁₆	1	48	yes	—
118	C₂×C₃×C₄	C₂²×C₄²	1	3	yes	—
119	C₂×C₃×C₄	C₂³×C₈	1	3	yes	—
120	C₃×C₈	C₄³	1	4	yes	—
121	C₃×C₈	C₂×C₄×C₈	1	24	yes	—
122	C₃×C₈	C₂²×C₄²	1	4	yes	—
123	C₂⁴	C₂²×C₃×C₈	1	1	yes	characteristic
124	C₂²×C₄	C₃×C₄×C₈	1	24	yes	—
125	C₂²×C₄	C₂×C₃×C₁₆	1	48	yes	—
126	C₂²×C₄	C₂×C₃×C₄²	1	6	yes	—
127	C₂²×C₄	C₂²×C₃×C₈	1	1	yes	characteristic
128	C₂²×C₄	C₂²×C₃×C₈	1	6	yes	—
129	C₂²×C₄	C₂²×C₃×C₈	1	12	yes	—
130	C₂²×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
131	C₂×C₈	C₃×C₄×C₈	1	48	yes	—
132	C₂×C₈	C₂×C₃×C₄²	1	6	yes	—
133	C₂×C₈	C₂×C₃×C₄²	1	12	yes	—
134	C₂×C₈	C₂×C₃×C₄²	1	16	yes	—
135	C₂×C₈	C₂²×C₃×C₈	1	24	yes	—
136	C₂×C₈	C₂³×C₃×C₄	1	6	yes	—
137	C₄²	C₃×C₄×C₈	1	24	yes	—
138	C₄²	C₂×C₃×C₁₆	1	64	yes	direct product factor
139	C₄²	C₂×C₃×C₄²	1	12	yes	—
140	C₄²	C₂²×C₃×C₈	1	12	yes	—
141	C₁₆	C₂×C₃×C₄²	1	32	yes	direct product factor
142	C₂²×C₃	C₄²×C₈	1	4	yes	—
143	C₂²×C₃	C₂×C₄×C₁₆	1	24	yes	—
144	C₂²×C₃	C₂²×C₄×C₈	1	3	yes	—
145	C₂²×C₃	C₂³×C₁₆	1	4	yes	—
146	C₃×C₄	C₄²×C₈	1	4	yes	—
147	C₃×C₄	C₂×C₄×C₁₆	1	48	yes	direct product factor
148	C₃×C₄	C₂×C₄³	1	1	yes	characteristic
149	C₃×C₄	C₂²×C₄×C₈	1	3	yes	—
150	C₂³	C₂×C₃×C₄×C₈	1	6	yes	—
151	C₂³	C₂²×C₃×C₁₆	1	8	yes	—
152	C₂³	C₂³×C₃×C₈	1	1	yes	characteristic
153	C₂×C₄	C₃×C₄×C₁₆	1	96	yes	direct product factor
154	C₂×C₄	C₃×C₄³	1	4	yes	—
155	C₂×C₄	C₂×C₃×C₄×C₈	1	6	yes	—
156	C₂×C₄	C₂×C₃×C₄×C₈	1	12	yes	—
157	C₂×C₄	C₂×C₃×C₄×C₈	1	24	yes	—
158	C₂×C₄	C₂²×C₃×C₁₆	1	48	yes	—
159	C₂×C₄	C₂²×C₃×C₄²	1	3	yes	—
160	C₂×C₄	C₂³×C₃×C₈	1	3	yes	—
161	C₈	C₃×C₄³	1	4	yes	—
162	C₈	C₂×C₃×C₄×C₈	1	24	yes	—
163	C₈	C₂²×C₃×C₄²	1	4	yes	—
164	C₂×C₃	C₄²×C₁₆	1	8	yes	direct product factor
165	C₂×C₃	C₂×C₄²×C₈	1	1	yes	characteristic
166	C₂×C₃	C₂²×C₄×C₁₆	1	6	yes	—
167	C₂²	C₃×C₄²×C₈	1	4	yes	—
168	C₂²	C₂×C₃×C₄×C₁₆	1	24	yes	—
169	C₂²	C₂²×C₃×C₄×C₈	1	3	yes	—
170	C₂²	C₂³×C₃×C₁₆	1	4	yes	—
171	C₄	C₃×C₄²×C₈	1	4	yes	—
172	C₄	C₂×C₃×C₄×C₁₆	1	48	yes	direct product factor
173	C₄	C₂×C₃×C₄³	1	1	yes	characteristic
174	C₄	C₂²×C₃×C₄×C₈	1	3	yes	—
175	C₃	C₂×C₄²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
176	C₂	C₃×C₄²×C₁₆	1	8	yes	direct product factor
177	C₂	C₂×C₃×C₄²×C₈	1	1	yes	characteristic
178	C₂	C₂²×C₃×C₄×C₁₆	1	6	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₈	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₂²×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₄²×C₁₆	C₂	direct (N × H)
4. C₂×C₄²×C₁₆	C₃	direct (N × H)
9. C₂×C₃×C₄×C₁₆	C₄	direct (N × H)
15. C₄²×C₁₆	C₂×C₃	direct (N × H)
29. C₃×C₄×C₁₆	C₂×C₄	direct (N × H)
34. C₂×C₄×C₁₆	C₃×C₄	direct (N × H)
46. C₂×C₃×C₄²	C₁₆	direct (N × H)
52. C₂×C₃×C₁₆	C₄²	direct (N × H)
70. C₄×C₁₆	C₂×C₃×C₄	direct (N × H)
86. C₃×C₄²	C₂×C₁₆	direct (N × H)
89. C₃×C₁₆	C₂×C₄²	direct (N × H)
98. C₂×C₄²	C₃×C₁₆	direct (N × H)
104. C₂×C₁₆	C₃×C₄²	direct (N × H)
112. C₂×C₃×C₄	C₄×C₁₆	direct (N × H)
138. C₄²	C₂×C₃×C₁₆	direct (N × H)
141. C₁₆	C₂×C₃×C₄²	direct (N × H)
147. C₃×C₄	C₂×C₄×C₁₆	direct (N × H)
153. C₂×C₄	C₃×C₄×C₁₆	direct (N × H)
164. C₂×C₃	C₄²×C₁₆	direct (N × H)
172. C₄	C₂×C₃×C₄×C₁₆	direct (N × H)
175. C₃	C₂×C₄²×C₁₆	direct (N × H)
176. C₂	C₃×C₄²×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×C₄×C₁₆	2
C₂×C₃×C₄²×C₈	2
C₃×C₄²×C₁₆	2
C₂×C₄²×C₁₆	3

Subgroups — C2×C3×C42×C16 #72181981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄²×C₁₆ #72181981