Subgroups — C₂×C₃×C₄×C₁₆ #72820

Order 384.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₁₆	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂²×C₃×C₈	C₄	1	1	yes	characteristic
6	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
8	C₂×C₃×C₁₆	C₄	1	8	yes	direct product factor
9	C₂×C₃×C₁₆	C₂²	1	4	yes	—
10	C₃×C₄×C₈	C₄	1	2	yes	—
11	C₃×C₄×C₈	C₂²	1	2	yes	—
12	C₂²×C₁₆	C₂×C₃	1	2	yes	—
13	C₂×C₄×C₈	C₂×C₃	1	1	yes	characteristic
14	C₄×C₁₆	C₂×C₃	1	4	yes	direct product factor
15	C₂²×C₃×C₄	C₈	1	2	yes	—
16	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₈	C₈	1	4	yes	—
18	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
21	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
22	C₃×C₄²	C₈	1	2	yes	—
23	C₃×C₄²	C₂×C₄	1	2	yes	—
24	C₃×C₁₆	C₂×C₄	1	8	yes	direct product factor
25	C₂²×C₈	C₃×C₄	1	1	yes	characteristic
26	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
27	C₂×C₄²	C₃×C₄	1	1	yes	characteristic
28	C₂×C₁₆	C₃×C₄	1	8	yes	direct product factor
29	C₂×C₁₆	C₂²×C₃	1	4	yes	—
30	C₄×C₈	C₃×C₄	1	2	yes	—
31	C₄×C₈	C₂²×C₃	1	2	yes	—
32	C₂³×C₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₃×C₄	C₁₆	1	8	yes	direct product factor
34	C₂×C₃×C₄	C₄²	1	2	yes	—
35	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
36	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
37	C₂×C₃×C₄	C₂×C₈	1	4	yes	—
38	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₃×C₈	C₄²	1	2	yes	—
40	C₃×C₈	C₂×C₈	1	4	yes	—
41	C₃×C₈	C₂²×C₄	1	2	yes	—
42	C₂²×C₄	C₃×C₈	1	2	yes	—
43	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
44	C₂×C₈	C₃×C₈	1	4	yes	—
45	C₂×C₈	C₂×C₃×C₄	1	1	yes	characteristic
46	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
47	C₂×C₈	C₂×C₃×C₄	1	4	yes	—
48	C₂×C₈	C₂³×C₃	1	1	yes	characteristic
49	C₄²	C₃×C₈	1	2	yes	—
50	C₄²	C₂×C₃×C₄	1	2	yes	—
51	C₁₆	C₂×C₃×C₄	1	8	yes	direct product factor
52	C₂²×C₃	C₄×C₈	1	2	yes	—
53	C₂²×C₃	C₂×C₁₆	1	4	yes	—
54	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
55	C₃×C₄	C₄×C₈	1	2	yes	—
56	C₃×C₄	C₂×C₁₆	1	8	yes	direct product factor
57	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
58	C₃×C₄	C₂²×C₈	1	1	yes	characteristic
59	C₂³	C₂×C₃×C₈	1	1	yes	characteristic
60	C₂×C₄	C₃×C₁₆	1	8	yes	direct product factor
61	C₂×C₄	C₃×C₄²	1	2	yes	—
62	C₂×C₄	C₂×C₃×C₈	1	1	yes	characteristic
63	C₂×C₄	C₂×C₃×C₈	1	2	yes	—
64	C₂×C₄	C₂×C₃×C₈	1	4	yes	—
65	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
66	C₈	C₃×C₄²	1	2	yes	—
67	C₈	C₂×C₃×C₈	1	4	yes	—
68	C₈	C₂²×C₃×C₄	1	2	yes	—
69	C₂×C₃	C₄×C₁₆	1	4	yes	direct product factor
70	C₂×C₃	C₂×C₄×C₈	1	1	yes	characteristic
71	C₂×C₃	C₂²×C₁₆	1	2	yes	—
72	C₂²	C₃×C₄×C₈	1	2	yes	—
73	C₂²	C₂×C₃×C₁₆	1	4	yes	—
74	C₂²	C₂²×C₃×C₈	1	1	yes	characteristic
75	C₄	C₃×C₄×C₈	1	2	yes	—
76	C₄	C₂×C₃×C₁₆	1	8	yes	direct product factor
77	C₄	C₂×C₃×C₄²	1	1	yes	characteristic
78	C₄	C₂²×C₃×C₈	1	1	yes	characteristic
79	C₃	C₂×C₄×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
80	C₂	C₃×C₄×C₁₆	1	4	yes	direct product factor
81	C₂	C₂×C₃×C₄×C₈	1	1	yes	characteristic
82	C₂	C₂²×C₃×C₁₆	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂×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×C₄×C₁₆	C₂	direct (N × H)
4. C₂×C₄×C₁₆	C₃	direct (N × H)
8. C₂×C₃×C₁₆	C₄	direct (N × H)
14. C₄×C₁₆	C₂×C₃	direct (N × H)
24. C₃×C₁₆	C₂×C₄	direct (N × H)
28. C₂×C₁₆	C₃×C₄	direct (N × H)
33. C₂×C₃×C₄	C₁₆	direct (N × H)
51. C₁₆	C₂×C₃×C₄	direct (N × H)
56. C₃×C₄	C₂×C₁₆	direct (N × H)
60. C₂×C₄	C₃×C₁₆	direct (N × H)
69. C₂×C₃	C₄×C₁₆	direct (N × H)
76. C₄	C₂×C₃×C₁₆	direct (N × H)
79. C₃	C₂×C₄×C₁₆	direct (N × H)
80. C₂	C₃×C₄×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×C₁₆	2
C₂×C₃×C₄×C₈	2
C₃×C₄×C₁₆	2
C₂×C₄×C₁₆	3

Subgroups — C2×C3×C4×C16 #72820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₄×C₁₆ #72820