Subgroups — C₂³×C₃×C₆₄ #72181971

Order 1536.

644 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₆₄	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₆₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂³×C₃×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₃×C₃₂	C₄	1	7	yes	—
6	C₂²×C₃×C₃₂	C₂²	1	7	yes	—
7	C₂×C₃×C₆₄	C₂²	1	28	yes	direct product factor
8	C₂³×C₃₂	C₂×C₃	1	1	yes	characteristic
9	C₂²×C₆₄	C₂×C₃	1	14	yes	direct product factor
10	C₂³×C₃×C₈	C₈	1	1	yes	characteristic
11	C₂²×C₃×C₁₆	C₈	1	7	yes	—
12	C₂²×C₃×C₁₆	C₂×C₄	1	7	yes	—
13	C₂×C₃×C₃₂	C₂×C₄	1	21	yes	—
14	C₂×C₃×C₃₂	C₂³	1	7	yes	—
15	C₃×C₆₄	C₂³	1	8	yes	direct product factor
16	C₂³×C₁₆	C₃×C₄	1	1	yes	characteristic
17	C₂²×C₃₂	C₃×C₄	1	7	yes	—
18	C₂²×C₃₂	C₂²×C₃	1	7	yes	—
19	C₂×C₆₄	C₂²×C₃	1	28	yes	direct product factor
20	C₂³×C₃×C₄	C₁₆	1	1	yes	characteristic
21	C₂²×C₃×C₈	C₁₆	1	7	yes	—
22	C₂²×C₃×C₈	C₂×C₈	1	7	yes	—
23	C₂×C₃×C₁₆	C₂×C₈	1	21	yes	—
24	C₂×C₃×C₁₆	C₂²×C₄	1	7	yes	—
25	C₃×C₃₂	C₂²×C₄	1	7	yes	—
26	C₃×C₃₂	C₂⁴	1	1	yes	characteristic
27	C₂³×C₈	C₃×C₈	1	1	yes	characteristic
28	C₂²×C₁₆	C₃×C₈	1	7	yes	—
29	C₂²×C₁₆	C₂×C₃×C₄	1	7	yes	—
30	C₂×C₃₂	C₂×C₃×C₄	1	21	yes	—
31	C₂×C₃₂	C₂³×C₃	1	7	yes	—
32	C₆₄	C₂³×C₃	1	8	yes	direct product factor
33	C₂⁴×C₃	C₃₂	1	1	yes	characteristic
34	C₂²×C₃×C₄	C₃₂	1	7	yes	—
35	C₂²×C₃×C₄	C₂×C₁₆	1	7	yes	—
36	C₂×C₃×C₈	C₂×C₁₆	1	21	yes	—
37	C₂×C₃×C₈	C₂²×C₈	1	7	yes	—
38	C₃×C₁₆	C₂²×C₈	1	7	yes	—
39	C₃×C₁₆	C₂³×C₄	1	1	yes	characteristic
40	C₂³×C₄	C₃×C₁₆	1	1	yes	characteristic
41	C₂²×C₈	C₃×C₁₆	1	7	yes	—
42	C₂²×C₈	C₂×C₃×C₈	1	7	yes	—
43	C₂×C₁₆	C₂×C₃×C₈	1	21	yes	—
44	C₂×C₁₆	C₂²×C₃×C₄	1	7	yes	—
45	C₃₂	C₂²×C₃×C₄	1	7	yes	—
46	C₃₂	C₂⁴×C₃	1	1	yes	characteristic
47	C₂³×C₃	C₆₄	1	8	yes	direct product factor
48	C₂³×C₃	C₂×C₃₂	1	7	yes	—
49	C₂×C₃×C₄	C₂×C₃₂	1	21	yes	—
50	C₂×C₃×C₄	C₂²×C₁₆	1	7	yes	—
51	C₃×C₈	C₂²×C₁₆	1	7	yes	—
52	C₃×C₈	C₂³×C₈	1	1	yes	characteristic
53	C₂⁴	C₃×C₃₂	1	1	yes	characteristic
54	C₂²×C₄	C₃×C₃₂	1	7	yes	—
55	C₂²×C₄	C₂×C₃×C₁₆	1	7	yes	—
56	C₂×C₈	C₂×C₃×C₁₆	1	21	yes	—
57	C₂×C₈	C₂²×C₃×C₈	1	7	yes	—
58	C₁₆	C₂²×C₃×C₈	1	7	yes	—
59	C₁₆	C₂³×C₃×C₄	1	1	yes	characteristic
60	C₂²×C₃	C₂×C₆₄	1	28	yes	direct product factor
61	C₂²×C₃	C₂²×C₃₂	1	7	yes	—
62	C₃×C₄	C₂²×C₃₂	1	7	yes	—
63	C₃×C₄	C₂³×C₁₆	1	1	yes	characteristic
64	C₂³	C₃×C₆₄	1	8	yes	direct product factor
65	C₂³	C₂×C₃×C₃₂	1	7	yes	—
66	C₂×C₄	C₂×C₃×C₃₂	1	21	yes	—
67	C₂×C₄	C₂²×C₃×C₁₆	1	7	yes	—
68	C₈	C₂²×C₃×C₁₆	1	7	yes	—
69	C₈	C₂³×C₃×C₈	1	1	yes	characteristic
70	C₂×C₃	C₂²×C₆₄	1	14	yes	direct product factor
71	C₂×C₃	C₂³×C₃₂	1	1	yes	characteristic
72	C₂²	C₂×C₃×C₆₄	1	28	yes	direct product factor
73	C₂²	C₂²×C₃×C₃₂	1	7	yes	—
74	C₄	C₂²×C₃×C₃₂	1	7	yes	—
75	C₄	C₂³×C₃×C₁₆	1	1	yes	characteristic
76	C₃	C₂³×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
77	C₂	C₂²×C₃×C₆₄	1	14	yes	direct product factor
78	C₂	C₂³×C₃×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₃₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₆₄	C₂	direct (N × H)
3. C₂³×C₆₄	C₃	direct (N × H)
7. C₂×C₃×C₆₄	C₂²	direct (N × H)
9. C₂²×C₆₄	C₂×C₃	direct (N × H)
15. C₃×C₆₄	C₂³	direct (N × H)
19. C₂×C₆₄	C₂²×C₃	direct (N × H)
32. C₆₄	C₂³×C₃	direct (N × H)
47. C₂³×C₃	C₆₄	direct (N × H)
60. C₂²×C₃	C₂×C₆₄	direct (N × H)
64. C₂³	C₃×C₆₄	direct (N × H)
70. C₂×C₃	C₂²×C₆₄	direct (N × H)
72. C₂²	C₂×C₃×C₆₄	direct (N × H)
76. C₃	C₂³×C₆₄	direct (N × H)
77. C₂	C₂²×C₃×C₆₄	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃×C₃₂	2
C₂²×C₃×C₆₄	2
C₂³×C₆₄	3

Subgroups — C23×C3×C64 #72181971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃×C₆₄ #72181971