Subgroups — C₂³×C₇×C₆₄ #72183229

Order 3584.

644 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×C₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×C₆₄	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₇×C₁₆	C₄	1	1	yes	characteristic
4	C₂²×C₇×C₃₂	C₄	1	7	yes	—
5	C₂²×C₇×C₃₂	C₂²	1	7	yes	—
6	C₂×C₇×C₆₄	C₂²	1	28	yes	direct product factor
7	C₂³×C₆₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₂³×C₇×C₈	C₈	1	1	yes	characteristic
9	C₂²×C₇×C₁₆	C₈	1	7	yes	—
10	C₂²×C₇×C₁₆	C₂×C₄	1	7	yes	—
11	C₂×C₇×C₃₂	C₂×C₄	1	21	yes	—
12	C₂×C₇×C₃₂	C₂³	1	7	yes	—
13	C₇×C₆₄	C₂³	1	8	yes	direct product factor
14	C₂³×C₃₂	C₂×C₇	1	1	yes	characteristic
15	C₂²×C₆₄	C₂×C₇	1	14	yes	direct product factor
16	C₂³×C₄×C₇	C₁₆	1	1	yes	characteristic
17	C₂²×C₇×C₈	C₁₆	1	7	yes	—
18	C₂²×C₇×C₈	C₂×C₈	1	7	yes	—
19	C₂×C₇×C₁₆	C₂×C₈	1	21	yes	—
20	C₂×C₇×C₁₆	C₂²×C₄	1	7	yes	—
21	C₇×C₃₂	C₂²×C₄	1	7	yes	—
22	C₇×C₃₂	C₂⁴	1	1	yes	characteristic
23	C₂³×C₁₆	C₄×C₇	1	1	yes	characteristic
24	C₂²×C₃₂	C₄×C₇	1	7	yes	—
25	C₂²×C₃₂	C₂²×C₇	1	7	yes	—
26	C₂×C₆₄	C₂²×C₇	1	28	yes	direct product factor
27	C₂⁴×C₇	C₃₂	1	1	yes	characteristic
28	C₂²×C₄×C₇	C₃₂	1	7	yes	—
29	C₂²×C₄×C₇	C₂×C₁₆	1	7	yes	—
30	C₂×C₇×C₈	C₂×C₁₆	1	21	yes	—
31	C₂×C₇×C₈	C₂²×C₈	1	7	yes	—
32	C₇×C₁₆	C₂²×C₈	1	7	yes	—
33	C₇×C₁₆	C₂³×C₄	1	1	yes	characteristic
34	C₂³×C₈	C₇×C₈	1	1	yes	characteristic
35	C₂²×C₁₆	C₇×C₈	1	7	yes	—
36	C₂²×C₁₆	C₂×C₄×C₇	1	7	yes	—
37	C₂×C₃₂	C₂×C₄×C₇	1	21	yes	—
38	C₂×C₃₂	C₂³×C₇	1	7	yes	—
39	C₆₄	C₂³×C₇	1	8	yes	direct product factor
40	C₂³×C₇	C₆₄	1	8	yes	direct product factor
41	C₂³×C₇	C₂×C₃₂	1	7	yes	—
42	C₂×C₄×C₇	C₂×C₃₂	1	21	yes	—
43	C₂×C₄×C₇	C₂²×C₁₆	1	7	yes	—
44	C₇×C₈	C₂²×C₁₆	1	7	yes	—
45	C₇×C₈	C₂³×C₈	1	1	yes	characteristic
46	C₂³×C₄	C₇×C₁₆	1	1	yes	characteristic
47	C₂²×C₈	C₇×C₁₆	1	7	yes	—
48	C₂²×C₈	C₂×C₇×C₈	1	7	yes	—
49	C₂×C₁₆	C₂×C₇×C₈	1	21	yes	—
50	C₂×C₁₆	C₂²×C₄×C₇	1	7	yes	—
51	C₃₂	C₂²×C₄×C₇	1	7	yes	—
52	C₃₂	C₂⁴×C₇	1	1	yes	characteristic
53	C₂²×C₇	C₂×C₆₄	1	28	yes	direct product factor
54	C₂²×C₇	C₂²×C₃₂	1	7	yes	—
55	C₄×C₇	C₂²×C₃₂	1	7	yes	—
56	C₄×C₇	C₂³×C₁₆	1	1	yes	characteristic
57	C₂⁴	C₇×C₃₂	1	1	yes	characteristic
58	C₂²×C₄	C₇×C₃₂	1	7	yes	—
59	C₂²×C₄	C₂×C₇×C₁₆	1	7	yes	—
60	C₂×C₈	C₂×C₇×C₁₆	1	21	yes	—
61	C₂×C₈	C₂²×C₇×C₈	1	7	yes	—
62	C₁₆	C₂²×C₇×C₈	1	7	yes	—
63	C₁₆	C₂³×C₄×C₇	1	1	yes	characteristic
64	C₂×C₇	C₂²×C₆₄	1	14	yes	direct product factor
65	C₂×C₇	C₂³×C₃₂	1	1	yes	characteristic
66	C₂³	C₇×C₆₄	1	8	yes	direct product factor
67	C₂³	C₂×C₇×C₃₂	1	7	yes	—
68	C₂×C₄	C₂×C₇×C₃₂	1	21	yes	—
69	C₂×C₄	C₂²×C₇×C₁₆	1	7	yes	—
70	C₈	C₂²×C₇×C₁₆	1	7	yes	—
71	C₈	C₂³×C₇×C₈	1	1	yes	characteristic
72	C₇	C₂³×C₆₄	1	1	yes	characteristic, direct product factor, Sylow
73	C₂²	C₂×C₇×C₆₄	1	28	yes	direct product factor
74	C₂²	C₂²×C₇×C₃₂	1	7	yes	—
75	C₄	C₂²×C₇×C₃₂	1	7	yes	—
76	C₄	C₂³×C₇×C₁₆	1	1	yes	characteristic
77	C₂	C₂²×C₇×C₆₄	1	14	yes	direct product factor
78	C₂	C₂³×C₇×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃₂	C₂⁴×C₇
socle, prime radical	C₂⁴×C₇	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×C₇×C₆₄	C₂	direct (N × H)
6. C₂×C₇×C₆₄	C₂²	direct (N × H)
7. C₂³×C₆₄	C₇	direct (N × H)
13. C₇×C₆₄	C₂³	direct (N × H)
15. C₂²×C₆₄	C₂×C₇	direct (N × H)
26. C₂×C₆₄	C₂²×C₇	direct (N × H)
39. C₆₄	C₂³×C₇	direct (N × H)
40. C₂³×C₇	C₆₄	direct (N × H)
53. C₂²×C₇	C₂×C₆₄	direct (N × H)
64. C₂×C₇	C₂²×C₆₄	direct (N × H)
66. C₂³	C₇×C₆₄	direct (N × H)
72. C₇	C₂³×C₆₄	direct (N × H)
73. C₂²	C₂×C₇×C₆₄	direct (N × H)
77. C₂	C₂²×C₇×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×C₇×C₃₂	2
C₂²×C₇×C₆₄	2
C₂³×C₆₄	7

Subgroups — C23×C7×C64 #72183229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₇×C₆₄ #72183229