Subgroups — C₃×C₇×M₆₄ #13051521

Order 1344.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₃₂	C₂	1	2	yes	maximal, complemented
3	C₇×M₆₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₇×C₈	C₄	1	1	yes	characteristic
5	C₃×C₇×C₁₆	C₄	1	1	yes	characteristic
6	C₃×C₇×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₁₆	C₂×C₃	1	1	yes	characteristic
8	C₇×C₃₂	C₂×C₃	1	2	yes	complemented
9	C₃×M₆₄	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₄×C₇	C₈	1	1	yes	characteristic
11	C₃×C₇×C₈	C₈	1	1	yes	characteristic
12	C₃×C₇×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂×C₇×C₈	C₃×C₄	1	1	yes	characteristic
14	C₇×C₁₆	C₃×C₄	1	1	yes	characteristic
15	C₇×C₁₆	C₂²×C₃	1	1	yes	characteristic
16	C₂×C₃×C₁₆	C₂×C₇	1	1	yes	characteristic
17	C₃×C₃₂	C₂×C₇	1	2	yes	complemented
18	C₂²×C₃×C₇	C₁₆	1	1	yes	characteristic
19	C₃×C₄×C₇	C₁₆	1	1	yes	characteristic
20	C₃×C₄×C₇	C₂×C₈	1	1	yes	characteristic
21	M₆₄	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
22	C₂×C₄×C₇	C₃×C₈	1	1	yes	characteristic
23	C₇×C₈	C₃×C₈	1	1	yes	characteristic
24	C₇×C₈	C₂×C₃×C₄	1	1	yes	characteristic
25	C₂×C₃×C₈	C₄×C₇	1	1	yes	characteristic
26	C₃×C₁₆	C₄×C₇	1	1	yes	characteristic
27	C₃×C₁₆	C₂²×C₇	1	1	yes	characteristic
28	C₂×C₃×C₇	—	2	1	no	—
29	C₂×C₃×C₇	C₂×C₁₆	1	1	yes	characteristic
30	C₂×C₁₆	C₂×C₃×C₇	1	1	yes	characteristic
31	C₃₂	C₂×C₃×C₇	1	2	yes	complemented
32	C₂²×C₇	C₃×C₁₆	1	1	yes	characteristic
33	C₄×C₇	C₃×C₁₆	1	1	yes	characteristic
34	C₄×C₇	C₂×C₃×C₈	1	1	yes	characteristic
35	C₂×C₃×C₄	C₇×C₈	1	1	yes	characteristic
36	C₃×C₈	C₇×C₈	1	1	yes	characteristic
37	C₃×C₈	C₂×C₄×C₇	1	1	yes	characteristic
38	C₃×C₇	M₆₄	1	1	yes	characteristic, direct product factor, Hall
39	C₂×C₈	C₃×C₄×C₇	1	1	yes	characteristic
40	C₁₆	C₃×C₄×C₇	1	1	yes	characteristic
41	C₁₆	C₂²×C₃×C₇	1	1	yes	characteristic
42	C₂×C₇	—	2	1	no	—
43	C₂×C₇	C₂×C₃×C₁₆	1	1	yes	characteristic
44	C₂²×C₃	C₇×C₁₆	1	1	yes	characteristic
45	C₃×C₄	C₇×C₁₆	1	1	yes	characteristic
46	C₃×C₄	C₂×C₇×C₈	1	1	yes	characteristic
47	C₂×C₄	C₃×C₇×C₈	1	1	yes	characteristic
48	C₈	C₃×C₇×C₈	1	1	yes	characteristic
49	C₈	C₂×C₃×C₄×C₇	1	1	yes	characteristic
50	C₇	C₃×M₆₄	1	1	yes	characteristic, direct product factor, Sylow
51	C₂×C₃	—	2	1	no	—
52	C₂×C₃	C₂×C₇×C₁₆	1	1	yes	characteristic
53	C₂²	C₃×C₇×C₁₆	1	1	yes	characteristic
54	C₄	C₃×C₇×C₁₆	1	1	yes	characteristic
55	C₄	C₂×C₃×C₇×C₈	1	1	yes	characteristic
56	C₃	C₇×M₆₄	1	1	yes	characteristic, direct product factor, Sylow
57	C₂	—	2	1	no	—
58	C₂	C₂×C₃×C₇×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₇×C₁₆	C₂²
commutator	C₂	C₂×C₃×C₇×C₁₆
Frattini, cosocle	C₁₆	C₂²×C₃×C₇
socle	C₂×C₃×C₇	C₂×C₁₆
prime radical	C₂²×C₃×C₇	C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₇×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×C₃₂	C₂	semidirect
3. C₇×M₆₄	C₃	direct (N × H)
8. C₇×C₃₂	C₂×C₃	semidirect
9. C₃×M₆₄	C₇	direct (N × H)
17. C₃×C₃₂	C₂×C₇	semidirect
21. M₆₄	C₃×C₇	direct (N × H)
31. C₃₂	C₂×C₃×C₇	semidirect
38. C₃×C₇	M₆₄	direct (N × H)
50. C₇	C₃×M₆₄	direct (N × H)
56. C₃	C₇×M₆₄	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×C₇×C₁₆	2
C₃×C₇×C₃₂	2
C₇×M₆₄	3
C₃×M₆₄	7

Subgroups — C3×C7×M64 #13051521

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×M₆₄ #13051521