Subgroups — C₂×C₇×C₃₂×C₇₃ #72209366

Order 32704.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₇×C₃₂×C₇₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×C₈×C₇₃	C₄	1	1	yes	characteristic
4	C₇×C₁₆×C₇₃	C₄	1	1	yes	characteristic
5	C₇×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₇×C₇₃	C₈	1	1	yes	characteristic
7	C₇×C₈×C₇₃	C₈	1	1	yes	characteristic
8	C₇×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₃₂×C₇₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₁₆×C₇₃	C₂×C₇	1	1	yes	characteristic
11	C₃₂×C₇₃	C₂×C₇	1	2	yes	direct product factor
12	C₂²×C₇×C₇₃	C₁₆	1	1	yes	characteristic
13	C₄×C₇×C₇₃	C₁₆	1	1	yes	characteristic
14	C₄×C₇×C₇₃	C₂×C₈	1	1	yes	characteristic
15	C₂×C₈×C₇₃	C₄×C₇	1	1	yes	characteristic
16	C₁₆×C₇₃	C₄×C₇	1	1	yes	characteristic
17	C₁₆×C₇₃	C₂²×C₇	1	1	yes	characteristic
18	C₂×C₇×C₇₃	C₃₂	1	2	yes	direct product factor
19	C₂×C₇×C₇₃	C₂×C₁₆	1	1	yes	characteristic
20	C₂×C₄×C₇₃	C₇×C₈	1	1	yes	characteristic
21	C₈×C₇₃	C₇×C₈	1	1	yes	characteristic
22	C₈×C₇₃	C₂×C₄×C₇	1	1	yes	characteristic
23	C₇×C₇₃	C₂×C₃₂	1	1	yes	characteristic, direct product factor, Hall
24	C₂×C₇×C₃₂	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂²×C₇₃	C₇×C₁₆	1	1	yes	characteristic
26	C₄×C₇₃	C₇×C₁₆	1	1	yes	characteristic
27	C₄×C₇₃	C₂×C₇×C₈	1	1	yes	characteristic
28	C₂×C₇×C₁₆	C₂×C₇₃	1	1	yes	characteristic
29	C₇×C₃₂	C₂×C₇₃	1	2	yes	direct product factor
30	C₂×C₇₃	C₇×C₃₂	1	2	yes	direct product factor
31	C₂×C₇₃	C₂×C₇×C₁₆	1	1	yes	characteristic
32	C₂×C₇×C₈	C₄×C₇₃	1	1	yes	characteristic
33	C₇×C₁₆	C₄×C₇₃	1	1	yes	characteristic
34	C₇×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
35	C₇₃	C₂×C₇×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
36	C₂×C₃₂	C₇×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×C₄×C₇	C₈×C₇₃	1	1	yes	characteristic
38	C₇×C₈	C₈×C₇₃	1	1	yes	characteristic
39	C₇×C₈	C₂×C₄×C₇₃	1	1	yes	characteristic
40	C₂×C₁₆	C₂×C₇×C₇₃	1	1	yes	characteristic
41	C₃₂	C₂×C₇×C₇₃	1	2	yes	direct product factor
42	C₂²×C₇	C₁₆×C₇₃	1	1	yes	characteristic
43	C₄×C₇	C₁₆×C₇₃	1	1	yes	characteristic
44	C₄×C₇	C₂×C₈×C₇₃	1	1	yes	characteristic
45	C₂×C₈	C₄×C₇×C₇₃	1	1	yes	characteristic
46	C₁₆	C₄×C₇×C₇₃	1	1	yes	characteristic
47	C₁₆	C₂²×C₇×C₇₃	1	1	yes	characteristic
48	C₂×C₇	C₃₂×C₇₃	1	2	yes	direct product factor
49	C₂×C₇	C₂×C₁₆×C₇₃	1	1	yes	characteristic
50	C₂×C₄	C₇×C₈×C₇₃	1	1	yes	characteristic
51	C₈	C₇×C₈×C₇₃	1	1	yes	characteristic
52	C₈	C₂×C₄×C₇×C₇₃	1	1	yes	characteristic
53	C₇	C₂×C₃₂×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
54	C₂²	C₇×C₁₆×C₇₃	1	1	yes	characteristic
55	C₄	C₇×C₁₆×C₇₃	1	1	yes	characteristic
56	C₄	C₂×C₇×C₈×C₇₃	1	1	yes	characteristic
57	C₂	C₇×C₃₂×C₇₃	1	2	yes	direct product factor
58	C₂	C₂×C₇×C₁₆×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₁₆	C₂²×C₇×C₇₃
socle, prime radical	C₂²×C₇×C₇₃	C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇×C₃₂×C₇₃	C₂	direct (N × H)
9. C₂×C₃₂×C₇₃	C₇	direct (N × H)
11. C₃₂×C₇₃	C₂×C₇	direct (N × H)
18. C₂×C₇×C₇₃	C₃₂	direct (N × H)
23. C₇×C₇₃	C₂×C₃₂	direct (N × H)
24. C₂×C₇×C₃₂	C₇₃	direct (N × H)
29. C₇×C₃₂	C₂×C₇₃	direct (N × H)
30. C₂×C₇₃	C₇×C₃₂	direct (N × H)
35. C₇₃	C₂×C₇×C₃₂	direct (N × H)
36. C₂×C₃₂	C₇×C₇₃	direct (N × H)
41. C₃₂	C₂×C₇×C₇₃	direct (N × H)
48. C₂×C₇	C₃₂×C₇₃	direct (N × H)
53. C₇	C₂×C₃₂×C₇₃	direct (N × H)
57. C₂	C₇×C₃₂×C₇₃	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×C₇×C₁₆×C₇₃	2
C₇×C₃₂×C₇₃	2
C₂×C₃₂×C₇₃	7
C₂×C₇×C₃₂	73

Subgroups — C2×C7×C32×C73 #72209366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×C₃₂×C₇₃ #72209366