Subgroups — C₄×C₇×C₁₆×C₇₃ #72209367

Order 32704.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×C₁₆×C₇₃	C₂	1	2	yes	maximal
3	C₄²×C₇×C₇₃	C₄	1	1	yes	characteristic
4	C₂×C₇×C₈×C₇₃	C₄	1	1	yes	characteristic
5	C₂×C₇×C₈×C₇₃	C₂²	1	1	yes	characteristic
6	C₇×C₁₆×C₇₃	C₄	1	4	yes	direct product factor
7	C₂×C₄×C₇×C₇₃	C₈	1	2	yes	—
8	C₂×C₄×C₇×C₇₃	C₂×C₄	1	1	yes	characteristic
9	C₇×C₈×C₇₃	C₈	1	2	yes	—
10	C₇×C₈×C₇₃	C₂×C₄	1	2	yes	—
11	C₄×C₁₆×C₇₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₁₆×C₇₃	C₂×C₇	1	2	yes	—
13	C₄×C₈×C₇₃	C₂×C₇	1	1	yes	characteristic
14	C₂²×C₇×C₇₃	C₂×C₈	1	1	yes	characteristic
15	C₄×C₇×C₇₃	C₁₆	1	4	yes	direct product factor
16	C₄×C₇×C₇₃	C₄²	1	1	yes	characteristic
17	C₄×C₇×C₇₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₈×C₇₃	C₄×C₇	1	1	yes	characteristic
19	C₂×C₈×C₇₃	C₂²×C₇	1	1	yes	characteristic
20	C₄²×C₇₃	C₄×C₇	1	1	yes	characteristic
21	C₁₆×C₇₃	C₄×C₇	1	4	yes	direct product factor
22	C₂×C₇×C₇₃	C₄×C₈	1	1	yes	characteristic
23	C₂×C₇×C₇₃	C₂×C₁₆	1	2	yes	—
24	C₂×C₄×C₇₃	C₇×C₈	1	2	yes	—
25	C₂×C₄×C₇₃	C₂×C₄×C₇	1	1	yes	characteristic
26	C₈×C₇₃	C₇×C₈	1	2	yes	—
27	C₈×C₇₃	C₂×C₄×C₇	1	2	yes	—
28	C₇×C₇₃	C₄×C₁₆	1	1	yes	characteristic, direct product factor, Hall
29	C₄×C₇×C₁₆	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂²×C₇₃	C₂×C₇×C₈	1	1	yes	characteristic
31	C₄×C₇₃	C₇×C₁₆	1	4	yes	direct product factor
32	C₄×C₇₃	C₄²×C₇	1	1	yes	characteristic
33	C₄×C₇₃	C₂×C₇×C₈	1	1	yes	characteristic
34	C₂×C₇×C₁₆	C₂×C₇₃	1	2	yes	—
35	C₄×C₇×C₈	C₂×C₇₃	1	1	yes	characteristic
36	C₂×C₇₃	C₄×C₇×C₈	1	1	yes	characteristic
37	C₂×C₇₃	C₂×C₇×C₁₆	1	2	yes	—
38	C₂×C₇×C₈	C₄×C₇₃	1	1	yes	characteristic
39	C₂×C₇×C₈	C₂²×C₇₃	1	1	yes	characteristic
40	C₄²×C₇	C₄×C₇₃	1	1	yes	characteristic
41	C₇×C₁₆	C₄×C₇₃	1	4	yes	direct product factor
42	C₇₃	C₄×C₇×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
43	C₄×C₁₆	C₇×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
44	C₂×C₄×C₇	C₈×C₇₃	1	2	yes	—
45	C₂×C₄×C₇	C₂×C₄×C₇₃	1	1	yes	characteristic
46	C₇×C₈	C₈×C₇₃	1	2	yes	—
47	C₇×C₈	C₂×C₄×C₇₃	1	2	yes	—
48	C₂×C₁₆	C₂×C₇×C₇₃	1	2	yes	—
49	C₄×C₈	C₂×C₇×C₇₃	1	1	yes	characteristic
50	C₂²×C₇	C₂×C₈×C₇₃	1	1	yes	characteristic
51	C₄×C₇	C₁₆×C₇₃	1	4	yes	direct product factor
52	C₄×C₇	C₄²×C₇₃	1	1	yes	characteristic
53	C₄×C₇	C₂×C₈×C₇₃	1	1	yes	characteristic
54	C₂×C₈	C₄×C₇×C₇₃	1	1	yes	characteristic
55	C₂×C₈	C₂²×C₇×C₇₃	1	1	yes	characteristic
56	C₄²	C₄×C₇×C₇₃	1	1	yes	characteristic
57	C₁₆	C₄×C₇×C₇₃	1	4	yes	direct product factor
58	C₂×C₇	C₄×C₈×C₇₃	1	1	yes	characteristic
59	C₂×C₇	C₂×C₁₆×C₇₃	1	2	yes	—
60	C₂×C₄	C₇×C₈×C₇₃	1	2	yes	—
61	C₂×C₄	C₂×C₄×C₇×C₇₃	1	1	yes	characteristic
62	C₈	C₇×C₈×C₇₃	1	2	yes	—
63	C₈	C₂×C₄×C₇×C₇₃	1	2	yes	—
64	C₇	C₄×C₁₆×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²	C₂×C₇×C₈×C₇₃	1	1	yes	characteristic
66	C₄	C₇×C₁₆×C₇₃	1	4	yes	direct product factor
67	C₄	C₂×C₇×C₈×C₇₃	1	1	yes	characteristic
68	C₄	C₄²×C₇×C₇₃	1	1	yes	characteristic
69	C₂	C₂×C₇×C₁₆×C₇₃	1	2	yes	—
70	C₂	C₄×C₇×C₈×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈	C₂²×C₇×C₇₃
socle, prime radical	C₂²×C₇×C₇₃	C₂×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₇×C₁₆×C₇₃	C₄	direct (N × H)
11. C₄×C₁₆×C₇₃	C₇	direct (N × H)
15. C₄×C₇×C₇₃	C₁₆	direct (N × H)
21. C₁₆×C₇₃	C₄×C₇	direct (N × H)
28. C₇×C₇₃	C₄×C₁₆	direct (N × H)
29. C₄×C₇×C₁₆	C₇₃	direct (N × H)
31. C₄×C₇₃	C₇×C₁₆	direct (N × H)
41. C₇×C₁₆	C₄×C₇₃	direct (N × H)
42. C₇₃	C₄×C₇×C₁₆	direct (N × H)
43. C₄×C₁₆	C₇×C₇₃	direct (N × H)
51. C₄×C₇	C₁₆×C₇₃	direct (N × H)
57. C₁₆	C₄×C₇×C₇₃	direct (N × H)
64. C₇	C₄×C₁₆×C₇₃	direct (N × H)
66. C₄	C₇×C₁₆×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
C₄×C₇×C₈×C₇₃	2
C₂×C₇×C₁₆×C₇₃	2
C₄×C₁₆×C₇₃	7
C₄×C₇×C₁₆	73

Subgroups — C4×C7×C16×C73 #72209367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₇×C₁₆×C₇₃ #72209367