Subgroups — C₂×C₃²×C₁₆×C₇₃ #72198613

Order 21024.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₁₆×C₇₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₁₆×C₇₃	C₃	1	4	yes	maximal, direct product factor
4	C₂×C₃²×C₄×C₇₃	C₄	1	1	yes	characteristic
5	C₃²×C₈×C₇₃	C₄	1	1	yes	characteristic
6	C₃²×C₈×C₇₃	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₈×C₇₃	C₂×C₃	1	4	yes	—
8	C₃×C₁₆×C₇₃	C₂×C₃	1	8	yes	direct product factor
9	C₂²×C₃²×C₇₃	C₈	1	1	yes	characteristic
10	C₃²×C₄×C₇₃	C₈	1	1	yes	characteristic
11	C₃²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
12	C₂×C₁₆×C₇₃	C₃²	1	1	yes	characteristic, direct product factor, Hall
13	C₂×C₃×C₄×C₇₃	C₃×C₄	1	4	yes	—
14	C₃×C₈×C₇₃	C₃×C₄	1	4	yes	—
15	C₃×C₈×C₇₃	C₂²×C₃	1	4	yes	—
16	C₂×C₃²×C₇₃	C₁₆	1	2	yes	direct product factor
17	C₂×C₃²×C₇₃	C₂×C₈	1	1	yes	characteristic
18	C₂×C₈×C₇₃	C₂×C₃²	1	1	yes	characteristic
19	C₁₆×C₇₃	C₂×C₃²	1	2	yes	direct product factor
20	C₂²×C₃×C₇₃	C₃×C₈	1	4	yes	—
21	C₃×C₄×C₇₃	C₃×C₈	1	4	yes	—
22	C₃×C₄×C₇₃	C₂×C₃×C₄	1	4	yes	—
23	C₃²×C₇₃	C₂×C₁₆	1	1	yes	characteristic, direct product factor, Hall
24	C₂×C₄×C₇₃	C₃²×C₄	1	1	yes	characteristic
25	C₈×C₇₃	C₃²×C₄	1	1	yes	characteristic
26	C₈×C₇₃	C₂²×C₃²	1	1	yes	characteristic
27	C₂×C₃×C₇₃	C₃×C₁₆	1	8	yes	direct product factor
28	C₂×C₃×C₇₃	C₂×C₃×C₈	1	4	yes	—
29	C₂²×C₇₃	C₃²×C₈	1	1	yes	characteristic
30	C₄×C₇₃	C₃²×C₈	1	1	yes	characteristic
31	C₄×C₇₃	C₂×C₃²×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₁₆	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
33	C₃×C₇₃	C₂×C₃×C₁₆	1	4	yes	direct product factor
34	C₂×C₇₃	C₃²×C₁₆	1	2	yes	direct product factor
35	C₂×C₇₃	C₂×C₃²×C₈	1	1	yes	characteristic
36	C₂×C₃²×C₈	C₂×C₇₃	1	1	yes	characteristic
37	C₃²×C₁₆	C₂×C₇₃	1	2	yes	direct product factor
38	C₂×C₃×C₁₆	C₃×C₇₃	1	4	yes	direct product factor
39	C₇₃	C₂×C₃²×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
40	C₂×C₃²×C₄	C₄×C₇₃	1	1	yes	characteristic
41	C₃²×C₈	C₄×C₇₃	1	1	yes	characteristic
42	C₃²×C₈	C₂²×C₇₃	1	1	yes	characteristic
43	C₂×C₃×C₈	C₂×C₃×C₇₃	1	4	yes	—
44	C₃×C₁₆	C₂×C₃×C₇₃	1	8	yes	direct product factor
45	C₂²×C₃²	C₈×C₇₃	1	1	yes	characteristic
46	C₃²×C₄	C₈×C₇₃	1	1	yes	characteristic
47	C₃²×C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
48	C₂×C₁₆	C₃²×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
49	C₂×C₃×C₄	C₃×C₄×C₇₃	1	4	yes	—
50	C₃×C₈	C₃×C₄×C₇₃	1	4	yes	—
51	C₃×C₈	C₂²×C₃×C₇₃	1	4	yes	—
52	C₂×C₃²	C₁₆×C₇₃	1	2	yes	direct product factor
53	C₂×C₃²	C₂×C₈×C₇₃	1	1	yes	characteristic
54	C₂×C₈	C₂×C₃²×C₇₃	1	1	yes	characteristic
55	C₁₆	C₂×C₃²×C₇₃	1	2	yes	direct product factor
56	C₂²×C₃	C₃×C₈×C₇₃	1	4	yes	—
57	C₃×C₄	C₃×C₈×C₇₃	1	4	yes	—
58	C₃×C₄	C₂×C₃×C₄×C₇₃	1	4	yes	—
59	C₃²	C₂×C₁₆×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×C₄	C₃²×C₄×C₇₃	1	1	yes	characteristic
61	C₈	C₃²×C₄×C₇₃	1	1	yes	characteristic
62	C₈	C₂²×C₃²×C₇₃	1	1	yes	characteristic
63	C₂×C₃	C₃×C₁₆×C₇₃	1	8	yes	direct product factor
64	C₂×C₃	C₂×C₃×C₈×C₇₃	1	4	yes	—
65	C₂²	C₃²×C₈×C₇₃	1	1	yes	characteristic
66	C₄	C₃²×C₈×C₇₃	1	1	yes	characteristic
67	C₄	C₂×C₃²×C₄×C₇₃	1	1	yes	characteristic
68	C₃	C₂×C₃×C₁₆×C₇₃	1	4	yes	direct product factor
69	C₂	C₃²×C₁₆×C₇₃	1	2	yes	direct product factor
70	C₂	C₂×C₃²×C₈×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₈	C₂²×C₃²×C₇₃
socle, prime radical	C₂²×C₃²×C₇₃	C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×C₁₆×C₇₃	C₂	direct (N × H)
3. C₂×C₃×C₁₆×C₇₃	C₃	direct (N × H)
8. C₃×C₁₆×C₇₃	C₂×C₃	direct (N × H)
12. C₂×C₁₆×C₇₃	C₃²	direct (N × H)
16. C₂×C₃²×C₇₃	C₁₆	direct (N × H)
19. C₁₆×C₇₃	C₂×C₃²	direct (N × H)
23. C₃²×C₇₃	C₂×C₁₆	direct (N × H)
27. C₂×C₃×C₇₃	C₃×C₁₆	direct (N × H)
32. C₂×C₃²×C₁₆	C₇₃	direct (N × H)
33. C₃×C₇₃	C₂×C₃×C₁₆	direct (N × H)
34. C₂×C₇₃	C₃²×C₁₆	direct (N × H)
37. C₃²×C₁₆	C₂×C₇₃	direct (N × H)
38. C₂×C₃×C₁₆	C₃×C₇₃	direct (N × H)
39. C₇₃	C₂×C₃²×C₁₆	direct (N × H)
44. C₃×C₁₆	C₂×C₃×C₇₃	direct (N × H)
48. C₂×C₁₆	C₃²×C₇₃	direct (N × H)
52. C₂×C₃²	C₁₆×C₇₃	direct (N × H)
55. C₁₆	C₂×C₃²×C₇₃	direct (N × H)
59. C₃²	C₂×C₁₆×C₇₃	direct (N × H)
63. C₂×C₃	C₃×C₁₆×C₇₃	direct (N × H)
68. C₃	C₂×C₃×C₁₆×C₇₃	direct (N × H)
69. C₂	C₃²×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
C₂×C₃²×C₈×C₇₃	2
C₃²×C₁₆×C₇₃	2
C₂×C₃×C₁₆×C₇₃	3
C₂×C₃²×C₁₆	73

Subgroups — C2×C32×C16×C73 #72198613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₁₆×C₇₃ #72198613