Subgroups — C₂×C₃×C₇×C₈×C₇₃ #72201781

Order 24528.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₇×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₈×C₇₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₇×C₈×C₇₃	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₄×C₇×C₇₃	C₂×C₃	1	1	yes	characteristic
5	C₇×C₈×C₇₃	C₂×C₃	1	2	yes	direct product factor
6	C₂×C₃×C₈×C₇₃	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₇×C₇₃	C₄	1	1	yes	characteristic
8	C₃×C₄×C₇×C₇₃	C₄	1	1	yes	characteristic
9	C₃×C₄×C₇×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×C₇₃	C₈	1	2	yes	direct product factor
11	C₂×C₃×C₇×C₇₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₇×C₇₃	C₃×C₄	1	1	yes	characteristic
13	C₄×C₇×C₇₃	C₃×C₄	1	1	yes	characteristic
14	C₄×C₇×C₇₃	C₂²×C₃	1	1	yes	characteristic
15	C₂×C₃×C₄×C₇₃	C₂×C₇	1	1	yes	characteristic
16	C₃×C₈×C₇₃	C₂×C₇	1	2	yes	direct product factor
17	C₃×C₇×C₇₃	C₂×C₈	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₈×C₇₃	C₃×C₇	1	1	yes	characteristic, direct product factor, Hall
19	C₂×C₇×C₇₃	C₃×C₈	1	2	yes	direct product factor
20	C₂×C₇×C₇₃	C₂×C₃×C₄	1	1	yes	characteristic
21	C₂²×C₃×C₇₃	C₄×C₇	1	1	yes	characteristic
22	C₃×C₄×C₇₃	C₄×C₇	1	1	yes	characteristic
23	C₃×C₄×C₇₃	C₂²×C₇	1	1	yes	characteristic
24	C₂×C₄×C₇₃	C₂×C₃×C₇	1	1	yes	characteristic
25	C₈×C₇₃	C₂×C₃×C₇	1	2	yes	direct product factor
26	C₇×C₇₃	C₂×C₃×C₈	1	1	yes	characteristic, direct product factor, Hall
27	C₂×C₃×C₇₃	C₇×C₈	1	2	yes	direct product factor
28	C₂×C₃×C₇₃	C₂×C₄×C₇	1	1	yes	characteristic
29	C₂×C₃×C₇×C₈	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂²×C₇₃	C₃×C₄×C₇	1	1	yes	characteristic
31	C₄×C₇₃	C₃×C₄×C₇	1	1	yes	characteristic
32	C₄×C₇₃	C₂²×C₃×C₇	1	1	yes	characteristic
33	C₃×C₇₃	C₂×C₇×C₈	1	1	yes	characteristic, direct product factor, Hall
34	C₂×C₃×C₄×C₇	C₂×C₇₃	1	1	yes	characteristic
35	C₃×C₇×C₈	C₂×C₇₃	1	2	yes	direct product factor
36	C₂×C₇₃	C₃×C₇×C₈	1	2	yes	direct product factor
37	C₂×C₇₃	C₂×C₃×C₄×C₇	1	1	yes	characteristic
38	C₂×C₇×C₈	C₃×C₇₃	1	1	yes	characteristic, direct product factor, Hall
39	C₂²×C₃×C₇	C₄×C₇₃	1	1	yes	characteristic
40	C₃×C₄×C₇	C₄×C₇₃	1	1	yes	characteristic
41	C₃×C₄×C₇	C₂²×C₇₃	1	1	yes	characteristic
42	C₇₃	C₂×C₃×C₇×C₈	1	1	yes	characteristic, direct product factor, Sylow
43	C₂×C₄×C₇	C₂×C₃×C₇₃	1	1	yes	characteristic
44	C₇×C₈	C₂×C₃×C₇₃	1	2	yes	direct product factor
45	C₂×C₃×C₈	C₇×C₇₃	1	1	yes	characteristic, direct product factor, Hall
46	C₂×C₃×C₇	C₈×C₇₃	1	2	yes	direct product factor
47	C₂×C₃×C₇	C₂×C₄×C₇₃	1	1	yes	characteristic
48	C₂²×C₇	C₃×C₄×C₇₃	1	1	yes	characteristic
49	C₄×C₇	C₃×C₄×C₇₃	1	1	yes	characteristic
50	C₄×C₇	C₂²×C₃×C₇₃	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂×C₇×C₇₃	1	1	yes	characteristic
52	C₃×C₈	C₂×C₇×C₇₃	1	2	yes	direct product factor
53	C₃×C₇	C₂×C₈×C₇₃	1	1	yes	characteristic, direct product factor, Hall
54	C₂×C₈	C₃×C₇×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
55	C₂×C₇	C₃×C₈×C₇₃	1	2	yes	direct product factor
56	C₂×C₇	C₂×C₃×C₄×C₇₃	1	1	yes	characteristic
57	C₂²×C₃	C₄×C₇×C₇₃	1	1	yes	characteristic
58	C₃×C₄	C₄×C₇×C₇₃	1	1	yes	characteristic
59	C₃×C₄	C₂²×C₇×C₇₃	1	1	yes	characteristic
60	C₂×C₄	C₂×C₃×C₇×C₇₃	1	1	yes	characteristic
61	C₈	C₂×C₃×C₇×C₇₃	1	2	yes	direct product factor
62	C₇	C₂×C₃×C₈×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₃	C₇×C₈×C₇₃	1	2	yes	direct product factor
64	C₂×C₃	C₂×C₄×C₇×C₇₃	1	1	yes	characteristic
65	C₂²	C₃×C₄×C₇×C₇₃	1	1	yes	characteristic
66	C₄	C₃×C₄×C₇×C₇₃	1	1	yes	characteristic
67	C₄	C₂²×C₃×C₇×C₇₃	1	1	yes	characteristic
68	C₃	C₂×C₇×C₈×C₇₃	1	1	yes	characteristic, direct product factor, Sylow
69	C₂	C₃×C₇×C₈×C₇₃	1	2	yes	direct product factor
70	C₂	C₂×C₃×C₄×C₇×C₇₃	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄	C₂²×C₃×C₇×C₇₃
socle, prime radical	C₂²×C₃×C₇×C₇₃	C₄
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×C₈×C₇₃	C₂	direct (N × H)
3. C₂×C₇×C₈×C₇₃	C₃	direct (N × H)
5. C₇×C₈×C₇₃	C₂×C₃	direct (N × H)
6. C₂×C₃×C₈×C₇₃	C₇	direct (N × H)
10. C₂×C₃×C₇×C₇₃	C₈	direct (N × H)
16. C₃×C₈×C₇₃	C₂×C₇	direct (N × H)
17. C₃×C₇×C₇₃	C₂×C₈	direct (N × H)
18. C₂×C₈×C₇₃	C₃×C₇	direct (N × H)
19. C₂×C₇×C₇₃	C₃×C₈	direct (N × H)
25. C₈×C₇₃	C₂×C₃×C₇	direct (N × H)
26. C₇×C₇₃	C₂×C₃×C₈	direct (N × H)
27. C₂×C₃×C₇₃	C₇×C₈	direct (N × H)
29. C₂×C₃×C₇×C₈	C₇₃	direct (N × H)
33. C₃×C₇₃	C₂×C₇×C₈	direct (N × H)
35. C₃×C₇×C₈	C₂×C₇₃	direct (N × H)
36. C₂×C₇₃	C₃×C₇×C₈	direct (N × H)
38. C₂×C₇×C₈	C₃×C₇₃	direct (N × H)
42. C₇₃	C₂×C₃×C₇×C₈	direct (N × H)
44. C₇×C₈	C₂×C₃×C₇₃	direct (N × H)
45. C₂×C₃×C₈	C₇×C₇₃	direct (N × H)
46. C₂×C₃×C₇	C₈×C₇₃	direct (N × H)
52. C₃×C₈	C₂×C₇×C₇₃	direct (N × H)
53. C₃×C₇	C₂×C₈×C₇₃	direct (N × H)
54. C₂×C₈	C₃×C₇×C₇₃	direct (N × H)
55. C₂×C₇	C₃×C₈×C₇₃	direct (N × H)
61. C₈	C₂×C₃×C₇×C₇₃	direct (N × H)
62. C₇	C₂×C₃×C₈×C₇₃	direct (N × H)
63. C₂×C₃	C₇×C₈×C₇₃	direct (N × H)
68. C₃	C₂×C₇×C₈×C₇₃	direct (N × H)
69. C₂	C₃×C₇×C₈×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×C₃×C₄×C₇×C₇₃	2
C₃×C₇×C₈×C₇₃	2
C₂×C₇×C₈×C₇₃	3
C₂×C₃×C₈×C₇₃	7
C₂×C₃×C₇×C₈	73

Subgroups — C2×C3×C7×C8×C73 #72201781

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₇×C₈×C₇₃ #72201781