Subgroups — C₂×C₃²×C₄×C₅×C₆₇ #72201410

Order 24120.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₆₇	C₃	1	4	yes	maximal, direct product factor
2	C₂×C₃²×C₄×C₆₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
3	C₂×C₄×C₅×C₆₇	C₃²	1	1	yes	characteristic, direct product factor, Hall
4	C₂²×C₃²×C₅×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₃²×C₄×C₅×C₆₇	C₂	1	2	yes	maximal, direct product factor
6	C₂×C₃²×C₅×C₆₇	C₄	1	2	yes	direct product factor
7	C₂×C₃²×C₅×C₆₇	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×C₆₇	C₂×C₃	1	4	yes	—
9	C₃×C₄×C₅×C₆₇	C₂×C₃	1	8	yes	direct product factor
10	C₃²×C₅×C₆₇	C₂×C₄	1	1	yes	characteristic, direct product factor, Hall
11	C₂²×C₃²×C₆₇	C₂×C₅	1	1	yes	characteristic
12	C₃²×C₄×C₆₇	C₂×C₅	1	2	yes	direct product factor
13	C₂×C₃×C₅×C₆₇	C₃×C₄	1	8	yes	direct product factor
14	C₂×C₃×C₅×C₆₇	C₂²×C₃	1	4	yes	—
15	C₂×C₃×C₄×C₆₇	C₃×C₅	1	4	yes	direct product factor
16	C₂²×C₅×C₆₇	C₂×C₃²	1	1	yes	characteristic
17	C₄×C₅×C₆₇	C₂×C₃²	1	2	yes	direct product factor
18	C₂×C₃²×C₆₇	C₄×C₅	1	2	yes	direct product factor
19	C₂×C₃²×C₆₇	C₂²×C₅	1	1	yes	characteristic
20	C₃×C₅×C₆₇	C₂×C₃×C₄	1	4	yes	direct product factor
21	C₂²×C₃×C₆₇	C₂×C₃×C₅	1	4	yes	—
22	C₃×C₄×C₆₇	C₂×C₃×C₅	1	8	yes	direct product factor
23	C₂×C₅×C₆₇	C₃²×C₄	1	2	yes	direct product factor
24	C₂×C₅×C₆₇	C₂²×C₃²	1	1	yes	characteristic
25	C₃²×C₆₇	C₂×C₄×C₅	1	1	yes	characteristic, direct product factor, Hall
26	C₂×C₄×C₆₇	C₃²×C₅	1	1	yes	characteristic, direct product factor, Hall
27	C₂×C₃×C₆₇	C₃×C₄×C₅	1	8	yes	direct product factor
28	C₂×C₃×C₆₇	C₂²×C₃×C₅	1	4	yes	—
29	C₂×C₃²×C₄×C₅	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₅×C₆₇	C₂×C₃²×C₄	1	1	yes	characteristic, direct product factor, Hall
31	C₂²×C₆₇	C₂×C₃²×C₅	1	1	yes	characteristic
32	C₄×C₆₇	C₂×C₃²×C₅	1	2	yes	direct product factor
33	C₃×C₆₇	C₂×C₃×C₄×C₅	1	4	yes	direct product factor
34	C₂²×C₃²×C₅	C₂×C₆₇	1	1	yes	characteristic
35	C₃²×C₄×C₅	C₂×C₆₇	1	2	yes	direct product factor
36	C₂×C₆₇	C₃²×C₄×C₅	1	2	yes	direct product factor
37	C₂×C₆₇	C₂²×C₃²×C₅	1	1	yes	characteristic
38	C₂×C₃×C₄×C₅	C₃×C₆₇	1	4	yes	direct product factor
39	C₂×C₃²×C₅	C₄×C₆₇	1	2	yes	direct product factor
40	C₂×C₃²×C₅	C₂²×C₆₇	1	1	yes	characteristic
41	C₂×C₃²×C₄	C₅×C₆₇	1	1	yes	characteristic, direct product factor, Hall
42	C₆₇	C₂×C₃²×C₄×C₅	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²×C₃×C₅	C₂×C₃×C₆₇	1	4	yes	—
44	C₃×C₄×C₅	C₂×C₃×C₆₇	1	8	yes	direct product factor
45	C₃²×C₅	C₂×C₄×C₆₇	1	1	yes	characteristic, direct product factor, Hall
46	C₂×C₄×C₅	C₃²×C₆₇	1	1	yes	characteristic, direct product factor, Hall
47	C₂²×C₃²	C₂×C₅×C₆₇	1	1	yes	characteristic
48	C₃²×C₄	C₂×C₅×C₆₇	1	2	yes	direct product factor
49	C₂×C₃×C₅	C₃×C₄×C₆₇	1	8	yes	direct product factor
50	C₂×C₃×C₅	C₂²×C₃×C₆₇	1	4	yes	—
51	C₂×C₃×C₄	C₃×C₅×C₆₇	1	4	yes	direct product factor
52	C₂²×C₅	C₂×C₃²×C₆₇	1	1	yes	characteristic
53	C₄×C₅	C₂×C₃²×C₆₇	1	2	yes	direct product factor
54	C₂×C₃²	C₄×C₅×C₆₇	1	2	yes	direct product factor
55	C₂×C₃²	C₂²×C₅×C₆₇	1	1	yes	characteristic
56	C₃×C₅	C₂×C₃×C₄×C₆₇	1	4	yes	direct product factor
57	C₂²×C₃	C₂×C₃×C₅×C₆₇	1	4	yes	—
58	C₃×C₄	C₂×C₃×C₅×C₆₇	1	8	yes	direct product factor
59	C₂×C₅	C₃²×C₄×C₆₇	1	2	yes	direct product factor
60	C₂×C₅	C₂²×C₃²×C₆₇	1	1	yes	characteristic
61	C₃²	C₂×C₄×C₅×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
62	C₂×C₄	C₃²×C₅×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₃	C₃×C₄×C₅×C₆₇	1	8	yes	direct product factor
64	C₂×C₃	C₂²×C₃×C₅×C₆₇	1	4	yes	—
65	C₅	C₂×C₃²×C₄×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
66	C₂²	C₂×C₃²×C₅×C₆₇	1	1	yes	characteristic
67	C₄	C₂×C₃²×C₅×C₆₇	1	2	yes	direct product factor
68	C₃	C₂×C₃×C₄×C₅×C₆₇	1	4	yes	direct product factor
69	C₂	C₃²×C₄×C₅×C₆₇	1	2	yes	direct product factor
70	C₂	C₂²×C₃²×C₅×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂²×C₃²×C₅×C₆₇
socle, prime radical	C₂²×C₃²×C₅×C₆₇	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₄×C₅×C₆₇	C₃	direct (N × H)
2. C₂×C₃²×C₄×C₆₇	C₅	direct (N × H)
3. C₂×C₄×C₅×C₆₇	C₃²	direct (N × H)
5. C₃²×C₄×C₅×C₆₇	C₂	direct (N × H)
6. C₂×C₃²×C₅×C₆₇	C₄	direct (N × H)
9. C₃×C₄×C₅×C₆₇	C₂×C₃	direct (N × H)
10. C₃²×C₅×C₆₇	C₂×C₄	direct (N × H)
12. C₃²×C₄×C₆₇	C₂×C₅	direct (N × H)
13. C₂×C₃×C₅×C₆₇	C₃×C₄	direct (N × H)
15. C₂×C₃×C₄×C₆₇	C₃×C₅	direct (N × H)
17. C₄×C₅×C₆₇	C₂×C₃²	direct (N × H)
18. C₂×C₃²×C₆₇	C₄×C₅	direct (N × H)
20. C₃×C₅×C₆₇	C₂×C₃×C₄	direct (N × H)
22. C₃×C₄×C₆₇	C₂×C₃×C₅	direct (N × H)
23. C₂×C₅×C₆₇	C₃²×C₄	direct (N × H)
25. C₃²×C₆₇	C₂×C₄×C₅	direct (N × H)
26. C₂×C₄×C₆₇	C₃²×C₅	direct (N × H)
27. C₂×C₃×C₆₇	C₃×C₄×C₅	direct (N × H)
29. C₂×C₃²×C₄×C₅	C₆₇	direct (N × H)
30. C₅×C₆₇	C₂×C₃²×C₄	direct (N × H)
32. C₄×C₆₇	C₂×C₃²×C₅	direct (N × H)
33. C₃×C₆₇	C₂×C₃×C₄×C₅	direct (N × H)
35. C₃²×C₄×C₅	C₂×C₆₇	direct (N × H)
36. C₂×C₆₇	C₃²×C₄×C₅	direct (N × H)
38. C₂×C₃×C₄×C₅	C₃×C₆₇	direct (N × H)
39. C₂×C₃²×C₅	C₄×C₆₇	direct (N × H)
41. C₂×C₃²×C₄	C₅×C₆₇	direct (N × H)
42. C₆₇	C₂×C₃²×C₄×C₅	direct (N × H)
44. C₃×C₄×C₅	C₂×C₃×C₆₇	direct (N × H)
45. C₃²×C₅	C₂×C₄×C₆₇	direct (N × H)
46. C₂×C₄×C₅	C₃²×C₆₇	direct (N × H)
48. C₃²×C₄	C₂×C₅×C₆₇	direct (N × H)
49. C₂×C₃×C₅	C₃×C₄×C₆₇	direct (N × H)
51. C₂×C₃×C₄	C₃×C₅×C₆₇	direct (N × H)
53. C₄×C₅	C₂×C₃²×C₆₇	direct (N × H)
54. C₂×C₃²	C₄×C₅×C₆₇	direct (N × H)
56. C₃×C₅	C₂×C₃×C₄×C₆₇	direct (N × H)
58. C₃×C₄	C₂×C₃×C₅×C₆₇	direct (N × H)
59. C₂×C₅	C₃²×C₄×C₆₇	direct (N × H)
61. C₃²	C₂×C₄×C₅×C₆₇	direct (N × H)
62. C₂×C₄	C₃²×C₅×C₆₇	direct (N × H)
63. C₂×C₃	C₃×C₄×C₅×C₆₇	direct (N × H)
65. C₅	C₂×C₃²×C₄×C₆₇	direct (N × H)
67. C₄	C₂×C₃²×C₅×C₆₇	direct (N × H)
68. C₃	C₂×C₃×C₄×C₅×C₆₇	direct (N × H)
69. C₂	C₃²×C₄×C₅×C₆₇	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×C₅×C₆₇	3
C₂×C₃²×C₄×C₆₇	5
C₂²×C₃²×C₅×C₆₇	2
C₃²×C₄×C₅×C₆₇	2
C₂×C₃²×C₄×C₅	67

Subgroups — C2×C32×C4×C5×C67 #72201410

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₄×C₅×C₆₇ #72201410