Subgroups — C₂×C₃×D₃×C₁₃×C₆₇ #44535824

Order 31356.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×C₁₃×C₆₇	C₂	1	2	yes	maximal, direct product factor
2	C₂×C₃²×C₁₃×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₃×C₁₃×C₆₇	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₂²×C₃×C₁₃×C₆₇	—	3	1	no	maximal
5	C₃²×C₁₃×C₆₇	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₃×C₁₃×C₆₇	C₂×C₃	1	2	yes	direct product factor
7	C₂×C₃×C₁₃×C₆₇	—	2	1	no	—
8	C₂×C₃×C₁₃×C₆₇	—	3	2	no	—
9	C₂×C₃×C₁₃×C₆₇	D₃	1	1	yes	characteristic, direct product factor
10	C₂×C₃×C₁₃×C₆₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂²×C₁₃×C₆₇	—	3	1	no	Hall
12	C₃×C₁₃×C₆₇	—	2	1	no	—
13	C₃×C₁₃×C₆₇	C₂×D₃	1	1	yes	characteristic, direct product factor
14	C₃×C₁₃×C₆₇	C₂²×C₃	1	1	yes	characteristic, complemented
15	C₂×C₃×D₃×C₆₇	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₂×C₁₃×C₆₇	—	3	2	no	—
17	C₂×C₁₃×C₆₇	C₃×D₃	1	1	yes	characteristic, direct product factor
18	C₂×C₃²×C₆₇	C₂×C₁₃	1	1	yes	characteristic, complemented
19	C₃×D₃×C₆₇	C₂×C₁₃	1	2	yes	direct product factor
20	C₁₃×C₆₇	C₂×C₃×D₃	1	1	yes	characteristic, direct product factor, Hall
21	C₂²×C₃×C₆₇	—	3	1	no	—
22	C₂×D₃×C₆₇	C₃×C₁₃	1	1	yes	characteristic, direct product factor
23	C₃²×C₆₇	C₂²×C₁₃	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×D₃×C₁₃	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
25	C₂×C₃×C₆₇	—	2	1	no	—
26	C₂×C₃×C₆₇	—	3	2	no	—
27	C₂×C₃×C₆₇	D₃×C₁₃	1	1	yes	characteristic, direct product factor
28	C₂×C₃×C₆₇	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented
29	D₃×C₆₇	C₂×C₃×C₁₃	1	2	yes	direct product factor
30	C₂²×C₆₇	—	3	1	no	Hall
31	C₂×C₃²×C₁₃	C₂×C₆₇	1	1	yes	characteristic, complemented
32	C₃×D₃×C₁₃	C₂×C₆₇	1	2	yes	direct product factor
33	C₃×C₆₇	—	2	1	no	—
34	C₃×C₆₇	C₂×D₃×C₁₃	1	1	yes	characteristic, direct product factor
35	C₃×C₆₇	C₂²×C₃×C₁₃	1	1	yes	characteristic, complemented
36	C₂²×C₃×C₁₃	—	3	1	no	—
37	C₂×D₃×C₁₃	C₃×C₆₇	1	1	yes	characteristic, direct product factor
38	C₂×C₆₇	—	3	2	no	—
39	C₂×C₆₇	C₃×D₃×C₁₃	1	1	yes	characteristic, direct product factor
40	C₃²×C₁₃	C₂²×C₆₇	1	1	yes	characteristic, complemented, Hall
41	C₂×C₃×C₁₃	—	2	1	no	—
42	C₂×C₃×C₁₃	—	3	2	no	—
43	C₂×C₃×C₁₃	D₃×C₆₇	1	1	yes	characteristic, direct product factor
44	C₂×C₃×C₁₃	C₂×C₃×C₆₇	1	1	yes	characteristic, complemented
45	D₃×C₁₃	C₂×C₃×C₆₇	1	2	yes	direct product factor
46	C₆₇	C₂×C₃×D₃×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
47	C₂²×C₁₃	—	3	1	no	Hall
48	C₃×C₁₃	—	2	1	no	—
49	C₃×C₁₃	C₂×D₃×C₆₇	1	1	yes	characteristic, direct product factor
50	C₃×C₁₃	C₂²×C₃×C₆₇	1	1	yes	characteristic, complemented
51	C₂×C₃×D₃	C₁₃×C₆₇	1	1	yes	characteristic, direct product factor, Hall
52	C₂×C₁₃	—	3	2	no	—
53	C₂×C₁₃	C₃×D₃×C₆₇	1	1	yes	characteristic, direct product factor
54	C₂×C₃²	C₂×C₁₃×C₆₇	1	1	yes	characteristic, complemented
55	C₃×D₃	C₂×C₁₃×C₆₇	1	2	yes	direct product factor
56	C₁₃	C₂×C₃×D₃×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²×C₃	—	3	1	no	—
58	C₂×D₃	C₃×C₁₃×C₆₇	1	1	yes	characteristic, direct product factor
59	C₃²	C₂²×C₁₃×C₆₇	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	—	3	2	no	—
62	C₂×C₃	C₂×C₃×C₁₃×C₆₇	1	1	yes	characteristic, complemented
63	C₂×C₃	D₃×C₁₃×C₆₇	1	1	yes	characteristic, direct product factor
64	D₃	C₂×C₃×C₁₃×C₆₇	1	2	yes	direct product factor
65	C₂²	—	3	1	no	Sylow
66	C₃	—	2	1	no	—
67	C₃	C₂²×C₃×C₁₃×C₆₇	1	1	yes	characteristic, complemented
68	C₃	C₂×D₃×C₁₃×C₆₇	1	1	yes	characteristic, direct product factor
69	C₂	—	3	2	no	—
70	C₂	C₃×D₃×C₁₃×C₆₇	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₃×C₆₇	D₃
commutator, cosocle, hypocenter	C₃	C₂²×C₃×C₁₃×C₆₇
Frattini	C₁	G
Fitting, socle	C₂×C₃²×C₁₃×C₆₇	C₂
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₁₃×C₆₇

Complements

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

N	H ≅ G/N	type
1. C₃×D₃×C₁₃×C₆₇	C₂	direct (N × H)
2. C₂×C₃²×C₁₃×C₆₇	C₂	semidirect
3. C₂×D₃×C₁₃×C₆₇	C₃	direct (N × H)
5. C₃²×C₁₃×C₆₇	C₂²	semidirect
6. D₃×C₁₃×C₆₇	C₂×C₃	direct (N × H)
9. C₂×C₃×C₁₃×C₆₇	D₃	direct (N × H)
10. C₂×C₃×C₁₃×C₆₇	C₂×C₃	semidirect
13. C₃×C₁₃×C₆₇	C₂×D₃	direct (N × H)
14. C₃×C₁₃×C₆₇	C₂²×C₃	semidirect
15. C₂×C₃×D₃×C₆₇	C₁₃	direct (N × H)
17. C₂×C₁₃×C₆₇	C₃×D₃	direct (N × H)
18. C₂×C₃²×C₆₇	C₂×C₁₃	semidirect
19. C₃×D₃×C₆₇	C₂×C₁₃	direct (N × H)
20. C₁₃×C₆₇	C₂×C₃×D₃	direct (N × H)
22. C₂×D₃×C₆₇	C₃×C₁₃	direct (N × H)
23. C₃²×C₆₇	C₂²×C₁₃	semidirect
24. C₂×C₃×D₃×C₁₃	C₆₇	direct (N × H)
27. C₂×C₃×C₆₇	D₃×C₁₃	direct (N × H)
28. C₂×C₃×C₆₇	C₂×C₃×C₁₃	semidirect
29. D₃×C₆₇	C₂×C₃×C₁₃	direct (N × H)
31. C₂×C₃²×C₁₃	C₂×C₆₇	semidirect
32. C₃×D₃×C₁₃	C₂×C₆₇	direct (N × H)
34. C₃×C₆₇	C₂×D₃×C₁₃	direct (N × H)
35. C₃×C₆₇	C₂²×C₃×C₁₃	semidirect
37. C₂×D₃×C₁₃	C₃×C₆₇	direct (N × H)
39. C₂×C₆₇	C₃×D₃×C₁₃	direct (N × H)
40. C₃²×C₁₃	C₂²×C₆₇	semidirect
43. C₂×C₃×C₁₃	D₃×C₆₇	direct (N × H)
44. C₂×C₃×C₁₃	C₂×C₃×C₆₇	semidirect
45. D₃×C₁₃	C₂×C₃×C₆₇	direct (N × H)
46. C₆₇	C₂×C₃×D₃×C₁₃	direct (N × H)
49. C₃×C₁₃	C₂×D₃×C₆₇	direct (N × H)
50. C₃×C₁₃	C₂²×C₃×C₆₇	semidirect
51. C₂×C₃×D₃	C₁₃×C₆₇	direct (N × H)
53. C₂×C₁₃	C₃×D₃×C₆₇	direct (N × H)
54. C₂×C₃²	C₂×C₁₃×C₆₇	semidirect
55. C₃×D₃	C₂×C₁₃×C₆₇	direct (N × H)
56. C₁₃	C₂×C₃×D₃×C₆₇	direct (N × H)
58. C₂×D₃	C₃×C₁₃×C₆₇	direct (N × H)
59. C₃²	C₂²×C₁₃×C₆₇	semidirect
62. C₂×C₃	C₂×C₃×C₁₃×C₆₇	semidirect
63. C₂×C₃	D₃×C₁₃×C₆₇	direct (N × H)
64. D₃	C₂×C₃×C₁₃×C₆₇	direct (N × H)
67. C₃	C₂²×C₃×C₁₃×C₆₇	semidirect
68. C₃	C₂×D₃×C₁₃×C₆₇	direct (N × H)
70. C₂	C₃×D₃×C₁₃×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
C₃×D₃×C₁₃×C₆₇	2
C₂×C₃²×C₁₃×C₆₇	2
C₂×D₃×C₁₃×C₆₇	3
C₂²×C₃×C₁₃×C₆₇	3
C₂×C₃×D₃×C₆₇	13
C₂×C₃×D₃×C₁₃	67

Subgroups — C2×C3×D3×C13×C67 #44535824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×D₃×C₁₃×C₆₇ #44535824