Subgroups — C₂×C₃×C₁₃×D₆₇⋊C₃ #44535749

Order 31356.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₁₃×C₆₇⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₁₃×D₆₇⋊C₃	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₁₃×D₆₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₃×D₆₇⋊C₃	C₃	1	3	yes	maximal, direct product factor
5	C₃×C₁₃×C₆₇⋊C₃	C₂²	1	1	yes	characteristic, complemented, Hall
6	C₁₃×D₆₇⋊C₃	C₂×C₃	1	6	yes	direct product factor
7	C₃×C₁₃×D₆₇	C₂×C₃	1	2	yes	complemented
8	C₂×C₁₃×C₆₇⋊C₃	C₂×C₃	1	3	yes	complemented
9	C₂×C₃×C₁₃×C₆₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₁₃×D₆₇	C₃²	1	1	yes	characteristic, complemented, Hall
11	C₃×C₁₃×C₆₇	C₂²×C₃	1	1	yes	characteristic, complemented
12	C₁₃×C₆₇⋊C₃	C₂²×C₃	1	3	yes	complemented
13	C₂×C₃×D₆₇⋊C₃	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
14	C₂×C₁₃×C₆₇	C₂×C₃²	1	1	yes	characteristic, complemented
15	C₁₃×D₆₇	C₂×C₃²	1	2	yes	complemented
16	C₂×C₃×C₆₇⋊C₃	C₂×C₁₃	1	1	yes	characteristic, complemented
17	C₃×D₆₇⋊C₃	C₂×C₁₃	1	2	yes	direct product factor
18	C₁₃×C₆₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
19	C₂×C₃×D₆₇	C₃×C₁₃	1	1	yes	characteristic, complemented
20	C₂×D₆₇⋊C₃	C₃×C₁₃	1	3	yes	direct product factor
21	C₃×C₆₇⋊C₃	C₂²×C₁₃	1	1	yes	characteristic, complemented, Hall
22	C₂²×C₃²×C₁₃	—	67	1	no	maximal, Hall
23	C₂×C₃×C₆₇	C₂×C₃×C₁₃	1	1	yes	characteristic, complemented
24	C₃×D₆₇	C₂×C₃×C₁₃	1	2	yes	complemented
25	C₂×C₆₇⋊C₃	C₂×C₃×C₁₃	1	3	yes	complemented
26	D₆₇⋊C₃	C₂×C₃×C₁₃	1	6	yes	direct product factor
27	C₂×D₆₇	C₃²×C₁₃	1	1	yes	characteristic, complemented, Hall
28	C₂×C₃²×C₁₃	—	67	2	no	—
29	C₂×C₃²×C₁₃	—	67	1	no	—
30	C₃×C₆₇	C₂²×C₃×C₁₃	1	1	yes	characteristic, complemented
31	C₆₇⋊C₃	C₂²×C₃×C₁₃	1	3	yes	complemented
32	C₂²×C₃×C₁₃	—	67	3	no	—
33	C₂²×C₃×C₁₃	—	67	1	no	—
34	C₂×C₆₇	C₂×C₃²×C₁₃	1	1	yes	characteristic, complemented
35	D₆₇	C₂×C₃²×C₁₃	1	2	yes	complemented
36	C₃²×C₁₃	—	67	1	no	Hall
37	C₂×C₃×C₁₃	—	67	6	no	—
38	C₂×C₃×C₁₃	—	67	2	no	—
39	C₂×C₃×C₁₃	—	67	3	no	—
40	C₂×C₃×C₁₃	D₆₇⋊C₃	1	1	yes	characteristic, direct product factor
41	C₆₇	C₂²×C₃²×C₁₃	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₁₃	—	67	1	no	Hall
43	C₃×C₁₃	—	67	3	no	—
44	C₃×C₁₃	C₂×D₆₇⋊C₃	1	1	yes	characteristic, direct product factor
45	C₂²×C₃²	—	67	1	no	Hall
46	C₂×C₁₃	—	67	2	no	—
47	C₂×C₁₃	C₃×D₆₇⋊C₃	1	1	yes	characteristic, direct product factor
48	C₂×C₃²	—	67	2	no	—
49	C₂×C₃²	—	67	1	no	—
50	C₁₃	C₂×C₃×D₆₇⋊C₃	1	1	yes	characteristic, direct product factor, Sylow
51	C₂²×C₃	—	67	3	no	—
52	C₂²×C₃	—	67	1	no	—
53	C₃²	—	67	1	no	Sylow
54	C₂×C₃	—	67	6	no	—
55	C₂×C₃	—	67	2	no	—
56	C₂×C₃	—	67	3	no	—
57	C₂×C₃	C₁₃×D₆₇⋊C₃	1	1	yes	characteristic, direct product factor
58	C₂²	—	67	1	no	Sylow
59	C₃	—	67	3	no	—
60	C₃	C₂×C₁₃×D₆₇⋊C₃	1	1	yes	characteristic, direct product factor
61	C₂	—	67	2	no	—
62	C₂	C₃×C₁₃×D₆₇⋊C₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃×C₁₃	D₆₇⋊C₃
commutator, cosocle, hypocenter	C₆₇	C₂²×C₃²×C₁₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₃×C₆₇	C₂×C₃
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₂×C₃×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₁₃×C₆₇⋊C₃	C₂	semidirect
2. C₃×C₁₃×D₆₇⋊C₃	C₂	direct (N × H)
3. C₂×C₃×C₁₃×D₆₇	C₃	semidirect
4. C₂×C₁₃×D₆₇⋊C₃	C₃	direct (N × H)
5. C₃×C₁₃×C₆₇⋊C₃	C₂²	semidirect
6. C₁₃×D₆₇⋊C₃	C₂×C₃	direct (N × H)
7. C₃×C₁₃×D₆₇	C₂×C₃	semidirect
8. C₂×C₁₃×C₆₇⋊C₃	C₂×C₃	semidirect
9. C₂×C₃×C₁₃×C₆₇	C₂×C₃	semidirect
10. C₂×C₁₃×D₆₇	C₃²	semidirect
11. C₃×C₁₃×C₆₇	C₂²×C₃	semidirect
12. C₁₃×C₆₇⋊C₃	C₂²×C₃	semidirect
13. C₂×C₃×D₆₇⋊C₃	C₁₃	direct (N × H)
14. C₂×C₁₃×C₆₇	C₂×C₃²	semidirect
15. C₁₃×D₆₇	C₂×C₃²	semidirect
16. C₂×C₃×C₆₇⋊C₃	C₂×C₁₃	semidirect
17. C₃×D₆₇⋊C₃	C₂×C₁₃	direct (N × H)
18. C₁₃×C₆₇	C₂²×C₃²	semidirect
19. C₂×C₃×D₆₇	C₃×C₁₃	semidirect
20. C₂×D₆₇⋊C₃	C₃×C₁₃	direct (N × H)
21. C₃×C₆₇⋊C₃	C₂²×C₁₃	semidirect
23. C₂×C₃×C₆₇	C₂×C₃×C₁₃	semidirect
24. C₃×D₆₇	C₂×C₃×C₁₃	semidirect
25. C₂×C₆₇⋊C₃	C₂×C₃×C₁₃	semidirect
26. D₆₇⋊C₃	C₂×C₃×C₁₃	direct (N × H)
27. C₂×D₆₇	C₃²×C₁₃	semidirect
30. C₃×C₆₇	C₂²×C₃×C₁₃	semidirect
31. C₆₇⋊C₃	C₂²×C₃×C₁₃	semidirect
34. C₂×C₆₇	C₂×C₃²×C₁₃	semidirect
35. D₆₇	C₂×C₃²×C₁₃	semidirect
40. C₂×C₃×C₁₃	D₆₇⋊C₃	direct (N × H)
41. C₆₇	C₂²×C₃²×C₁₃	semidirect
44. C₃×C₁₃	C₂×D₆₇⋊C₃	direct (N × H)
47. C₂×C₁₃	C₃×D₆₇⋊C₃	direct (N × H)
50. C₁₃	C₂×C₃×D₆₇⋊C₃	direct (N × H)
57. C₂×C₃	C₁₃×D₆₇⋊C₃	direct (N × H)
60. C₃	C₂×C₁₃×D₆₇⋊C₃	direct (N × H)
62. C₂	C₃×C₁₃×D₆₇⋊C₃	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×C₁₃×C₆₇⋊C₃	2
C₃×C₁₃×D₆₇⋊C₃	2
C₂×C₃×C₁₃×D₆₇	3
C₂×C₁₃×D₆₇⋊C₃	3
C₂×C₃×D₆₇⋊C₃	13
C₂²×C₃²×C₁₃	67

Subgroups — C2×C3×C13×D67⋊C3 #44535749

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×C₁₃×D₆₇⋊C₃ #44535749