Subgroups — C₂×C₃²×C₁₃×D₆₇ #44535821

Order 31356.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₃²×C₁₃×D₆₇	C₂	direct (N × H)
2. C₂×C₃²×C₁₃×C₆₇	C₂	semidirect
3. C₂×C₃×C₁₃×D₆₇	C₃	direct (N × H)
4. C₃²×C₁₃×C₆₇	C₂²	semidirect
5. C₃×C₁₃×D₆₇	C₂×C₃	direct (N × H)
6. C₂×C₃×C₁₃×C₆₇	C₂×C₃	semidirect
7. C₂×C₁₃×D₆₇	C₃²	direct (N × H)
8. C₃×C₁₃×C₆₇	C₂²×C₃	semidirect
9. C₂×C₃²×D₆₇	C₁₃	direct (N × H)
10. C₂×C₁₃×C₆₇	C₂×C₃²	semidirect
11. C₁₃×D₆₇	C₂×C₃²	direct (N × H)
12. C₂×C₃²×C₆₇	C₂×C₁₃	semidirect
13. C₃²×D₆₇	C₂×C₁₃	direct (N × H)
14. C₁₃×C₆₇	C₂²×C₃²	semidirect
15. C₂×C₃×D₆₇	C₃×C₁₃	direct (N × H)
16. C₃²×C₆₇	C₂²×C₁₃	semidirect
18. C₂×C₃×C₆₇	C₂×C₃×C₁₃	semidirect
19. C₃×D₆₇	C₂×C₃×C₁₃	direct (N × H)
20. C₂×D₆₇	C₃²×C₁₃	direct (N × H)
22. C₂×C₃²×C₁₃	D₆₇	direct (N × H)
23. C₃×C₆₇	C₂²×C₃×C₁₃	semidirect
25. C₂×C₆₇	C₂×C₃²×C₁₃	semidirect
26. D₆₇	C₂×C₃²×C₁₃	direct (N × H)
27. C₃²×C₁₃	C₂×D₆₇	direct (N × H)
29. C₂×C₃×C₁₃	C₃×D₆₇	direct (N × H)
30. C₆₇	C₂²×C₃²×C₁₃	semidirect
32. C₃×C₁₃	C₂×C₃×D₆₇	direct (N × H)
35. C₂×C₁₃	C₃²×D₆₇	direct (N × H)
37. C₂×C₃²	C₁₃×D₆₇	direct (N × H)
38. C₁₃	C₂×C₃²×D₆₇	direct (N × H)
40. C₃²	C₂×C₁₃×D₆₇	direct (N × H)
42. C₂×C₃	C₃×C₁₃×D₆₇	direct (N × H)
44. C₃	C₂×C₃×C₁₃×D₆₇	direct (N × H)
46. C₂	C₃²×C₁₃×D₆₇	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃²×C₁₃×D₆₇	2
C₂×C₃²×C₁₃×C₆₇	2
C₂×C₃×C₁₃×D₆₇	3
C₂×C₃²×D₆₇	13
C₂²×C₃²×C₁₃	67

Subgroups — C2×C32×C13×D67 #44535821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₁₃×D₆₇ #44535821