Subgroups — C₂²×C₃×C₁₃×C₆₇⋊C₃ #44535817

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

	subgroup	G/H
center, hypercenter	C₂²×C₃×C₁₃	C₆₇⋊C₃
commutator, cosocle, hypocenter	C₆₇	C₂²×C₃²×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₁₃×C₆₇	C₃
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₂²×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	direct (N × H)
2. C₂²×C₁₃×C₆₇⋊C₃	C₃	direct (N × H)
3. C₂²×C₃×C₁₃×C₆₇	C₃	semidirect
4. C₃×C₁₃×C₆₇⋊C₃	C₂²	direct (N × H)
5. C₂×C₁₃×C₆₇⋊C₃	C₂×C₃	direct (N × H)
6. C₂×C₃×C₁₃×C₆₇	C₂×C₃	semidirect
7. C₂²×C₁₃×C₆₇	C₃²	semidirect
8. C₃×C₁₃×C₆₇	C₂²×C₃	semidirect
9. C₁₃×C₆₇⋊C₃	C₂²×C₃	direct (N × H)
10. C₂²×C₃×C₆₇⋊C₃	C₁₃	direct (N × H)
11. C₂×C₁₃×C₆₇	C₂×C₃²	semidirect
12. C₂×C₃×C₆₇⋊C₃	C₂×C₁₃	direct (N × H)
13. C₁₃×C₆₇	C₂²×C₃²	semidirect
14. C₂²×C₃×C₆₇	C₃×C₁₃	semidirect
15. C₂²×C₆₇⋊C₃	C₃×C₁₃	direct (N × H)
16. C₃×C₆₇⋊C₃	C₂²×C₁₃	direct (N × H)
18. C₂×C₃×C₆₇	C₂×C₃×C₁₃	semidirect
19. C₂×C₆₇⋊C₃	C₂×C₃×C₁₃	direct (N × H)
20. C₂²×C₆₇	C₃²×C₁₃	semidirect
22. C₃×C₆₇	C₂²×C₃×C₁₃	semidirect
23. C₆₇⋊C₃	C₂²×C₃×C₁₃	direct (N × H)
25. C₂²×C₃×C₁₃	C₆₇⋊C₃	direct (N × H)
26. C₂×C₆₇	C₂×C₃²×C₁₃	semidirect
29. C₂×C₃×C₁₃	C₂×C₆₇⋊C₃	direct (N × H)
30. C₆₇	C₂²×C₃²×C₁₃	semidirect
31. C₂²×C₁₃	C₃×C₆₇⋊C₃	direct (N × H)
33. C₃×C₁₃	C₂²×C₆₇⋊C₃	direct (N × H)
35. C₂×C₁₃	C₂×C₃×C₆₇⋊C₃	direct (N × H)
37. C₁₃	C₂²×C₃×C₆₇⋊C₃	direct (N × H)
39. C₂²×C₃	C₁₃×C₆₇⋊C₃	direct (N × H)
42. C₂×C₃	C₂×C₁₃×C₆₇⋊C₃	direct (N × H)
43. C₂²	C₃×C₁₃×C₆₇⋊C₃	direct (N × H)
45. C₃	C₂²×C₁₃×C₆₇⋊C₃	direct (N × H)
46. C₂	C₂×C₃×C₁₃×C₆₇⋊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
C₂²×C₁₃×C₆₇⋊C₃	3
C₂²×C₃×C₁₃×C₆₇	3
C₂²×C₃×C₆₇⋊C₃	13
C₂²×C₃²×C₁₃	67

Subgroups — C22×C3×C13×C67⋊C3 #44535817

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×C₁₃×C₆₇⋊C₃ #44535817