Subgroups — C₂²×C₃×#14579732 #44535820

Order 31356.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#14579732	C₂	1	3	yes	maximal, direct product factor
2	C₂²×#14579732	C₃	1	3	yes	maximal, direct product factor
3	C₂²×C₃×C₁₃×C₆₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#14579732	C₂²	1	1	yes	characteristic, direct product factor, Hall
5	C₂×#14579732	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	#14579732	C₂²×C₃	1	3	yes	direct product factor
10	C₂²×C₃×C₆₇⋊C₃	—	13	1	no	maximal, Hall
11	C₂×C₁₃×C₆₇	C₂×C₃²	1	3	yes	complemented
12	C₂×C₃×C₆₇⋊C₃	—	13	3	no	—
13	C₁₃×C₆₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
14	C₂²×C₃×C₆₇	C₁₃⋊C₃	1	1	yes	characteristic, complemented
15	C₂²×C₆₇⋊C₃	—	13	3	no	—
16	C₃×C₆₇⋊C₃	—	13	1	no	Hall
17	C₂²×C₃×C₁₃⋊C₃	—	67	1	no	maximal, Hall
18	C₂×C₃×C₆₇	C₂×C₁₃⋊C₃	1	3	yes	complemented
19	C₂×C₆₇⋊C₃	—	13	9	no	—
20	C₂²×C₆₇	C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
21	C₂×C₃×C₁₃⋊C₃	—	67	3	no	—
22	C₃×C₆₇	C₂²×C₁₃⋊C₃	1	1	yes	characteristic, complemented
23	C₆₇⋊C₃	—	13	3	no	—
24	C₂²×C₃×C₁₃	C₆₇⋊C₃	1	1	yes	characteristic, complemented
25	C₂²×C₁₃⋊C₃	—	67	3	no	—
26	C₂×C₆₇	C₂×C₃×C₁₃⋊C₃	1	3	yes	complemented
27	C₃×C₁₃⋊C₃	—	67	1	no	Hall
28	C₂×C₃×C₁₃	C₂×C₆₇⋊C₃	1	3	yes	complemented
29	C₂×C₁₃⋊C₃	—	67	9	no	—
30	C₆₇	C₂²×C₃×C₁₃⋊C₃	1	1	yes	characteristic, complemented, Sylow
31	C₂²×C₁₃	C₃×C₆₇⋊C₃	1	1	yes	characteristic, complemented, Hall
32	C₃×C₁₃	C₂²×C₆₇⋊C₃	1	1	yes	characteristic, complemented
33	C₁₃⋊C₃	—	67	3	no	—
34	C₂²×C₃²	—	871	1	no	Hall
35	C₂×C₁₃	C₂×C₃×C₆₇⋊C₃	1	3	yes	complemented
36	C₂×C₃²	—	871	3	no	—
37	C₁₃	C₂²×C₃×C₆₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
38	C₂²×C₃	—	871	3	no	—
39	C₂²×C₃	#14579732	1	1	yes	characteristic, direct product factor
40	C₃²	—	871	1	no	Sylow
41	C₂×C₃	—	871	9	no	—
42	C₂×C₃	C₂×#14579732	1	3	yes	direct product factor
43	C₂²	C₃×#14579732	1	1	yes	characteristic, direct product factor, Sylow
44	C₃	—	871	3	no	—
45	C₃	C₂²×#14579732	1	1	yes	characteristic, direct product factor
46	C₂	C₂×C₃×#14579732	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₃	#14579732
commutator, cosocle, hypocenter	C₁₃×C₆₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₁₃×C₆₇	C₃
LCS	G ▷ C₁₃×C₆₇
UCS	C₁ ◁ C₂²×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#14579732	C₂	direct (N × H)
2. C₂²×#14579732	C₃	direct (N × H)
3. C₂²×C₃×C₁₃×C₆₇	C₃	semidirect
4. C₃×#14579732	C₂²	direct (N × H)
5. C₂×#14579732	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. #14579732	C₂²×C₃	direct (N × H)
11. C₂×C₁₃×C₆₇	C₂×C₃²	semidirect
13. C₁₃×C₆₇	C₂²×C₃²	semidirect
14. C₂²×C₃×C₆₇	C₁₃⋊C₃	semidirect
18. C₂×C₃×C₆₇	C₂×C₁₃⋊C₃	semidirect
20. C₂²×C₆₇	C₃×C₁₃⋊C₃	semidirect
22. C₃×C₆₇	C₂²×C₁₃⋊C₃	semidirect
24. C₂²×C₃×C₁₃	C₆₇⋊C₃	semidirect
26. C₂×C₆₇	C₂×C₃×C₁₃⋊C₃	semidirect
28. C₂×C₃×C₁₃	C₂×C₆₇⋊C₃	semidirect
30. C₆₇	C₂²×C₃×C₁₃⋊C₃	semidirect
31. C₂²×C₁₃	C₃×C₆₇⋊C₃	semidirect
32. C₃×C₁₃	C₂²×C₆₇⋊C₃	semidirect
35. C₂×C₁₃	C₂×C₃×C₆₇⋊C₃	semidirect
37. C₁₃	C₂²×C₃×C₆₇⋊C₃	semidirect
39. C₂²×C₃	#14579732	direct (N × H)
42. C₂×C₃	C₂×#14579732	direct (N × H)
43. C₂²	C₃×#14579732	direct (N × H)
45. C₃	C₂²×#14579732	direct (N × H)
46. C₂	C₂×C₃×#14579732	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×#14579732	2
C₂²×#14579732	3
C₂²×C₃×C₁₃×C₆₇	3
C₂²×C₃×C₆₇⋊C₃	13
C₂²×C₃×C₁₃⋊C₃	67

Subgroups — C22×C3×#14579732 #44535820

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃×#14579732 #44535820