Subgroups — D₃²×C₁₃×C₆₇ #44535794

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, complemented
2	C₁₃×C₃⋊D₃×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₃×C₁₃×C₆₇	—	3	2	no	maximal
4	C₃²×C₁₃×C₆₇	C₂²	1	1	yes	characteristic, complemented, Hall
5	D₃×C₁₃×C₆₇	—	3	2	no	—
6	D₃×C₁₃×C₆₇	—	6	1	no	—
7	D₃×C₁₃×C₆₇	D₃	1	2	yes	direct product factor
8	C₂×C₃×C₁₃×C₆₇	—	3	2	no	—
9	C₂²×C₁₃×C₆₇	—	9	1	no	Hall
10	C₃×C₁₃×C₆₇	—	2	1	no	—
11	C₃×C₁₃×C₆₇	C₂×D₃	1	2	yes	complemented
12	D₃²×C₆₇	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₁₃×C₆₇	—	3	2	no	—
14	C₂×C₁₃×C₆₇	—	9	1	no	—
15	C₃⋊D₃×C₆₇	C₂×C₁₃	1	1	yes	characteristic, complemented
16	C₃×D₃×C₆₇	C₂×C₁₃	1	2	yes	complemented
17	C₁₃×C₆₇	D₃²	1	1	yes	characteristic, direct product factor, Hall
18	C₂×D₃×C₆₇	—	3	2	no	—
19	C₃²×C₆₇	C₂²×C₁₃	1	1	yes	characteristic, complemented, Hall
20	D₃²×C₁₃	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Hall
21	C₂×C₃×C₆₇	—	3	2	no	—
22	D₃×C₆₇	—	3	2	no	—
23	D₃×C₆₇	—	6	1	no	—
24	D₃×C₆₇	D₃×C₁₃	1	2	yes	direct product factor
25	C₂²×C₆₇	—	9	1	no	Hall
26	C₁₃×C₃⋊D₃	C₂×C₆₇	1	1	yes	characteristic, complemented
27	C₃×D₃×C₁₃	C₂×C₆₇	1	2	yes	complemented
28	C₃×C₆₇	—	2	1	no	—
29	C₃×C₆₇	C₂×D₃×C₁₃	1	2	yes	complemented
30	C₂×D₃×C₁₃	—	3	2	no	—
31	C₂×C₆₇	—	3	2	no	—
32	C₂×C₆₇	—	9	1	no	—
33	C₃²×C₁₃	C₂²×C₆₇	1	1	yes	characteristic, complemented, Hall
34	C₂×C₃×C₁₃	—	3	2	no	—
35	D₃×C₁₃	—	3	2	no	—
36	D₃×C₁₃	—	6	1	no	—
37	D₃×C₁₃	D₃×C₆₇	1	2	yes	direct product factor
38	C₆₇	D₃²×C₁₃	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₁₃	—	9	1	no	Hall
40	C₃×C₁₃	—	2	1	no	—
41	C₃×C₁₃	C₂×D₃×C₆₇	1	2	yes	complemented
42	D₃²	C₁₃×C₆₇	1	1	yes	characteristic, direct product factor, Hall
43	C₂×C₁₃	—	3	2	no	—
44	C₂×C₁₃	—	9	1	no	—
45	C₃⋊D₃	C₂×C₁₃×C₆₇	1	1	yes	characteristic, complemented
46	C₃×D₃	C₂×C₁₃×C₆₇	1	2	yes	complemented
47	C₁₃	D₃²×C₆₇	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×D₃	—	3	2	no	—
49	C₃²	C₂²×C₁₃×C₆₇	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃	—	3	2	no	—
51	D₃	—	3	2	no	—
52	D₃	—	6	1	no	—
53	D₃	D₃×C₁₃×C₆₇	1	2	yes	direct product factor
54	C₂²	—	9	1	no	Sylow
55	C₃	—	2	1	no	—
56	C₃	C₂×D₃×C₁₃×C₆₇	1	2	yes	complemented
57	C₂	—	3	2	no	—
58	C₂	—	9	1	no	—

	subgroup	G/H
center, hypercenter	C₁₃×C₆₇	D₃²
commutator, cosocle, hypocenter	C₃²	C₂²×C₁₃×C₆₇
Frattini	C₁	G
Fitting, socle	C₃²×C₁₃×C₆₇	C₂²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₁₃×C₆₇

Complements

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

N	H ≅ G/N	type
1. C₃×D₃×C₁₃×C₆₇	C₂	semidirect
2. C₁₃×C₃⋊D₃×C₆₇	C₂	semidirect
4. C₃²×C₁₃×C₆₇	C₂²	semidirect
7. D₃×C₁₃×C₆₇	D₃	direct (N × H)
11. C₃×C₁₃×C₆₇	C₂×D₃	semidirect
12. D₃²×C₆₇	C₁₃	direct (N × H)
15. C₃⋊D₃×C₆₇	C₂×C₁₃	semidirect
16. C₃×D₃×C₆₇	C₂×C₁₃	semidirect
17. C₁₃×C₆₇	D₃²	direct (N × H)
19. C₃²×C₆₇	C₂²×C₁₃	semidirect
20. D₃²×C₁₃	C₆₇	direct (N × H)
24. D₃×C₆₇	D₃×C₁₃	direct (N × H)
26. C₁₃×C₃⋊D₃	C₂×C₆₇	semidirect
27. C₃×D₃×C₁₃	C₂×C₆₇	semidirect
29. C₃×C₆₇	C₂×D₃×C₁₃	semidirect
33. C₃²×C₁₃	C₂²×C₆₇	semidirect
37. D₃×C₁₃	D₃×C₆₇	direct (N × H)
38. C₆₇	D₃²×C₁₃	direct (N × H)
41. C₃×C₁₃	C₂×D₃×C₆₇	semidirect
42. D₃²	C₁₃×C₆₇	direct (N × H)
45. C₃⋊D₃	C₂×C₁₃×C₆₇	semidirect
46. C₃×D₃	C₂×C₁₃×C₆₇	semidirect
47. C₁₃	D₃²×C₆₇	direct (N × H)
49. C₃²	C₂²×C₁₃×C₆₇	semidirect
53. D₃	D₃×C₁₃×C₆₇	direct (N × H)
56. C₃	C₂×D₃×C₁₃×C₆₇	semidirect

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
C₁₃×C₃⋊D₃×C₆₇	2
C₂×D₃×C₁₃×C₆₇	3
D₃²×C₆₇	13
D₃²×C₁₃	67

Subgroups — D32×C13×C67 #44535794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃²×C₁₃×C₆₇ #44535794