Subgroups — C₂×C₃×#67 #3824

Order 144.

124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃×D₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#67	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×D₄	—	3	1	no	maximal
6	C₂×#67	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃²	—	2	2	no	—
8	C₂²×C₃²	C₂²	1	1	yes	characteristic
9	C₂²×C₃²	C₂²	1	2	yes	complemented
10	C₂×C₃×D₃	—	2	2	no	—
11	C₂×C₃×D₃	C₂²	1	2	yes	complemented
12	C₃×Dic₃	C₂²	1	2	yes	complemented
13	C₂³×C₃	—	2	1	no	—
14	C₂³×C₃	—	3	1	no	—
15	C₂³×C₃	D₃	1	1	yes	characteristic, complemented
16	C₂³×C₃	C₂×C₃	1	1	yes	characteristic, complemented
17	C₂²×D₃	C₂×C₃	1	1	yes	characteristic, complemented
18	C₃×D₄	—	3	4	no	—
19	C₂×C₃×C₄	—	3	1	no	—
20	#67	C₂×C₃	1	4	yes	direct product factor
21	C₂×Dic₃	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃²	—	2	2	no	—
23	C₂×C₃²	D₄	1	2	yes	complemented
24	C₂×C₃²	C₂³	1	1	yes	characteristic
25	C₃×D₃	—	2	2	no	—
26	C₂×D₄	—	3	1	no	Sylow
27	C₂²×C₃	—	2	4	no	—
28	C₂²×C₃	—	2	2	no	—
29	C₂²×C₃	—	2	2	no	—
30	C₂²×C₃	—	2	1	no	—
31	C₂²×C₃	—	2	2	no	—
32	C₂²×C₃	—	3	2	no	—
33	C₂²×C₃	—	6	2	no	—
34	C₂²×C₃	C₂×D₃	1	1	yes	characteristic
35	C₂²×C₃	C₂×D₃	1	2	yes	complemented
36	C₂²×C₃	C₂²×C₃	1	1	yes	characteristic
37	C₂²×C₃	C₂²×C₃	1	2	yes	complemented
38	C₂×D₃	—	2	2	no	—
39	C₂×D₃	C₂²×C₃	1	2	yes	complemented
40	C₃×C₄	—	3	2	no	—
41	Dic₃	C₂²×C₃	1	2	yes	complemented
42	C₃²	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
43	C₂³	—	3	1	no	—
44	C₂³	C₃×D₃	1	1	yes	characteristic, complemented
45	D₄	—	3	4	no	—
46	C₂×C₄	—	3	1	no	—
47	C₂×C₃	—	2	4	no	—
48	C₂×C₃	—	2	2	no	—
49	C₂×C₃	—	2	2	no	—
50	C₂×C₃	—	2	2	no	—
51	C₂×C₃	—	2	1	no	—
52	C₂×C₃	—	6	2	no	—
53	C₂×C₃	#67	1	2	yes	direct product factor
54	C₂×C₃	C₃×D₄	1	2	yes	complemented
55	C₂×C₃	C₂²×D₃	1	1	yes	characteristic
56	C₂×C₃	C₂³×C₃	1	1	yes	characteristic
57	D₃	—	2	2	no	—
58	C₂²	—	2	2	no	—
59	C₂²	—	3	2	no	—
60	C₂²	—	6	2	no	—
61	C₂²	C₂×C₃×D₃	1	1	yes	characteristic
62	C₂²	C₂×C₃×D₃	1	2	yes	complemented
63	C₄	—	3	2	no	—
64	C₃	—	2	1	no	—
65	C₃	C₂×#67	1	1	yes	characteristic, direct product factor
66	C₃	C₂×C₃×D₄	1	1	yes	characteristic, complemented
67	C₂	—	2	2	no	—
68	C₂	—	6	2	no	—
69	C₂	C₃×#67	1	2	yes	direct product factor
70	C₂	C₂²×C₃×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂×D₃
commutator, cosocle	C₂×C₃	C₂³×C₃
Frattini	C₂	C₂²×C₃×D₃
Fitting	C₂³×C₃²	C₂
socle	C₂²×C₃²	C₂²
hypercenter	C₂³×C₃	D₃
hypocenter	C₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃²	C₂	semidirect
2. C₂²×C₃×D₃	C₂	semidirect
3. C₃×#67	C₂	direct (N × H)
4. C₂×C₃×Dic₃	C₂	semidirect
6. C₂×#67	C₃	direct (N × H)
9. C₂²×C₃²	C₂²	semidirect
11. C₂×C₃×D₃	C₂²	semidirect
12. C₃×Dic₃	C₂²	semidirect
15. C₂³×C₃	D₃	semidirect
16. C₂³×C₃	C₂×C₃	semidirect
17. C₂²×D₃	C₂×C₃	semidirect
20. #67	C₂×C₃	direct (N × H)
21. C₂×Dic₃	C₂×C₃	semidirect
23. C₂×C₃²	D₄	semidirect
35. C₂²×C₃	C₂×D₃	semidirect
37. C₂²×C₃	C₂²×C₃	semidirect
39. C₂×D₃	C₂²×C₃	semidirect
41. Dic₃	C₂²×C₃	semidirect
42. C₃²	C₂×D₄	semidirect
44. C₂³	C₃×D₃	semidirect
53. C₂×C₃	#67	direct (N × H)
54. C₂×C₃	C₃×D₄	semidirect
62. C₂²	C₂×C₃×D₃	semidirect
65. C₃	C₂×#67	direct (N × H)
66. C₃	C₂×C₃×D₄	semidirect
69. C₂	C₃×#67	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃²	2
C₂²×C₃×D₃	2
C₃×#67	2
C₂×C₃×Dic₃	2
C₂×C₃×D₄	3
C₂×#67	3

Subgroups — C2×C3×#67 #3824

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#67 #3824