Subgroups — C₃×#560 #5890

Order 192.

252 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#126	C₂	1	2	yes	maximal, complemented
3	C₃×#123	C₂	1	3	yes	maximal, complemented
4	C₃×#121	C₂	1	6	yes	maximal, complemented
5	C₃×#120	C₂	1	3	yes	maximal, complemented
6	#560	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂⁴×C₃	C₂²	1	1	yes	characteristic
8	C₂×C₃×D₄	C₂²	1	6	yes	—
9	C₂×C₃×D₄	C₂²	1	3	yes	—
10	C₂×C₃×D₄	C₂²	1	3	yes	complemented
11	C₂²×C₃×C₄	C₂²	1	3	yes	complemented
12	C₃×C₄⋊C₄	C₂²	1	6	yes	complemented
13	C₃×C₂²⋊C₄	C₂²	1	6	yes	—
14	C₃×C₂²⋊C₄	C₂²	1	3	yes	—
15	C₃×C₂²⋊C₄	C₂²	1	3	yes	complemented
16	C₃×C₄²	C₂²	1	1	yes	characteristic, complemented
17	#127	C₂×C₃	1	1	yes	characteristic, complemented
18	#126	C₂×C₃	1	2	yes	complemented
19	#123	C₂×C₃	1	3	yes	complemented
20	#121	C₂×C₃	1	6	yes	complemented
21	#120	C₂×C₃	1	3	yes	complemented
22	C₂³×C₃	—	4	3	no	—
23	C₂³×C₃	C₂³	1	2	yes	—
24	C₂³×C₃	C₂³	1	1	yes	characteristic
25	C₂³×C₃	C₂³	1	3	yes	—
26	C₃×D₄	—	4	6	no	—
27	C₃×D₄	—	4	3	no	—
28	C₃×D₄	—	4	3	no	—
29	C₂×C₃×C₄	—	4	3	no	—
30	C₂×C₃×C₄	C₂³	1	3	yes	—
31	C₂×C₃×C₄	C₂³	1	6	yes	—
32	C₂⁴	C₂²×C₃	1	1	yes	characteristic
33	C₂×D₄	C₂²×C₃	1	6	yes	—
34	C₂×D₄	C₂²×C₃	1	3	yes	—
35	C₂×D₄	C₂²×C₃	1	3	yes	complemented
36	C₂²×C₄	C₂²×C₃	1	3	yes	complemented
37	C₄⋊C₄	C₂²×C₃	1	6	yes	complemented
38	C₂²⋊C₄	C₂²×C₃	1	6	yes	—
39	C₂²⋊C₄	C₂²×C₃	1	3	yes	—
40	C₂²⋊C₄	C₂²×C₃	1	3	yes	complemented
41	C₄²	C₂²×C₃	1	1	yes	characteristic, complemented
42	C₂²×C₃	—	2	3	no	—
43	C₂²×C₃	—	2	6	no	—
44	C₂²×C₃	—	2	6	no	—
45	C₂²×C₃	—	2	3	no	—
46	C₂²×C₃	—	4	4	no	—
47	C₂²×C₃	C₂⁴	1	1	yes	characteristic
48	C₃×C₄	—	2	6	no	—
49	C₃×C₄	—	2	3	no	—
50	C₂³	—	4	3	no	—
51	C₂³	C₂³×C₃	1	2	yes	—
52	C₂³	C₂³×C₃	1	1	yes	characteristic
53	C₂³	C₂³×C₃	1	3	yes	—
54	D₄	—	4	6	no	—
55	D₄	—	4	3	no	—
56	D₄	—	4	3	no	—
57	C₂×C₄	—	4	3	no	—
58	C₂×C₄	C₂³×C₃	1	3	yes	—
59	C₂×C₄	C₂³×C₃	1	6	yes	—
60	C₂×C₃	—	4	3	no	—
61	C₂×C₃	—	4	2	no	—
62	C₂×C₃	—	4	1	no	—
63	C₂×C₃	H(5,𝔽₂)	1	3	yes	—
64	C₂²	—	2	3	no	—
65	C₂²	—	2	6	no	—
66	C₂²	—	2	6	no	—
67	C₂²	—	2	3	no	—
68	C₂²	—	4	4	no	—
69	C₂²	C₂⁴×C₃	1	1	yes	characteristic
70	C₄	—	2	6	no	—
71	C₄	—	2	3	no	—
72	C₃	#560	1	1	yes	characteristic, direct product factor, Sylow
73	C₂	—	4	2	no	—
74	C₂	—	4	1	no	—
75	C₂	—	4	3	no	—
76	C₂	C₃×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₃	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×#127	C₂	semidirect
2. C₃×#126	C₂	semidirect
3. C₃×#123	C₂	semidirect
4. C₃×#121	C₂	semidirect
5. C₃×#120	C₂	semidirect
6. #560	C₃	direct (N × H)
10. C₂×C₃×D₄	C₂²	semidirect
11. C₂²×C₃×C₄	C₂²	semidirect
12. C₃×C₄⋊C₄	C₂²	semidirect
15. C₃×C₂²⋊C₄	C₂²	semidirect
16. C₃×C₄²	C₂²	semidirect
17. #127	C₂×C₃	semidirect
18. #126	C₂×C₃	semidirect
19. #123	C₂×C₃	semidirect
20. #121	C₂×C₃	semidirect
21. #120	C₂×C₃	semidirect
35. C₂×D₄	C₂²×C₃	semidirect
36. C₂²×C₄	C₂²×C₃	semidirect
37. C₄⋊C₄	C₂²×C₃	semidirect
40. C₂²⋊C₄	C₂²×C₃	semidirect
41. C₄²	C₂²×C₃	semidirect
72. C₃	#560	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×#127	2
C₃×#126	2
C₃×#123	2
C₃×#121	2
C₃×#120	2
#560	3

Subgroups — C3×#560 #5890

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#560 #5890