Subgroups — C₃×#458 #5332

Order 192.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₃×#123	C₂	1	3	yes	maximal, complemented
3	C₃×#99	C₂	1	3	yes	maximal, complemented
4	#458	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₃×#41	—	2	3	no	—
6	C₂×C₃×D₄	—	2	3	no	—
7	C₂×C₃×D₄	C₂²	1	3	yes	—
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic, complemented
9	C₃×C₄⋊C₄	—	2	3	no	—
10	C₃×C₂²⋊C₄	—	2	3	no	—
11	C₃×C₂²⋊C₄	C₂²	1	3	yes	complemented
12	H(5,𝔽₂)	C₂×C₃	1	1	yes	characteristic
13	#123	C₂×C₃	1	3	yes	complemented
14	#99	C₂×C₃	1	3	yes	complemented
15	C₂³×C₃	—	2	1	no	—
16	C₂³×C₃	D₄	1	3	yes	—
17	C₂³×C₃	C₂³	1	1	yes	characteristic
18	C₃×Q₈	—	2	1	no	—
19	C₃×D₄	—	2	3	no	—
20	C₃×D₄	—	2	6	no	—
21	C₂×C₃×C₄	—	2	3	no	—
22	C₂×C₃×C₄	—	2	3	no	—
23	C₂×C₃×C₄	—	2	3	no	—
24	C₂×C₃×C₄	D₄	1	3	yes	—
25	#41	—	2	3	no	—
26	C₂×D₄	—	2	3	no	—
27	C₂×D₄	C₂²×C₃	1	3	yes	—
28	C₂²×C₄	C₂²×C₃	1	1	yes	characteristic, complemented
29	C₄⋊C₄	—	2	3	no	—
30	C₂²⋊C₄	—	2	3	no	—
31	C₂²⋊C₄	C₂²×C₃	1	3	yes	complemented
32	C₂²×C₃	—	2	3	no	—
33	C₂²×C₃	—	4	1	no	—
34	C₂²×C₃	—	4	3	no	—
35	C₂²×C₃	—	8	1	no	—
36	C₂²×C₃	C₂×D₄	1	3	yes	—
37	C₃×C₄	—	2	3	no	—
38	C₃×C₄	—	4	3	no	—
39	C₃×C₄	—	4	1	no	—
40	C₂³	—	2	1	no	—
41	C₂³	C₃×D₄	1	3	yes	—
42	C₂³	C₂³×C₃	1	1	yes	characteristic
43	Q₈	—	2	1	no	—
44	D₄	—	2	3	no	—
45	D₄	—	2	6	no	—
46	C₂×C₄	—	2	3	no	—
47	C₂×C₄	—	2	3	no	—
48	C₂×C₄	—	2	3	no	—
49	C₂×C₄	C₃×D₄	1	3	yes	—
50	C₂×C₃	—	2	3	no	—
51	C₂×C₃	—	4	3	no	—
52	C₂×C₃	#120	1	1	yes	characteristic
53	C₂²	—	2	3	no	—
54	C₂²	—	4	1	no	—
55	C₂²	—	4	3	no	—
56	C₂²	—	8	1	no	—
57	C₂²	C₂×C₃×D₄	1	3	yes	—
58	C₄	—	2	3	no	—
59	C₄	—	4	3	no	—
60	C₄	—	4	1	no	—
61	C₃	#458	1	1	yes	characteristic, direct product factor, Sylow
62	C₂	—	2	3	no	—
63	C₂	—	4	3	no	—
64	C₂	C₃×#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	#120
commutator, Frattini, cosocle	C₂³	C₂³×C₃
prime radical	C₃×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂³×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#123	C₂	semidirect
3. C₃×#99	C₂	semidirect
4. #458	C₃	direct (N × H)
8. C₂²×C₃×C₄	C₂²	semidirect
11. C₃×C₂²⋊C₄	C₂²	semidirect
13. #123	C₂×C₃	semidirect
14. #99	C₂×C₃	semidirect
28. C₂²×C₄	C₂²×C₃	semidirect
31. C₂²⋊C₄	C₂²×C₃	semidirect
61. C₃	#458	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×H(5,𝔽₂)	2
C₃×#123	2
C₃×#99	2
#458	3

Subgroups — C3×#458 #5332

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#458 #5332