Subgroups — C₂×#6910 #92371

Order 486.

136 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6910	C₂	1	1	yes	maximal, characteristic, direct product factor, Sylow
2	C₂×C₃×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
3	C₂×C₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₂×#731	C₃	1	9	yes	maximal, complemented
6	C₃×3₋¹⁺²	C₂×C₃	1	2	yes	complemented
7	C₃×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic, complemented
8	C₃×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
9	#731	C₂×C₃	1	9	yes	complemented
10	C₂×C₃³	—	3	1	no	—
11	C₂×C₃³	C₃²	1	1	yes	characteristic
12	C₂×3₋¹⁺²	—	3	2	no	—
13	C₂×3₋¹⁺²	—	3	2	no	—
14	C₂×3₋¹⁺²	—	3	2	no	—
15	C₂×3₋¹⁺²	—	3	1	no	—
16	C₂×3₋¹⁺²	C₃²	1	6	yes	complemented
17	C₂×H(3,𝔽₃)	—	3	2	no	—
18	C₂×H(3,𝔽₃)	C₃²	1	3	yes	—
19	C₂×C₃×C₉	—	3	2	no	—
20	C₂×C₃×C₉	C₃²	1	3	yes	complemented
21	C₃³	—	3	1	no	—
22	C₃³	C₂×C₃²	1	1	yes	characteristic
23	3₋¹⁺²	—	3	2	no	—
24	3₋¹⁺²	—	3	2	no	—
25	3₋¹⁺²	—	3	2	no	—
26	3₋¹⁺²	—	3	1	no	—
27	3₋¹⁺²	C₂×C₃²	1	6	yes	complemented
28	H(3,𝔽₃)	—	3	2	no	—
29	H(3,𝔽₃)	C₂×C₃²	1	3	yes	—
30	C₃×C₉	—	3	2	no	—
31	C₃×C₉	C₂×C₃²	1	3	yes	complemented
32	C₂×C₃²	—	3	3	no	—
33	C₂×C₃²	—	9	1	no	—
34	C₂×C₃²	—	9	3	no	—
35	C₂×C₃²	H(3,𝔽₃)	1	2	yes	—
36	C₂×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
37	C₂×C₃²	C₃³	1	1	yes	characteristic
38	C₂×C₉	—	3	6	no	—
39	C₂×C₉	—	3	3	no	—
40	C₃²	—	3	3	no	—
41	C₃²	—	9	1	no	—
42	C₃²	—	9	3	no	—
43	C₃²	C₂×H(3,𝔽₃)	1	2	yes	—
44	C₃²	C₂×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃²	C₂×C₃³	1	1	yes	characteristic
46	C₉	—	3	6	no	—
47	C₉	—	3	3	no	—
48	C₂×C₃	—	3	1	no	—
49	C₂×C₃	—	3	2	no	—
50	C₂×C₃	—	3	1	no	—
51	C₂×C₃	—	9	3	no	—
52	C₂×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃	—	3	1	no	—
54	C₃	—	3	2	no	—
55	C₃	—	3	1	no	—
56	C₃	—	9	3	no	—
57	C₃	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₂	#6910	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, socle	C₂×C₃	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂×C₃³
prime radical	C₂×C₃×H(3,𝔽₃)	C₃
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃³ ◁ G

Complements

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

N	H ≅ G/N	type
1. #6910	C₂	direct (N × H)
2. C₂×C₃×3₋¹⁺²	C₃	semidirect
3. C₂×C₃×3₋¹⁺²	C₃	semidirect
5. C₂×#731	C₃	semidirect
6. C₃×3₋¹⁺²	C₂×C₃	semidirect
7. C₃×3₋¹⁺²	C₂×C₃	semidirect
9. #731	C₂×C₃	semidirect
16. C₂×3₋¹⁺²	C₃²	semidirect
20. C₂×C₃×C₉	C₃²	semidirect
27. 3₋¹⁺²	C₂×C₃²	semidirect
31. C₃×C₉	C₂×C₃²	semidirect
58. C₂	#6910	direct (N × H)

Maximal subgroups

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

subgroup	index
#6910	2
C₂×C₃×3₋¹⁺²	3
C₂×C₃×H(3,𝔽₃)	3
C₂×#731	3

Subgroups — C2×#6910 #92371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#6910 #92371