Subgroups — C₂×#4183 #66385

Order 324.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AΣL(1,𝔽₂₇)	C₂	1	1	yes	maximal, characteristic, complemented
2	#4183	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₃⋊D₃	—	3	1	no	maximal
4	C₂×AGL(1,ℤ/9)	—	3	2	no	maximal
5	C₂×(C₃⋊D₃)⋊C₃	—	3	1	no	maximal
6	AΣL(1,𝔽₂₇)	C₂²	1	1	yes	characteristic, complemented, Sylow
7	C₂×C₃³	D₃	1	1	yes	characteristic, complemented
8	C₃×C₃⋊D₃	—	3	2	no	—
9	C₂×3₋¹⁺²	D₃	1	2	yes	complemented
10	C₂×H(3,𝔽₃)	D₃	1	1	yes	characteristic, complemented
11	AGL(1,ℤ/9)	—	3	4	no	—
12	(C₃⋊D₃)⋊C₃	—	3	2	no	—
13	C₂×C₃⋊D₃	—	3	1	no	—
14	C₂×C₃⋊D₃	—	3	1	no	—
15	C₂×C₃×D₃	—	9	3	no	—
16	C₂×C₃×D₃	—	9	1	no	—
17	C₂×D₉	—	3	2	no	—
18	C₃³	C₂×D₃	1	1	yes	characteristic, complemented
19	3₋¹⁺²	C₂×D₃	1	2	yes	complemented
20	H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic, complemented
21	C₂×C₃²	—	3	1	no	—
22	C₂×C₃²	—	3	3	no	—
23	C₂×C₃²	—	3	1	no	—
24	C₂×C₃²	C₃⋊D₃	1	1	yes	characteristic
25	C₃⋊D₃	—	3	2	no	—
26	C₃⋊D₃	—	3	2	no	—
27	C₃×D₃	—	9	6	no	—
28	C₃×D₃	—	9	2	no	—
29	C₂×C₉	—	3	2	no	—
30	D₉	—	3	4	no	—
31	C₂²×C₃	—	27	1	no	—
32	C₂×D₃	—	9	3	no	—
33	C₂×D₃	—	9	1	no	—
34	C₂×D₃	—	27	1	no	—
35	C₃²	—	3	3	no	—
36	C₃²	—	3	1	no	—
37	C₃²	—	3	1	no	—
38	C₃²	C₂×C₃⋊D₃	1	1	yes	characteristic
39	C₉	—	3	2	no	—
40	C₂×C₃	—	3	3	no	—
41	C₂×C₃	—	3	1	no	—
42	C₂×C₃	—	9	1	no	—
43	C₂×C₃	—	27	2	no	—
44	C₂×C₃	#272	1	1	yes	characteristic
45	D₃	—	9	6	no	—
46	D₃	—	9	2	no	—
47	D₃	—	27	2	no	—
48	C₂²	—	27	1	no	Sylow
49	C₃	—	3	3	no	—
50	C₃	—	3	1	no	—
51	C₃	—	9	1	no	—
52	C₃	C₂×#272	1	1	yes	characteristic
53	C₂	—	27	2	no	—
54	C₂	#4183	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#4183
commutator, cosocle, hypocenter	AΣL(1,𝔽₂₇)	C₂²
Frattini	C₃²	C₂×C₃⋊D₃
Fitting	C₂×AΣL(1,𝔽₂₇)	C₂
socle	C₂×C₃	#272
LCS	G ▷ AΣL(1,𝔽₂₇)
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×AΣL(1,𝔽₂₇)	C₂	semidirect
2. #4183	C₂	direct (N × H)
6. AΣL(1,𝔽₂₇)	C₂²	semidirect
7. C₂×C₃³	D₃	semidirect
9. C₂×3₋¹⁺²	D₃	semidirect
10. C₂×H(3,𝔽₃)	D₃	semidirect
18. C₃³	C₂×D₃	semidirect
19. 3₋¹⁺²	C₂×D₃	semidirect
20. H(3,𝔽₃)	C₂×D₃	semidirect
54. C₂	#4183	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×AΣL(1,𝔽₂₇)	2
#4183	2
C₂×C₃×C₃⋊D₃	3
C₂×AGL(1,ℤ/9)	3
C₂×(C₃⋊D₃)⋊C₃	3

Subgroups — C2×#4183 #66385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4183 #66385