Subgroups — C₂×#4174 #66376

Order 324.

84 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	#4174	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃²×D₃	—	3	1	no	maximal
4	C₂×#272	C₃	1	1	yes	maximal, characteristic, complemented
5	AΣL(1,𝔽₂₇)	C₂²	1	1	yes	characteristic, complemented, Sylow
6	C₂×C₃³	D₃	1	1	yes	characteristic, complemented
7	C₃²×D₃	—	3	2	no	—
8	C₂×3₋¹⁺²	—	2	1	no	—
9	C₂×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic, complemented
10	#272	C₂×C₃	1	2	yes	complemented
11	C₂²×C₃²	—	9	1	no	—
12	C₂×C₃×D₃	—	3	3	no	—
13	C₂×C₃×D₃	—	3	1	no	—
14	C₂×C₃×D₃	—	9	1	no	—
15	C₃³	C₂×D₃	1	1	yes	characteristic, complemented
16	3₋¹⁺²	—	2	1	no	—
17	H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic, complemented
18	C₂×C₃²	—	3	1	no	—
19	C₂×C₃²	—	3	3	no	—
20	C₂×C₃²	—	3	1	no	—
21	C₂×C₃²	—	9	2	no	—
22	C₂×C₃²	C₃×D₃	1	1	yes	characteristic
23	C₃×D₃	—	3	6	no	—
24	C₃×D₃	—	3	2	no	—
25	C₃×D₃	—	9	2	no	—
26	C₂×C₉	—	6	1	no	—
27	C₂²×C₃	—	9	3	no	—
28	C₂²×C₃	—	9	1	no	—
29	C₂×D₃	—	3	1	no	—
30	C₂×D₃	—	9	1	no	—
31	C₃²	—	3	3	no	—
32	C₃²	—	3	1	no	—
33	C₃²	—	3	1	no	—
34	C₃²	C₂×C₃×D₃	1	1	yes	characteristic
35	C₉	—	6	1	no	—
36	C₂×C₃	—	3	1	no	—
37	C₂×C₃	—	3	3	no	—
38	C₂×C₃	—	9	6	no	—
39	C₂×C₃	—	9	2	no	—
40	C₂×C₃	—	9	1	no	—
41	C₂×C₃	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
42	D₃	—	3	2	no	—
43	D₃	—	9	2	no	—
44	C₂²	—	9	1	no	Sylow
45	C₃	—	3	1	no	—
46	C₃	—	3	3	no	—
47	C₃	—	9	1	no	—
48	C₃	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
49	C₂	—	9	2	no	—
50	C₂	#4174	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, socle, hypercenter	C₂×C₃	(C₃⋊D₃)⋊C₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂²×C₃
Frattini	C₃²	C₂×C₃×D₃
Fitting	C₂×AΣL(1,𝔽₂₇)	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×AΣL(1,𝔽₂₇)	C₂	semidirect
2. #4174	C₂	direct (N × H)
4. C₂×#272	C₃	semidirect
5. AΣL(1,𝔽₂₇)	C₂²	semidirect
6. C₂×C₃³	D₃	semidirect
9. C₂×H(3,𝔽₃)	C₂×C₃	semidirect
10. #272	C₂×C₃	semidirect
15. C₃³	C₂×D₃	semidirect
17. H(3,𝔽₃)	C₂²×C₃	semidirect
50. C₂	#4174	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
#4174	2
C₂×C₃²×D₃	3
C₂×#272	3

Subgroups — C2×#4174 #66376

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4174 #66376