Subgroups — C₃⋊#4174 #92321

Order 486.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×AΣL(1,𝔽₂₇)	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃²×C₃⋊D₃	—	3	1	no	maximal
3	C₃⋊#272	C₃	1	1	yes	maximal, characteristic, complemented
4	#4174	—	3	3	no	maximal
5	C₃⁴	D₃	1	1	yes	characteristic, complemented
6	C₃×3₋¹⁺²	—	2	1	no	—
7	C₃×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic, complemented
8	AΣL(1,𝔽₂₇)	—	2	3	no	—
9	AΣL(1,𝔽₂₇)	D₃	1	3	yes	complemented
10	C₃×C₃⋊D₃	—	3	3	no	—
11	C₃×C₃⋊D₃	—	3	1	no	—
12	C₃×C₃⋊D₃	—	9	1	no	—
13	C₃²×D₃	—	9	2	no	—
14	C₃²×D₃	—	9	1	no	—
15	C₃²×D₃	—	9	1	no	—
16	#272	—	3	3	no	—
17	#272	—	9	2	no	—
18	C₃³	—	2	1	no	—
19	C₃³	—	3	3	no	—
20	C₃³	—	3	2	no	—
21	C₃³	—	3	1	no	—
22	C₃³	—	3	1	no	—
23	C₃³	—	6	3	no	—
24	C₃³	C₃×D₃	1	1	yes	characteristic
25	C₃³	C₃⋊D₃	1	1	yes	characteristic, complemented
26	3₋¹⁺²	—	2	3	no	—
27	3₋¹⁺²	—	6	2	no	—
28	H(3,𝔽₃)	—	3	2	no	—
29	H(3,𝔽₃)	C₃×D₃	1	3	yes	complemented
30	C₃×C₉	—	6	1	no	—
31	C₂×C₃²	—	27	1	no	—
32	C₃⋊D₃	—	3	1	no	—
33	C₃⋊D₃	—	9	1	no	—
34	C₃×D₃	—	9	3	no	—
35	C₃×D₃	—	9	6	no	—
36	C₃×D₃	—	9	3	no	—
37	C₃×D₃	—	9	2	no	—
38	C₃×D₃	—	9	1	no	—
39	C₃×D₃	—	9	1	no	—
40	C₃×D₃	—	27	3	no	—
41	C₃²	—	3	3	no	—
42	C₃²	—	3	1	no	—
43	C₃²	—	3	1	no	—
44	C₃²	—	3	6	no	—
45	C₃²	—	3	3	no	—
46	C₃²	—	3	3	no	—
47	C₃²	—	6	1	no	—
48	C₃²	—	6	3	no	—
49	C₃²	—	6	6	no	—
50	C₃²	—	6	3	no	—
51	C₃²	—	6	1	no	—
52	C₃²	—	9	1	no	—
53	C₃²	—	18	1	no	—
54	C₃²	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
55	C₃²	(C₃⋊D₃)⋊C₃	1	2	yes	—
56	C₃²	C₃×C₃⋊D₃	1	1	yes	characteristic
57	C₉	—	6	3	no	—
58	C₂×C₃	—	27	3	no	—
59	C₂×C₃	—	27	1	no	—
60	D₃	—	9	2	no	—
61	D₃	—	9	1	no	—
62	D₃	—	9	1	no	—
63	D₃	—	27	3	no	—
64	C₃	—	2	1	no	—
65	C₃	—	3	2	no	—
66	C₃	—	3	1	no	—
67	C₃	—	3	3	no	—
68	C₃	—	6	3	no	—
69	C₃	—	9	3	no	—
70	C₃	#4174	1	1	yes	characteristic, complemented
71	C₃	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
72	C₂	—	27	1	no	Sylow

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

Complements

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

N	H ≅ G/N	type
1. C₃×AΣL(1,𝔽₂₇)	C₂	semidirect
3. C₃⋊#272	C₃	semidirect
5. C₃⁴	D₃	semidirect
7. C₃×H(3,𝔽₃)	C₂×C₃	semidirect
9. AΣL(1,𝔽₂₇)	D₃	semidirect
25. C₃³	C₃⋊D₃	semidirect
29. H(3,𝔽₃)	C₃×D₃	semidirect
70. C₃	#4174	semidirect

Maximal subgroups

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

subgroup	index
C₃×AΣL(1,𝔽₂₇)	2
C₃²×C₃⋊D₃	3
C₃⋊#272	3
#4174	3

Subgroups — C3⋊#4174 #92321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊#4174 #92321