Subgroups — #13091646

Order 1458.

650 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624930	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	#92404	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊#272	—	3	1	no	maximal
4	C₃⋊#4174	—	3	12	no	maximal
5	3₋¹⁺⁴	—	2	1	no	—
6	H(5,𝔽₃)	C₂×C₃	1	1	yes	characteristic, complemented
7	C₃²×H(3,𝔽₃)	D₃	1	1	yes	characteristic, complemented
8	#6909	—	2	12	no	—
9	C₃×AΣL(1,𝔽₂₇)	D₃	1	12	yes	complemented
10	C₃²×C₃⋊D₃	—	9	4	no	—
11	C₃⋊#272	—	3	1	no	—
12	C₃⋊#272	—	3	12	no	—
13	C₃⋊#272	—	9	9	no	—
14	C₃×#272	—	9	9	no	—
15	#4174	—	9	9	no	—
16	C₃⁴	C₃⋊D₃	1	4	yes	complemented
17	#737	—	6	9	no	—
18	C₃×3₋¹⁺²	—	2	12	no	—
19	C₃×H(3,𝔽₃)	—	2	4	no	—
20	C₃×H(3,𝔽₃)	—	3	9	no	—
21	C₃×H(3,𝔽₃)	—	3	9	no	—
22	C₃×H(3,𝔽₃)	C₃×D₃	1	1	yes	characteristic
23	C₃×H(3,𝔽₃)	C₃×D₃	1	12	yes	complemented
24	AΣL(1,𝔽₂₇)	—	2	36	no	—
25	AΣL(1,𝔽₂₇)	C₃⋊D₃	1	9	yes	complemented
26	C₃×C₃⋊D₃	—	9	12	no	—
27	C₃×C₃⋊D₃	—	9	4	no	—
28	C₃×C₃⋊D₃	—	27	12	no	—
29	C₃²×D₃	—	27	12	no	—
30	C₃²×D₃	—	27	1	no	—
31	#272	—	9	9	no	—
32	#272	—	9	9	no	—
33	#272	—	27	24	no	—
34	C₃³	—	2	4	no	—
35	C₃³	—	3	12	no	—
36	C₃³	—	3	12	no	—
37	C₃³	—	9	12	no	—
38	C₃³	C₃×C₃⋊D₃	1	4	yes	—
39	C₃³	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
40	3₋¹⁺²	—	2	9	no	—
41	3₋¹⁺²	—	6	24	no	—
42	H(3,𝔽₃)	—	3	24	no	—
43	H(3,𝔽₃)	—	6	12	no	—
44	H(3,𝔽₃)	(C₃⋊D₃)⋊C₃	1	9	yes	—
45	H(3,𝔽₃)	C₃×C₃⋊D₃	1	9	yes	—
46	C₃×C₉	—	6	12	no	—
47	C₂×C₃²	—	81	1	no	—
48	C₃⋊D₃	—	9	4	no	—
49	C₃⋊D₃	—	27	12	no	—
50	C₃×D₃	—	27	3	no	—
51	C₃×D₃	—	27	36	no	—
52	C₃×D₃	—	27	12	no	—
53	C₃×D₃	—	27	1	no	—
54	C₃×D₃	—	81	9	no	—
55	C₃²	—	3	3	no	—
56	C₃²	—	3	1	no	—
57	C₃²	—	3	9	no	—
58	C₃²	—	6	4	no	—
59	C₃²	—	9	4	no	—
60	C₃²	—	9	36	no	—
61	C₃²	—	18	4	no	—
62	C₃²	—	27	12	no	—
63	C₃²	C₃⋊((C₃⋊D₃)⋊C₃)	1	12	yes	—
64	C₃²	C₃×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
65	C₉	—	6	9	no	—
66	C₂×C₃	—	81	3	no	—
67	C₂×C₃	—	81	1	no	—
68	D₃	—	27	1	no	—
69	D₃	—	27	12	no	—
70	D₃	—	81	9	no	—
71	C₃	—	3	1	no	—
72	C₃	—	3	12	no	—
73	C₃	—	3	3	no	—
74	C₃	—	9	9	no	—
75	C₃	—	18	4	no	—
76	C₃	C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))	1	1	yes	characteristic
77	C₂	—	81	1	no	Sylow

	subgroup	G/H
center, socle, hypercenter	C₃	C₃⋊(C₃⋊((C₃⋊D₃)⋊C₃))
commutator, cosocle, hypocenter	H(5,𝔽₃)	C₂×C₃
Frattini	C₃²	C₃×C₃⋊(C₃⋊D₃)
Fitting	#10624930	C₂
LCS	G ▷ H(5,𝔽₃)
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. #10624930	C₂	semidirect
2. #92404	C₃	semidirect
6. H(5,𝔽₃)	C₂×C₃	semidirect
7. C₃²×H(3,𝔽₃)	D₃	semidirect
9. C₃×AΣL(1,𝔽₂₇)	D₃	semidirect
16. C₃⁴	C₃⋊D₃	semidirect
23. C₃×H(3,𝔽₃)	C₃×D₃	semidirect
25. AΣL(1,𝔽₂₇)	C₃⋊D₃	semidirect

Maximal subgroups

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

subgroup	index
#10624930	2
#92404	3
C₃×C₃⋊#272	3
C₃⋊#4174	3