Subgroups — C₃×#10624477 #14550821

Order 2187.

506 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#732	C₃	1	2	yes	maximal, complemented
2	C₃×#6867	C₃	1	2	yes	maximal, complemented
3	#10624477	C₃	1	9	yes	maximal, direct product factor
4	C₃²×H(3,𝔽₃)	—	3	2	no	—
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₃×#732	—	3	12	no	—
7	C₃×#732	—	3	6	no	—
8	C₃×#732	—	3	4	no	—
9	C₃×#732	C₃²	1	6	yes	complemented
10	C₃×C₃²⋊C₉	—	3	2	no	—
11	#6867	—	3	4	no	—
12	#6867	C₃²	1	6	yes	complemented
13	C₃⁴	—	9	2	no	—
14	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
15	C₃×H(3,𝔽₃)	—	3	12	no	—
16	C₃×H(3,𝔽₃)	—	3	6	no	—
17	C₃×H(3,𝔽₃)	—	3	6	no	—
18	C₃×H(3,𝔽₃)	—	9	4	no	—
19	C₃×H(3,𝔽₃)	—	9	8	no	—
20	C₃×H(3,𝔽₃)	—	9	4	no	—
21	C₃²×C₉	—	3	4	no	—
22	C₃²×C₉	—	3	2	no	—
23	C₃²×C₉	—	3	4	no	—
24	C₃²×C₉	—	3	2	no	—
25	C₃²×C₉	—	9	2	no	—
26	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
27	C₃²×C₉	C₃³	1	1	yes	characteristic
28	#732	—	3	18	no	—
29	#732	—	3	36	no	—
30	C₃²⋊C₉	—	3	6	no	—
31	C₃²⋊C₉	—	9	4	no	—
32	C₃³	—	9	3	no	—
33	C₃³	—	9	12	no	—
34	C₃³	—	9	6	no	—
35	C₃³	—	9	6	no	—
36	C₃³	—	27	18	no	—
37	C₃³	AΣL(1,𝔽₂₇)	1	2	yes	—
38	C₃³	AΣL(1,𝔽₂₇)	1	4	yes	—
39	C₃³	#732	1	2	yes	—
40	C₃³	#732	1	4	yes	—
41	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
42	H(3,𝔽₃)	—	3	18	no	—
43	H(3,𝔽₃)	—	9	24	no	—
44	H(3,𝔽₃)	—	9	12	no	—
45	H(3,𝔽₃)	—	9	12	no	—
46	C₃×C₉	—	3	4	no	—
47	C₃×C₉	—	3	2	no	—
48	C₃×C₉	—	3	4	no	—
49	C₃×C₉	—	3	2	no	—
50	C₃×C₉	—	9	4	no	—
51	C₃×C₉	—	9	4	no	—
52	C₃×C₉	—	9	1	no	—
53	C₃×C₉	—	9	6	no	—
54	C₃×C₉	—	27	6	no	—
55	C₃²	—	3	6	no	—
56	C₃²	—	3	6	no	—
57	C₃²	—	9	4	no	—
58	C₃²	—	9	4	no	—
59	C₃²	—	9	1	no	—
60	C₃²	—	9	18	no	—
61	C₃²	—	27	36	no	—
62	C₃²	—	27	18	no	—
63	C₃²	—	27	18	no	—
64	C₃²	#6856	1	4	yes	—
65	C₃²	#6856	1	4	yes	—
66	C₃²	#6856	1	1	yes	characteristic
67	C₃²	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
68	C₃²	C₃×#732	1	2	yes	—
69	C₉	—	9	2	no	—
70	C₉	—	9	1	no	—
71	C₉	—	27	6	no	—
72	C₃	—	3	2	no	—
73	C₃	—	3	4	no	—
74	C₃	—	3	2	no	—
75	C₃	—	3	4	no	—
76	C₃	—	27	18	no	—
77	C₃	#10624477	1	3	yes	direct product factor
78	C₃	C₃×#6856	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	#6856
commutator, Frattini, cosocle	C₃²×C₉	C₃³
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃²×#732	C₃	semidirect
2. C₃×#6867	C₃	semidirect
3. #10624477	C₃	direct (N × H)
9. C₃×#732	C₃²	semidirect
12. #6867	C₃²	semidirect
77. C₃	#10624477	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#732	3
C₃×#6867	3
#10624477	3

Subgroups — C3×#10624477 #14550821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10624477 #14550821