Subgroups — #14546316

Order 2187.

253 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	#10624444	C₃	1	3	yes	maximal, complemented
3	C₃²×3₋¹⁺²	—	3	1	no	—
4	C₃²×H(3,𝔽₃)	C₉	1	1	yes	characteristic
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₉⋊H(3,𝔽₃)	—	3	9	no	—
7	C₃×C₃²⋊C₉	—	3	1	no	—
8	C₃×C₃²⋊C₉	—	3	1	no	—
9	C₃×C₃²⋊C₉	C₉	1	2	yes	—
10	C₃²⋊C₂₇	—	3	3	no	—
11	C₃⁴	—	3	1	no	—
12	C₃⁴	C₃×C₉	1	1	yes	characteristic
13	C₃×3₋¹⁺²	—	3	9	no	—
14	C₃×3₋¹⁺²	—	9	9	no	—
15	C₃×H(3,𝔽₃)	—	3	3	no	—
16	C₃×H(3,𝔽₃)	—	9	3	no	—
17	C₃²×C₉	—	3	2	no	—
18	C₃²×C₉	—	3	3	no	—
19	C₃²×C₉	—	9	3	no	—
20	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃²×C₉	3₋¹⁺²	1	1	yes	characteristic
22	C₃²×C₉	3₋¹⁺²	1	1	yes	characteristic
23	C₃×C₂₇	—	9	3	no	—
24	C₃²⋊C₉	—	3	9	no	—
25	C₃²⋊C₉	—	3	9	no	—
26	C₃²⋊C₉	—	3	6	no	—
27	C₃³	—	3	3	no	—
28	C₃³	—	3	1	no	—
29	C₃³	—	3	3	no	—
30	C₃³	—	3	3	no	—
31	C₃³	—	9	1	no	—
32	C₃³	—	9	1	no	—
33	C₃³	—	9	3	no	—
34	C₃³	—	9	3	no	—
35	C₃³	—	9	3	no	—
36	C₃³	—	9	3	no	—
37	C₃³	C₃²⋊C₉	1	1	yes	characteristic
38	3₋¹⁺²	—	9	27	no	—
39	H(3,𝔽₃)	—	9	9	no	—
40	C₃×C₉	—	3	1	no	—
41	C₃×C₉	—	3	3	no	—
42	C₃×C₉	—	3	1	no	—
43	C₃×C₉	—	3	6	no	—
44	C₃×C₉	—	9	1	no	—
45	C₃×C₉	—	9	9	no	—
46	C₃×C₉	—	9	6	no	—
47	C₃×C₉	—	9	1	no	—
48	C₃×C₉	AΣL(1,𝔽₂₇)	1	3	yes	—
49	C₂₇	—	27	3	no	—
50	C₃²	—	3	1	no	—
51	C₃²	—	3	1	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	3	3	no	—
54	C₃²	—	9	9	no	—
55	C₃²	—	9	1	no	—
56	C₃²	—	9	1	no	—
57	C₃²	—	9	1	no	—
58	C₃²	—	9	3	no	—
59	C₃²	—	9	3	no	—
60	C₃²	—	9	3	no	—
61	C₃²	—	9	3	no	—
62	C₃²	—	27	9	no	—
63	C₃²	#6866	1	1	yes	characteristic
64	C₃²	#6872	1	3	yes	—
65	C₉	—	3	3	no	—
66	C₉	—	3	3	no	—
67	C₉	—	9	1	no	—
68	C₉	—	9	6	no	—
69	C₃	—	3	3	no	—
70	C₃	—	9	3	no	—
71	C₃	—	27	3	no	—
72	C₃	#10624456	1	1	yes	characteristic
73	C₃	#10624507	1	1	yes	characteristic
74	C₃	#10624508	1	1	yes	characteristic
75	C₃	#10624511	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	#6866
commutator	C₃⁴	C₃×C₉
Frattini, cosocle	C₃³×C₉	C₃²
prime radical	C₃²×H(3,𝔽₃)	C₉
LCS	G ▷ C₃⁴ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃³×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624444	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉⋊H(3,𝔽₃)	3
#10624444	3