Subgroups — #14548137

Order 2187.

380 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6895	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624733	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624732	C₃	1	2	yes	maximal, complemented
4	#10624636	C₃	1	9	yes	maximal, complemented
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₃×#733	—	3	3	no	—
7	C₃×#732	—	3	6	no	—
8	#6895	—	3	9	no	—
9	#6891	C₃²	1	3	yes	—
10	C₃²⋊H(3,𝔽₃)	C₃²	1	6	yes	—
11	C₃×C₉⋊C₉	—	3	3	no	—
12	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
13	#6856	—	3	27	no	—
14	C₃⁴	C₃³	1	1	yes	characteristic
15	C₃×3₋¹⁺²	—	3	3	no	—
16	C₃×H(3,𝔽₃)	—	3	6	no	—
17	C₃×H(3,𝔽₃)	—	3	18	no	—
18	C₃²×C₉	—	3	3	no	—
19	C₃²×C₉	—	3	3	no	—
20	C₃²×C₉	—	9	3	no	—
21	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
22	#733	—	9	9	no	—
23	#732	—	9	18	no	—
24	C₉⋊C₉	—	3	27	no	—
25	C₃²⋊C₉	—	3	6	no	—
26	C₃²⋊C₉	—	3	9	no	—
27	C₃²⋊C₉	H(3,𝔽₃)	1	9	yes	—
28	C₃³	—	3	3	no	—
29	C₃³	—	9	3	no	—
30	C₃³	—	9	6	no	—
31	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
32	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	3₋¹⁺²	—	9	9	no	—
34	H(3,𝔽₃)	—	9	18	no	—
35	H(3,𝔽₃)	—	27	18	no	—
36	C₃×C₉	—	3	9	no	—
37	C₃×C₉	—	3	3	no	—
38	C₃×C₉	—	3	6	no	—
39	C₃×C₉	—	3	3	no	—
40	C₃×C₉	—	9	9	no	—
41	C₃×C₉	—	9	9	no	—
42	C₃×C₉	—	9	3	no	—
43	C₃²	—	3	3	no	—
44	C₃²	—	3	3	no	—
45	C₃²	—	3	6	no	—
46	C₃²	—	3	3	no	—
47	C₃²	—	9	9	no	—
48	C₃²	—	27	6	no	—
49	C₃²	—	27	18	no	—
50	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
51	C₃²	#6911	1	3	yes	—
52	C₉	—	9	3	no	—
53	C₉	—	9	9	no	—
54	C₉	—	27	3	no	—
55	C₃	—	3	3	no	—
56	C₃	—	9	3	no	—
57	C₃	—	81	6	no	—
58	C₃	#10624737	1	1	yes	characteristic
59	C₃	#10624757	1	3	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃×#6895	C₃	semidirect
2. #10624733	C₃	semidirect
3. #10624732	C₃	semidirect
4. #10624636	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6895	3
#10624733	3
#10624732	3
#10624636	3