Subgroups — #14552167

Order 2187.

2792 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#737	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₉×H(3,𝔽₃)	C₃	1	12	yes	maximal, complemented
3	#10624564	C₃	1	2	yes	maximal, complemented
4	#10624563	C₃	1	24	yes	maximal, complemented
5	#10624562	C₃	1	1	yes	maximal, characteristic
6	C₃×#737	—	3	36	no	—
7	C₃²×3₋¹⁺²	C₃²	1	8	yes	—
8	C₃²×H(3,𝔽₃)	C₃²	1	12	yes	—
9	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
10	C₃³×C₉	C₃²	1	9	yes	—
11	C₃³×C₉	C₃²	1	4	yes	—
12	C₉×H(3,𝔽₃)	—	3	108	no	—
13	C₃×C₃²⋊C₉	C₃²	1	72	yes	—
14	C₃×C₃²⋊C₉	C₃²	1	24	yes	—
15	C₃⁴	C₃³	1	4	yes	—
16	C₃⁴	C₃³	1	9	yes	—
17	#737	—	3	243	no	—
18	C₃×3₋¹⁺²	—	3	24	no	—
19	C₃×3₋¹⁺²	—	3	72	no	—
20	C₃×H(3,𝔽₃)	—	3	12	no	—
21	C₃×H(3,𝔽₃)	—	3	36	no	—
22	C₃×H(3,𝔽₃)	—	3	108	no	—
23	C₃²×C₉	—	3	36	no	—
24	C₃²×C₉	—	3	108	no	—
25	C₃²×C₉	H(3,𝔽₃)	1	36	yes	complemented
26	C₃²×C₉	C₃³	1	18	yes	—
27	C₃²×C₉	C₃³	1	8	yes	—
28	C₃²×C₉	C₃³	1	1	yes	characteristic
29	C₃²⋊C₉	—	3	72	no	—
30	C₃²⋊C₉	—	3	216	no	—
31	C₃³	—	3	36	no	—
32	C₃³	—	3	12	no	—
33	C₃³	—	3	36	no	—
34	C₃³	—	3	72	no	—
35	C₃³	C₃×H(3,𝔽₃)	1	12	yes	—
36	C₃³	#737	1	27	yes	complemented
37	C₃³	C₃⁴	1	1	yes	characteristic
38	3₋¹⁺²	—	3	216	no	—
39	H(3,𝔽₃)	—	3	324	no	—
40	H(3,𝔽₃)	—	3	27	no	—
41	C₃×C₉	—	3	24	no	—
42	C₃×C₉	—	3	108	no	—
43	C₃×C₉	—	3	72	no	—
44	C₃×C₉	—	9	81	no	—
45	C₃×C₉	C₃×H(3,𝔽₃)	1	24	yes	—
46	C₃×C₉	C₃×H(3,𝔽₃)	1	12	yes	—
47	C₃²	—	3	12	no	—
48	C₃²	—	3	36	no	—
49	C₃²	—	3	108	no	—
50	C₃²	—	9	9	no	—
51	C₃²	—	9	72	no	—
52	C₃²	—	9	36	no	—
53	C₃²	C₃²×H(3,𝔽₃)	1	4	yes	—
54	C₃²	C₃×#737	1	9	yes	—
55	C₉	—	3	24	no	—
56	C₉	—	9	18	no	—
57	C₉	C₃²⋊H(3,𝔽₃)	1	9	yes	—
58	C₃	—	9	12	no	—
59	C₃	—	9	27	no	—
60	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
61	C₃	#10624869	1	12	yes	—

	subgroup	G/H
center	C₃²×C₉	C₃³
commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	#10624562	C₃
LCS	G ▷ C₃³ ▷ C₁
UCS	C₁ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃²×#737	C₃	semidirect
2. C₃×C₉×H(3,𝔽₃)	C₃	semidirect
3. #10624564	C₃	semidirect
4. #10624563	C₃	semidirect
25. C₃²×C₉	H(3,𝔽₃)	semidirect
36. C₃³	#737	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#737	3
C₃×C₉×H(3,𝔽₃)	3
#10624564	3
#10624563	3
#10624562	3