Subgroups — #14552055

Order 2187.

2024 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
2	C₃×C₉×H(3,𝔽₃)	C₃	1	3	yes	maximal
3	#10624558	C₃	1	24	yes	maximal, complemented
4	#10624549	C₃	1	8	yes	maximal, complemented
5	C₃×#6855	C₃	1	4	yes	maximal, complemented
6	C₃×#737	—	3	9	no	—
7	C₃×#737	—	3	18	no	—
8	C₃×#737	C₉	1	27	yes	complemented
9	C₃²×3₋¹⁺²	C₃²	1	8	yes	—
10	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
11	C₃³×C₉	C₃²	1	4	yes	—
12	C₃³×C₉	C₃²	1	12	yes	—
13	C₉×H(3,𝔽₃)	—	3	27	no	—
14	C₃×C₃²⋊C₉	C₃²	1	24	yes	—
15	#6855	C₃²	1	1	yes	characteristic
16	#6855	C₃²	1	8	yes	—
17	#6855	C₃²	1	24	yes	—
18	#6855	C₃²	1	48	yes	—
19	C₃⁴	C₃³	1	4	yes	—
20	#737	—	3	243	no	—
21	C₃×3₋¹⁺²	—	3	24	no	—
22	C₃×3₋¹⁺²	—	3	48	no	—
23	C₃×3₋¹⁺²	C₃×C₉	1	72	yes	complemented
24	C₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	—
25	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
26	C₃×H(3,𝔽₃)	3₋¹⁺²	1	18	yes	—
27	C₃²×C₉	—	3	12	no	—
28	C₃²×C₉	—	3	24	no	—
29	C₃²×C₉	—	3	36	no	—
30	C₃²×C₉	—	3	36	no	—
31	C₃²×C₉	—	9	24	no	—
32	C₃²×C₉	C₃×C₉	1	36	yes	complemented
33	C₃²×C₉	C₃³	1	24	yes	—
34	C₃²×C₉	C₃³	1	1	yes	characteristic
35	C₃²×C₉	C₃³	1	8	yes	—
36	C₃²×C₉	C₃³	1	3	yes	—
37	C₃²⋊C₉	—	3	72	no	—
38	C₃³	—	3	12	no	—
39	C₃³	—	3	24	no	—
40	C₃³	C₃²×C₉	1	12	yes	—
41	C₃³	C₃×H(3,𝔽₃)	1	12	yes	—
42	C₃³	C₃×3₋¹⁺²	1	24	yes	—
43	C₃³	C₃⁴	1	1	yes	characteristic
44	3₋¹⁺²	—	3	216	no	—
45	H(3,𝔽₃)	C₃²⋊C₉	1	81	yes	—
46	C₃×C₉	—	3	1	no	—
47	C₃×C₉	—	3	3	no	—
48	C₃×C₉	—	3	8	no	—
49	C₃×C₉	—	3	2	no	—
50	C₃×C₉	—	3	16	no	—
51	C₃×C₉	—	3	6	no	—
52	C₃×C₉	—	3	108	no	—
53	C₃×C₉	—	3	24	no	—
54	C₃×C₉	—	3	24	no	—
55	C₃×C₉	—	3	48	no	—
56	C₃×C₉	—	9	108	no	—
57	C₃×C₉	C₃²×C₉	1	3	yes	—
58	C₃×C₉	C₃²×C₉	1	24	yes	—
59	C₃×C₉	#737	1	9	yes	complemented
60	C₃²	—	3	12	no	—
61	C₃²	—	3	24	no	—
62	C₃²	—	3	36	no	—
63	C₃²	—	3	72	no	—
64	C₃²	C₃×C₃²⋊C₉	1	36	yes	—
65	C₃²	C₃³×C₉	1	1	yes	characteristic
66	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
67	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
68	C₃²	C₃×#737	1	3	yes	—
69	C₃²	3₋¹⁺⁴	1	6	yes	—
70	C₉	—	3	3	no	—
71	C₉	—	3	24	no	—
72	C₉	—	3	9	no	—
73	C₉	—	9	24	no	—
74	C₃	—	3	36	no	—
75	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic
76	C₃	C₉×#737	1	1	yes	characteristic
77	C₃	#10624864	1	2	yes	—
78	C₃	#10624894	1	3	yes	—
79	C₃	#10624900	1	6	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #10624558	C₃	semidirect
4. #10624549	C₃	semidirect
5. C₃×#6855	C₃	semidirect
8. C₃×#737	C₉	semidirect
23. C₃×3₋¹⁺²	C₃×C₉	semidirect
32. C₃²×C₉	C₃×C₉	semidirect
59. C₃×C₉	#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
#10624558	3
#10624549	3
C₃×#6855	3