Subgroups — #14550301

Order 2187.

1377 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	#10624698	C₃	1	8	yes	maximal, complemented
3	C₃×H(3,𝔽₃)⋊C₉	C₃	1	4	yes	maximal, complemented
4	H(5,𝔽₃)	—	3	24	no	—
5	H(5,𝔽₃)	C₉	1	9	yes	complemented
6	C₃²×H(3,𝔽₃)	—	3	12	no	—
7	C₃²×H(3,𝔽₃)	C₃²	1	4	yes	—
8	C₃³×C₉	C₃²	1	1	yes	characteristic, complemented
9	C₃×C₃²⋊C₉	C₃²	1	8	yes	complemented
10	H(3,𝔽₃)⋊C₉	—	3	12	no	—
11	H(3,𝔽₃)⋊C₉	—	3	24	no	—
12	C₃⁴	—	3	9	no	—
13	C₃⁴	—	3	4	no	—
14	C₃⁴	C₃³	1	1	yes	characteristic
15	C₃×H(3,𝔽₃)	—	3	18	no	—
16	C₃×H(3,𝔽₃)	—	3	216	no	—
17	C₃×H(3,𝔽₃)	—	3	8	no	—
18	C₃×H(3,𝔽₃)	—	3	16	no	—
19	C₃×H(3,𝔽₃)	—	3	8	no	—
20	C₃×H(3,𝔽₃)	—	3	108	no	—
21	C₃×H(3,𝔽₃)	C₃×C₉	1	12	yes	—
22	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	12	yes	—
23	C₃×H(3,𝔽₃)	3₋¹⁺²	1	24	yes	—
24	C₃²×C₉	—	3	4	no	—
25	C₃²×C₉	—	3	8	no	—
26	C₃²×C₉	—	9	3	no	—
27	C₃²⋊C₉	—	3	8	no	—
28	C₃²⋊C₉	—	3	16	no	—
29	C₃³	—	3	81	no	—
30	C₃³	—	3	12	no	—
31	C₃³	—	3	12	no	—
32	C₃³	—	3	24	no	—
33	C₃³	—	9	1	no	—
34	C₃³	—	9	2	no	—
35	C₃³	—	27	4	no	—
36	C₃³	—	27	4	no	—
37	C₃³	—	27	4	no	—
38	C₃³	—	27	18	no	—
39	C₃³	—	27	9	no	—
40	C₃³	C₃²×C₉	1	1	yes	characteristic
41	C₃³	C₃×H(3,𝔽₃)	1	4	yes	—
42	C₃³	C₃×3₋¹⁺²	1	8	yes	—
43	H(3,𝔽₃)	—	3	162	no	—
44	H(3,𝔽₃)	—	3	24	no	—
45	H(3,𝔽₃)	—	3	24	no	—
46	H(3,𝔽₃)	—	3	48	no	—
47	H(3,𝔽₃)	C₃²⋊C₉	1	36	yes	—
48	C₃×C₉	—	9	1	no	—
49	C₃×C₉	—	9	12	no	—
50	C₃×C₉	—	9	8	no	—
51	C₃×C₉	—	27	8	no	—
52	C₃²	—	3	4	no	—
53	C₃²	—	3	4	no	—
54	C₃²	—	3	4	no	—
55	C₃²	—	3	36	no	—
56	C₃²	—	9	1	no	—
57	C₃²	—	9	8	no	—
58	C₃²	—	9	12	no	—
59	C₃²	—	9	12	no	—
60	C₃²	—	9	12	no	—
61	C₃²	—	27	12	no	—
62	C₃²	—	27	24	no	—
63	C₃²	—	27	81	no	—
64	C₃²	C₃×C₃²⋊C₉	1	4	yes	—
65	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
66	C₃²	#6891	1	8	yes	—
67	C₉	—	9	3	no	—
68	C₉	—	27	8	no	—
69	C₃	—	3	4	no	—
70	C₃	—	3	8	no	—
71	C₃	—	9	36	no	—
72	C₃	#10624547	1	1	yes	characteristic
73	C₃	#10624760	1	2	yes	—
74	C₃	#10624761	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃×H(5,𝔽₃)	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624698	C₃	semidirect
3. C₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
5. H(5,𝔽₃)	C₉	semidirect
8. C₃³×C₉	C₃²	semidirect
9. C₃×C₃²⋊C₉	C₃²	semidirect

Maximal subgroups

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

subgroup	index
C₃×H(5,𝔽₃)	3
#10624698	3
C₃×H(3,𝔽₃)⋊C₉	3