Subgroups — #14550770

Order 2187.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624872	C₃	1	2	yes	maximal
2	#10624688	C₃	1	2	yes	maximal
3	#10624470	C₃	1	4	yes	maximal, complemented
4	#10624470	C₃	1	4	yes	maximal, complemented
5	H(3,ℤ/9)	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	#6895	—	3	4	no	—
8	#6894	—	3	12	no	—
9	#6891	—	3	2	no	—
10	C₉×3₋¹⁺²	—	3	6	no	—
11	#6887	C₃²	1	2	yes	—
12	#6887	C₃²	1	4	yes	—
13	#6887	C₃²	1	4	yes	—
14	#6887	C₃²	1	2	yes	—
15	#6868	—	3	2	no	—
16	#6867	—	3	2	no	—
17	#6866	—	3	2	no	—
18	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃×3₋¹⁺²	—	3	6	no	—
20	C₃²×C₉	—	3	2	no	—
21	C₃²×C₉	—	3	4	no	—
22	C₃²×C₉	—	3	4	no	—
23	C₃²×C₉	—	3	2	no	—
24	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
25	C₃²×C₉	C₃³	1	1	yes	characteristic
26	C₉⋊C₉	—	3	36	no	—
27	C₉⋊C₉	—	3	4	no	—
28	C₉⋊C₉	—	3	4	no	—
29	C₉⋊C₉	—	3	4	no	—
30	C₉⋊C₉	—	3	8	no	—
31	C₉⋊C₉	3₋¹⁺²	1	12	yes	complemented
32	C₃²⋊C₉	—	3	12	no	—
33	C₃²⋊C₉	—	3	4	no	—
34	C₃²⋊C₉	—	3	2	no	—
35	C₃²⋊C₉	—	3	2	no	—
36	C₃²⋊C₉	—	3	4	no	—
37	C₉²	—	3	4	no	—
38	C₉²	—	3	2	no	—
39	C₉²	—	3	4	no	—
40	C₉²	3₋¹⁺²	1	6	yes	complemented
41	C₃³	—	3	2	no	—
42	C₃³	—	3	2	no	—
43	C₃³	—	9	3	no	—
44	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	3₋¹⁺²	—	9	18	no	—
46	C₃×C₉	—	3	12	no	—
47	C₃×C₉	—	3	6	no	—
48	C₃×C₉	—	3	4	no	—
49	C₃×C₉	—	3	4	no	—
50	C₃×C₉	—	9	4	no	—
51	C₃×C₉	—	9	4	no	—
52	C₃×C₉	—	9	1	no	—
53	C₃×C₉	—	9	2	no	—
54	C₃×C₉	—	9	4	no	—
55	C₃×C₉	C₃×3₋¹⁺²	1	4	yes	—
56	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
57	C₃×C₉	#737	1	4	yes	—
58	C₃×C₉	#737	1	2	yes	—
59	C₃²	—	3	6	no	—
60	C₃²	—	9	1	no	—
61	C₃²	—	9	6	no	—
62	C₃²	—	9	4	no	—
63	C₃²	—	9	1	no	—
64	C₃²	#6891	1	2	yes	—
65	C₃²	#6893	1	2	yes	—
66	C₉	—	9	12	no	—
67	C₉	—	9	6	no	—
68	C₉	—	9	2	no	—
69	C₉	—	27	2	no	—
70	C₉	—	27	4	no	—
71	C₉	#6896	1	4	yes	—
72	C₉	#6896	1	4	yes	—
73	C₉	#6896	1	1	yes	characteristic
74	C₃	—	3	2	no	—
75	C₃	—	3	2	no	—
76	C₃	—	9	3	no	—
77	C₃	#10624567	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#6896
commutator, Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃	#10624567
prime radical	C₃⁴	H(3,𝔽₃)
LCS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #10624470	C₃	semidirect
4. #10624470	C₃	semidirect
5. H(3,ℤ/9)	C₃	semidirect
31. C₉⋊C₉	3₋¹⁺²	semidirect
40. C₉²	3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
#10624872	3
#10624688	3
#10624470	3
H(3,ℤ/9)	3