Subgroups — #14550338

Order 2187.

311 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6887	C₃	1	1	yes	maximal, characteristic
2	#10624688	C₃	1	3	yes	maximal
3	#10624470	C₃	1	6	yes	maximal, complemented
4	H(3,ℤ/9)	C₃	1	3	yes	maximal, complemented
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	#6887	—	3	1	no	—
7	#6887	—	3	1	no	—
8	#6887	—	3	2	no	—
9	#6887	—	3	2	no	—
10	#6887	C₉	1	6	yes	complemented
11	#6887	C₃²	1	9	yes	—
12	#6887	C₃²	1	2	yes	—
13	#6887	C₃²	1	1	yes	characteristic
14	C₃×C₉⋊C₉	—	3	2	no	—
15	C₃×C₃²⋊C₉	—	3	1	no	—
16	C₃×C₉²	C₉	1	3	yes	complemented
17	#6868	—	3	3	no	—
18	#6867	—	3	3	no	—
19	#6866	—	3	3	no	—
20	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃²×C₉	—	3	6	no	—
22	C₃²×C₉	—	3	3	no	—
23	C₃²×C₉	—	3	3	no	—
24	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
25	C₃²×C₉	C₃×C₉	1	2	yes	—
26	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
27	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
28	C₃²×C₉	3₋¹⁺²	1	2	yes	—
29	C₃²×C₉	3₋¹⁺²	1	2	yes	—
30	C₃²×C₉	3₋¹⁺²	1	2	yes	—
31	C₃²×C₉	C₃³	1	1	yes	characteristic
32	C₉⋊C₉	—	3	6	no	—
33	C₉⋊C₉	—	3	6	no	—
34	C₉⋊C₉	—	3	4	no	—
35	C₉⋊C₉	—	3	2	no	—
36	C₉⋊C₉	—	3	18	no	—
37	C₃²⋊C₉	—	3	6	no	—
38	C₃²⋊C₉	—	3	2	no	—
39	C₃²⋊C₉	—	3	1	no	—
40	C₃²⋊C₉	—	3	9	no	—
41	C₉²	—	3	3	no	—
42	C₉²	—	3	3	no	—
43	C₉²	—	3	9	no	—
44	C₉²	C₃×C₉	1	6	yes	—
45	C₉²	C₃×C₉	1	3	yes	—
46	C₃³	—	3	3	no	—
47	C₃³	—	9	3	no	—
48	C₃³	C₃²⋊C₉	1	1	yes	characteristic
49	C₃³	C₃²⋊C₉	1	2	yes	—
50	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃×C₉	—	3	3	no	—
52	C₃×C₉	—	3	3	no	—
53	C₃×C₉	—	3	2	no	—
54	C₃×C₉	—	3	4	no	—
55	C₃×C₉	—	3	1	no	—
56	C₃×C₉	—	3	2	no	—
57	C₃×C₉	—	3	6	no	—
58	C₃×C₉	—	3	6	no	—
59	C₃×C₉	—	3	6	no	—
60	C₃×C₉	—	3	6	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	9	9	no	—
63	C₃×C₉	—	9	9	no	—
64	C₃×C₉	C₃²⋊C₉	1	1	yes	characteristic
65	C₃×C₉	C₃²⋊C₉	1	1	yes	characteristic
66	C₃×C₉	C₃²⋊C₉	1	2	yes	—
67	C₃×C₉	C₃²⋊C₉	1	2	yes	—
68	C₃×C₉	C₃²×C₉	1	1	yes	characteristic
69	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
70	C₃×C₉	#737	1	6	yes	—
71	C₃×C₉	#737	1	3	yes	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	1	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	9	1	no	—
76	C₃²	—	9	9	no	—
77	C₃²	—	9	2	no	—
78	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
79	C₃²	#6893	1	3	yes	—
80	C₉	—	3	1	no	—
81	C₉	—	3	1	no	—
82	C₉	—	3	2	no	—
83	C₉	—	3	2	no	—
84	C₉	—	9	6	no	—
85	C₉	—	9	3	no	—
86	C₉	—	27	9	no	—
87	C₉	#6887	1	2	yes	—
88	C₉	#6887	1	1	yes	characteristic
89	C₉	#6895	1	6	yes	—
90	C₃	—	3	1	no	—
91	C₃	—	3	3	no	—
92	C₃	—	3	3	no	—
93	C₃	—	9	2	no	—
94	C₃	#10624549	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#6887
commutator	C₃×C₉	C₃²×C₉
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃	#10624549
prime radical	C₃⁴	H(3,𝔽₃)
LCS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉ ◁ #6887 ◁ 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. H(3,ℤ/9)	C₃	semidirect
10. #6887	C₉	semidirect
16. C₃×C₉²	C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6887	3
#10624688	3
#10624470	3
H(3,ℤ/9)	3