Subgroups — #14554482

Order 2187.

773 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624883	C₃	1	2	yes	maximal
2	C₃×#6893	C₃	1	1	yes	maximal, characteristic
3	#10624862	C₃	1	1	yes	maximal, characteristic
4	#10624807	C₃	1	18	yes	maximal, complemented
5	#10624806	C₃	1	9	yes	maximal, complemented
6	#10624691	C₃	1	6	yes	maximal, complemented
7	C₃×#6868	C₃	1	3	yes	maximal, complemented
8	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
9	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
10	C₃³×C₉	C₃²	1	1	yes	characteristic
11	#6893	—	3	9	no	—
12	C₉⋊H(3,𝔽₃)	—	3	18	no	—
13	#6891	C₃²	1	18	yes	—
14	C₉×H(3,𝔽₃)	—	3	9	no	—
15	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
16	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
17	C₃×C₃²⋊C₉	C₃²	1	9	yes	—
18	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
19	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
20	#6868	C₃²	1	9	yes	—
21	#6868	C₃²	1	54	yes	—
22	#6868	C₃²	1	18	yes	—
23	C₃⁴	C₃³	1	3	yes	—
24	C₃⁴	C₃³	1	1	yes	characteristic
25	C₃×3₋¹⁺²	—	3	6	no	—
26	C₃×3₋¹⁺²	—	3	18	no	—
27	C₃×3₋¹⁺²	—	3	18	no	—
28	C₃×H(3,𝔽₃)	—	3	9	no	—
29	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
30	C₃²×C₉	—	3	3	no	—
31	C₃²×C₉	—	3	9	no	—
32	C₃²×C₉	—	3	9	no	—
33	C₃²×C₉	—	3	3	no	—
34	C₃²×C₉	—	3	6	no	—
35	C₃²×C₉	C₃³	1	1	yes	characteristic
36	C₃²×C₉	C₃³	1	2	yes	—
37	C₃²×C₉	C₃³	1	6	yes	—
38	C₃²⋊C₉	—	3	18	no	—
39	C₃²⋊C₉	—	3	6	no	—
40	C₃²⋊C₉	—	3	12	no	—
41	C₃²⋊C₉	—	3	9	no	—
42	C₃²⋊C₉	—	3	18	no	—
43	C₃²⋊C₉	—	3	36	no	—
44	C₃²⋊C₉	C₃³	1	9	yes	—
45	C₃²⋊C₉	C₃³	1	18	yes	—
46	C₃³	—	3	9	no	—
47	C₃³	—	9	1	no	—
48	C₃³	—	9	2	no	—
49	C₃³	—	9	3	no	—
50	C₃³	—	9	6	no	—
51	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
52	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
53	C₃³	#737	1	9	yes	—
54	C₃³	C₃⁴	1	1	yes	characteristic
55	3₋¹⁺²	—	9	18	no	—
56	3₋¹⁺²	—	27	18	no	—
57	H(3,𝔽₃)	—	3	27	no	—
58	C₃×C₉	—	3	1	no	—
59	C₃×C₉	—	3	2	no	—
60	C₃×C₉	—	3	6	no	—
61	C₃×C₉	—	3	9	no	—
62	C₃×C₉	—	3	18	no	—
63	C₃×C₉	—	3	3	no	—
64	C₃×C₉	—	3	6	no	—
65	C₃×C₉	—	3	18	no	—
66	C₃×C₉	—	9	9	no	—
67	C₃×C₉	—	27	3	no	—
68	C₃×C₉	—	27	9	no	—
69	C₃×C₉	—	27	6	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	2	no	—
73	C₃²	—	3	6	no	—
74	C₃²	—	3	9	no	—
75	C₃²	—	3	9	no	—
76	C₃²	—	3	18	no	—
77	C₃²	—	9	9	no	—
78	C₃²	—	27	9	no	—
79	C₃²	#6893	1	9	yes	—
80	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
81	C₃²	C₃×#737	1	3	yes	—
82	C₉	—	9	1	no	—
83	C₉	—	9	2	no	—
84	C₉	—	9	6	no	—
85	C₉	—	9	9	no	—
86	C₉	—	9	18	no	—
87	C₃	—	3	3	no	—
88	C₃	—	3	9	no	—
89	C₃	—	9	9	no	—
90	C₃	C₃×#6893	1	1	yes	characteristic
91	C₃	#10624932	1	1	yes	characteristic
92	C₃	#10624933	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #10624807	C₃	semidirect
5. #10624806	C₃	semidirect
6. #10624691	C₃	semidirect
7. C₃×#6868	C₃	semidirect

Maximal subgroups

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

subgroup	index
#10624883	3
C₃×#6893	3
#10624862	3
#10624807	3
#10624806	3
#10624691	3
C₃×#6868	3