Subgroups — #14548110

Order 2187.

813 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624765	C₃	1	2	yes	maximal, complemented
3	#10624730	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624632	C₃	1	3	yes	maximal, complemented
5	#10624630	C₃	1	3	yes	maximal, complemented
6	#10624628	C₃	1	3	yes	maximal, complemented
7	C₃⁵	C₃²	1	1	yes	characteristic
8	C₃²×H(3,𝔽₃)	—	3	3	no	—
9	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
10	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
11	#6891	C₃²	1	6	yes	—
12	#6891	C₃²	1	1	yes	characteristic
13	#6891	C₃²	1	1	yes	characteristic
14	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
15	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
16	#6866	—	3	6	no	—
17	#6859	—	3	9	no	—
18	#6857	—	3	9	no	—
19	#6856	—	3	9	no	—
20	C₃⁴	—	3	3	no	—
21	C₃⁴	—	3	3	no	—
22	C₃⁴	—	9	3	no	—
23	C₃⁴	—	9	9	no	—
24	C₃⁴	H(3,𝔽₃)	1	3	yes	—
25	C₃⁴	C₃³	1	1	yes	characteristic
26	C₃×3₋¹⁺²	—	3	1	no	—
27	C₃×3₋¹⁺²	—	3	1	no	—
28	C₃×3₋¹⁺²	—	3	6	no	—
29	C₃×H(3,𝔽₃)	—	3	27	no	—
30	C₃×H(3,𝔽₃)	—	3	6	no	—
31	C₃×H(3,𝔽₃)	—	3	1	no	—
32	C₃×H(3,𝔽₃)	—	3	3	no	—
33	C₃×H(3,𝔽₃)	—	9	9	no	—
34	C₃×H(3,𝔽₃)	—	9	27	no	—
35	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
36	AΣL(1,𝔽₂₇)	—	9	9	no	—
37	C₃²⋊C₉	—	3	6	no	—
38	C₃²⋊C₉	—	3	3	no	—
39	C₃²⋊C₉	—	3	6	no	—
40	C₃²⋊C₉	—	3	3	no	—
41	C₃²⋊C₉	—	3	6	no	—
42	C₃³	—	3	9	no	—
43	C₃³	—	3	3	no	—
44	C₃³	—	3	3	no	—
45	C₃³	—	3	9	no	—
46	C₃³	—	3	3	no	—
47	C₃³	—	3	6	no	—
48	C₃³	—	9	3	no	—
49	C₃³	—	9	9	no	—
50	C₃³	—	9	18	no	—
51	C₃³	—	9	6	no	—
52	C₃³	—	9	9	no	—
53	C₃³	—	9	27	no	—
54	C₃³	—	9	81	no	—
55	C₃³	—	9	9	no	—
56	C₃³	—	9	1	no	—
57	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
58	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	3₋¹⁺²	—	9	3	no	—
60	3₋¹⁺²	—	9	3	no	—
61	3₋¹⁺²	—	27	6	no	—
62	H(3,𝔽₃)	—	9	9	no	—
63	H(3,𝔽₃)	—	9	18	no	—
64	H(3,𝔽₃)	—	9	81	no	—
65	H(3,𝔽₃)	—	9	3	no	—
66	H(3,𝔽₃)	—	27	3	no	—
67	C₃×C₉	—	9	1	no	—
68	C₃×C₉	—	9	1	no	—
69	C₃×C₉	—	9	6	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	9	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	3	6	no	—
75	C₃²	—	9	9	no	—
76	C₃²	—	9	18	no	—
77	C₃²	—	9	9	no	—
78	C₃²	—	9	81	no	—
79	C₃²	—	9	3	no	—
80	C₃²	—	9	6	no	—
81	C₃²	—	9	9	no	—
82	C₃²	—	9	27	no	—
83	C₃²	—	27	27	no	—
84	C₃²	—	27	1	no	—
85	C₃²	—	27	3	no	—
86	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
87	C₃²	#6909	1	3	yes	—
88	C₉	—	27	1	no	—
89	C₉	—	27	1	no	—
90	C₉	—	27	6	no	—
91	C₃	—	3	3	no	—
92	C₃	—	9	3	no	—
93	C₃	—	9	9	no	—
94	C₃	—	27	9	no	—
95	C₃	—	81	1	no	—
96	C₃	#10624735	1	1	yes	characteristic
97	C₃	#10624754	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
2. #10624765	C₃	semidirect
3. #10624730	C₃	semidirect
4. #10624632	C₃	semidirect
5. #10624630	C₃	semidirect
6. #10624628	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊H(3,𝔽₃)	3
#10624765	3
#10624730	3
#10624632	3
#10624630	3
#10624628	3