Subgroups — #14550736

Order 2187.

258 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624887	C₃	1	1	yes	maximal, characteristic
2	#10624709	C₃	1	3	yes	maximal, complemented
3	#10624704	C₃	1	3	yes	maximal
4	#10624472	C₃	1	3	yes	maximal, complemented
5	#10624465	C₃	1	2	yes	maximal, complemented
6	#10624465	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#737	C₃²	1	1	yes	characteristic
8	C₃×C₂₇⋊C₃	C₃²	1	6	yes	—
9	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
10	#6900	—	3	1	no	—
11	#6899	—	3	2	no	—
12	#6896	—	3	3	no	—
13	#6894	—	3	2	no	—
14	#6894	—	3	3	no	—
15	C₉⋊H(3,𝔽₃)	—	3	1	no	—
16	#6887	C₃²	1	1	yes	characteristic
17	#6887	C₃²	1	1	yes	characteristic
18	C₃×C₉⋊C₉	C₃²	1	1	yes	characteristic
19	#6875	—	3	9	no	—
20	#6873	—	3	9	no	—
21	#737	—	3	3	no	—
22	#737	—	3	6	no	—
23	C₃×3₋¹⁺²	—	3	3	no	—
24	C₃×3₋¹⁺²	—	3	1	no	—
25	C₃×3₋¹⁺²	—	3	1	no	—
26	C₃×3₋¹⁺²	3₋¹⁺²	1	2	yes	—
27	C₃×H(3,𝔽₃)	3₋¹⁺²	1	1	yes	characteristic
28	C₃²×C₉	—	3	1	no	—
29	C₃²×C₉	—	3	1	no	—
30	C₃²×C₉	C₃³	1	1	yes	characteristic
31	C₂₇⋊C₃	—	3	3	no	—
32	C₂₇⋊C₃	—	3	6	no	—
33	C₂₇⋊C₃	—	3	6	no	—
34	C₂₇⋊C₃	—	3	6	no	—
35	C₂₇⋊C₃	—	3	6	no	—
36	C₃×C₂₇	—	3	6	no	—
37	C₃×C₂₇	3₋¹⁺²	1	9	yes	complemented
38	C₉⋊C₉	—	3	12	no	—
39	C₉⋊C₉	—	3	6	no	—
40	C₉⋊C₉	—	3	1	no	—
41	C₉⋊C₉	—	3	1	no	—
42	C₉⋊C₉	—	3	1	no	—
43	C₉⋊C₉	—	3	1	no	—
44	C₉⋊C₉	—	3	1	no	—
45	C₉⋊C₉	H(3,𝔽₃)	1	3	yes	—
46	C₉⋊C₉	3₋¹⁺²	1	2	yes	—
47	C₉⋊C₉	3₋¹⁺²	1	1	yes	characteristic
48	C₃²⋊C₉	—	3	6	no	—
49	C₃²⋊C₉	—	3	1	no	—
50	C₃²⋊C₉	—	3	1	no	—
51	C₉²	—	3	1	no	—
52	C₉²	3₋¹⁺²	1	3	yes	—
53	C₃³	—	3	1	no	—
54	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
55	3₋¹⁺²	—	3	6	no	—
56	3₋¹⁺²	—	9	3	no	—
57	3₋¹⁺²	—	9	2	no	—
58	3₋¹⁺²	—	9	1	no	—
59	3₋¹⁺²	—	27	3	no	—
60	H(3,𝔽₃)	—	3	3	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	3	3	no	—
63	C₃×C₉	—	3	2	no	—
64	C₃×C₉	—	3	2	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	9	3	no	—
67	C₃×C₉	—	27	1	no	—
68	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
69	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
70	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
71	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
72	C₃×C₉	#737	1	6	yes	—
73	C₂₇	—	3	9	no	—
74	C₂₇	—	9	6	no	—
75	C₃²	—	9	1	no	—
76	C₃²	—	9	3	no	—
77	C₃²	—	9	1	no	—
78	C₃²	#6891	1	1	yes	characteristic
79	C₃²	#6894	1	3	yes	—
80	C₉	—	9	2	no	—
81	C₉	—	9	1	no	—
82	C₉	—	9	3	no	—
83	C₉	—	9	3	no	—
84	C₉	—	9	2	no	—
85	C₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
86	C₉	#6896	1	3	yes	—
87	C₉	#6899	1	2	yes	—
88	C₉	#6900	1	1	yes	characteristic
89	C₃	—	3	1	no	—
90	C₃	—	3	3	no	—
91	C₃	—	27	1	no	—
92	C₃	#10624565	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	#10624565
commutator, Frattini, cosocle	C₃²×C₉	C₃³
prime radical	C₃×H(3,𝔽₃)	3₋¹⁺²
LCS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ 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
2. #10624709	C₃	semidirect
4. #10624472	C₃	semidirect
5. #10624465	C₃	semidirect
6. #10624465	C₃	semidirect
37. C₃×C₂₇	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
#10624887	3
#10624709	3
#10624704	3
#10624472	3
#10624465	3