Subgroups — #14550563

Order 2187.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624879	C₃	1	1	yes	maximal, characteristic
2	#10624706	C₃	1	2	yes	maximal, complemented
3	C₃×#6872	C₃	1	3	yes	maximal
4	C₃×#6864	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624469	C₃	1	6	yes	maximal, complemented
6	C₃×#737	C₃²	1	1	yes	characteristic
7	C₃×C₂₇⋊C₃	C₃²	1	6	yes	—
8	C₃²×C₂₇	C₃²	1	3	yes	—
9	#6900	—	3	1	no	—
10	#6899	—	3	2	no	—
11	#6896	—	3	3	no	—
12	#6894	C₉	1	6	yes	—
13	C₉⋊H(3,𝔽₃)	C₉	1	3	yes	—
14	C₉×3₋¹⁺²	—	3	3	no	—
15	#6887	C₃²	1	2	yes	—
16	C₃×C₉²	C₃²	1	1	yes	characteristic
17	C₉⋊C₂₇	—	3	6	no	—
18	#6872	—	3	9	no	—
19	#6864	—	3	3	no	—
20	#737	—	3	3	no	—
21	#737	—	3	6	no	—
22	C₃×3₋¹⁺²	—	3	3	no	—
23	C₃×3₋¹⁺²	—	3	1	no	—
24	C₃×3₋¹⁺²	—	3	1	no	—
25	C₃×3₋¹⁺²	C₃×C₉	1	2	yes	—
26	C₃×H(3,𝔽₃)	C₃×C₉	1	1	yes	characteristic
27	C₃²×C₉	—	3	1	no	—
28	C₃²×C₉	C₃×C₉	1	3	yes	—
29	C₃²×C₉	C₃³	1	1	yes	characteristic
30	C₂₇⋊C₃	—	3	6	no	—
31	C₂₇⋊C₃	—	3	12	no	—
32	C₃×C₂₇	—	3	3	no	—
33	C₃×C₂₇	—	3	6	no	—
34	C₃×C₂₇	—	3	6	no	—
35	C₃×C₂₇	3₋¹⁺²	1	9	yes	complemented
36	C₉⋊C₉	—	3	12	no	—
37	C₉⋊C₉	—	3	6	no	—
38	C₉⋊C₉	—	3	2	no	—
39	C₉⋊C₉	—	3	2	no	—
40	C₃²⋊C₉	—	3	6	no	—
41	C₃²⋊C₉	C₃×C₉	1	6	yes	—
42	C₉²	—	3	2	no	—
43	C₉²	H(3,𝔽₃)	1	3	yes	—
44	C₉²	3₋¹⁺²	1	3	yes	—
45	C₉²	3₋¹⁺²	1	2	yes	—
46	C₉²	3₋¹⁺²	1	1	yes	characteristic
47	C₃³	—	3	1	no	—
48	C₃³	C₃²×C₉	1	1	yes	characteristic
49	3₋¹⁺²	—	3	6	no	—
50	3₋¹⁺²	—	9	9	no	—
51	3₋¹⁺²	—	9	3	no	—
52	3₋¹⁺²	—	9	2	no	—
53	3₋¹⁺²	—	9	1	no	—
54	H(3,𝔽₃)	—	3	3	no	—
55	C₃×C₉	—	3	6	no	—
56	C₃×C₉	—	3	2	no	—
57	C₃×C₉	—	9	3	no	—
58	C₃×C₉	—	9	3	no	—
59	C₃×C₉	C₃²⋊C₉	1	6	yes	—
60	C₃×C₉	C₃²⋊C₉	1	3	yes	—
61	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
62	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
63	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
64	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
65	C₃×C₉	#737	1	6	yes	—
66	C₂₇	—	3	9	no	—
67	C₂₇	—	9	6	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	9	1	no	—
70	C₃²	—	9	3	no	—
71	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
72	C₃²	C₉×3₋¹⁺²	1	3	yes	—
73	C₉	—	3	6	no	—
74	C₉	—	3	3	no	—
75	C₉	—	9	2	no	—
76	C₉	—	9	6	no	—
77	C₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
78	C₉	#6896	1	3	yes	—
79	C₉	#6899	1	2	yes	—
80	C₉	#6900	1	1	yes	characteristic
81	C₃	—	3	3	no	—
82	C₃	—	27	1	no	—
83	C₃	#10624560	1	1	yes	characteristic
84	C₃	#10624702	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃×H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃²	C₃×C₃²⋊C₉
prime radical	C₃×H(3,𝔽₃)	C₃×C₉
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃×C₉² ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624706	C₃	semidirect
4. C₃×#6864	C₃	semidirect
5. #10624469	C₃	semidirect
35. 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
#10624879	3
#10624706	3
C₃×#6872	3
C₃×#6864	3
#10624469	3