Subgroups — #14554108

Order 2187.

1610 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#737	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₂₇⋊C₃	C₃	1	3	yes	maximal, complemented
3	#10624684	C₃	1	27	yes	maximal, complemented
4	#10624683	C₃	1	9	yes	maximal, complemented
5	C₃×#737	—	3	9	no	—
6	C₃×#737	—	3	18	no	—
7	C₃×#737	C₉	1	18	yes	—
8	C₃×#737	C₃²	1	9	yes	—
9	C₃²×3₋¹⁺²	C₉	1	2	yes	—
10	C₃²×3₋¹⁺²	C₉	1	6	yes	—
11	C₃²×H(3,𝔽₃)	C₉	1	1	yes	characteristic
12	C₃³×C₉	C₃²	1	3	yes	—
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	C₃×C₂₇⋊C₃	—	3	27	no	—
15	C₃×C₂₇⋊C₃	C₃²	1	9	yes	—
16	C₃×C₂₇⋊C₃	C₃²	1	18	yes	complemented
17	C₃²×C₂₇	C₃²	1	9	yes	complemented
18	C₃²⋊C₂₇	C₃²	1	81	yes	complemented
19	C₃⁴	C₃×C₉	1	3	yes	—
20	C₃⁴	C₃×C₉	1	1	yes	characteristic
21	#737	—	3	162	no	—
22	#737	—	3	27	no	—
23	#737	—	3	54	no	—
24	C₃×3₋¹⁺²	—	3	18	no	—
25	C₃×3₋¹⁺²	—	3	54	no	—
26	C₃×3₋¹⁺²	C₃×C₉	1	18	yes	—
27	C₃×3₋¹⁺²	C₃×C₉	1	54	yes	—
28	C₃×H(3,𝔽₃)	—	3	9	no	—
29	C₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	—
30	C₃²×C₉	—	3	9	no	—
31	C₃²×C₉	—	3	18	no	—
32	C₃²×C₉	C₃×C₉	1	6	yes	—
33	C₃²×C₉	C₃×C₉	1	18	yes	—
34	C₃²×C₉	C₃×C₉	1	2	yes	—
35	C₃²×C₉	C₃×C₉	1	6	yes	—
36	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
37	C₃²×C₉	3₋¹⁺²	1	18	yes	—
38	C₃²×C₉	C₃³	1	9	yes	—
39	C₃²×C₉	C₃³	1	1	yes	characteristic
40	C₃²×C₉	C₃³	1	3	yes	—
41	C₂₇⋊C₃	—	3	27	no	—
42	C₂₇⋊C₃	—	3	54	no	—
43	C₃×C₂₇	—	3	27	no	—
44	C₃×C₂₇	C₃³	1	27	yes	—
45	C₃³	—	3	9	no	—
46	C₃³	—	9	3	no	—
47	C₃³	—	9	9	no	—
48	C₃³	C₃²⋊C₉	1	9	yes	—
49	C₃³	C₃²×C₉	1	9	yes	—
50	C₃³	C₃²×C₉	1	1	yes	characteristic
51	C₃³	C₃²×C₉	1	3	yes	—
52	3₋¹⁺²	—	3	54	no	—
53	3₋¹⁺²	—	3	162	no	—
54	H(3,𝔽₃)	—	3	27	no	—
55	C₃×C₉	—	3	18	no	—
56	C₃×C₉	—	3	9	no	—
57	C₃×C₉	—	3	18	no	—
58	C₃×C₉	—	3	54	no	—
59	C₃×C₉	C₃²⋊C₉	1	18	yes	—
60	C₃×C₉	C₃²⋊C₉	1	54	yes	—
61	C₃×C₉	C₃²×C₉	1	18	yes	—
62	C₃×C₉	C₃²×C₉	1	2	yes	—
63	C₃×C₉	C₃²×C₉	1	6	yes	—
64	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
65	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
66	C₃×C₉	C₃×3₋¹⁺²	1	6	yes	—
67	C₃×C₉	C₃×3₋¹⁺²	1	18	yes	—
68	C₃×C₉	C₃⁴	1	1	yes	characteristic
69	C₂₇	—	3	27	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	9	no	—
72	C₃²	—	3	9	no	—
73	C₃²	—	3	27	no	—
74	C₃²	—	9	9	no	—
75	C₃²	—	9	27	no	—
76	C₃²	C₃×C₃²⋊C₉	1	3	yes	—
77	C₃²	C₃×C₃²⋊C₉	1	9	yes	—
78	C₃²	C₃³×C₉	1	1	yes	characteristic
79	C₉	—	3	18	no	—
80	C₉	C₃×C₃²⋊C₉	1	6	yes	—
81	C₉	C₃×C₃²⋊C₉	1	18	yes	—
82	C₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
83	C₉	C₃²×3₋¹⁺²	1	2	yes	—
84	C₃	—	3	3	no	—
85	C₃	—	3	9	no	—
86	C₃	—	9	9	no	—
87	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic
88	C₃	#10624918	1	3	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₃×C₉	C₃⁴
commutator, socle	C₃²	C₃³×C₉
prime radical	C₃²×H(3,𝔽₃)	C₉
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ G
LECS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₂₇⋊C₃	C₃	semidirect
3. #10624684	C₃	semidirect
4. #10624683	C₃	semidirect
16. C₃×C₂₇⋊C₃	C₃²	semidirect
17. C₃²×C₂₇	C₃²	semidirect
18. C₃²⋊C₂₇	C₃²	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#737	3
C₃²×C₂₇⋊C₃	3
#10624684	3
#10624683	3