Subgroups — C₃²×#6873 #14554133

Order 2187.

1628 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	2	yes	maximal, complemented
3	C₃³×C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#6873	C₃	1	36	yes	maximal, direct product factor
5	C₃×#737	—	3	8	no	—
6	C₃×#737	—	3	16	no	—
7	C₃×#737	C₉	1	24	yes	—
8	C₃×#737	C₃²	1	12	yes	—
9	C₃²×3₋¹⁺²	—	3	2	no	—
10	C₃²×3₋¹⁺²	C₉	1	2	yes	—
11	C₃²×H(3,𝔽₃)	C₉	1	1	yes	characteristic
12	C₃³×C₉	—	3	1	no	—
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	C₃×C₂₇⋊C₃	—	3	16	no	—
15	C₃×C₂₇⋊C₃	C₃²	1	24	yes	complemented
16	C₃²×C₂₇	—	3	8	no	—
17	C₃²×C₂₇	—	3	1	no	—
18	C₃²×C₂₇	—	3	2	no	—
19	C₃²×C₂₇	C₃²	1	12	yes	complemented
20	#6873	C₃²	1	81	yes	direct product factor
21	C₃⁴	—	3	1	no	—
22	C₃⁴	C₃×C₉	1	1	yes	characteristic
23	#737	—	3	144	no	—
24	#737	—	3	24	no	—
25	#737	—	3	48	no	—
26	#737	C₃×C₉	1	72	yes	—
27	#737	C₃³	1	9	yes	—
28	C₃×3₋¹⁺²	—	3	16	no	—
29	C₃×3₋¹⁺²	—	3	48	no	—
30	C₃×3₋¹⁺²	—	3	24	no	—
31	C₃×3₋¹⁺²	C₃×C₉	1	24	yes	—
32	C₃×H(3,𝔽₃)	—	3	8	no	—
33	C₃×H(3,𝔽₃)	C₃×C₉	1	12	yes	—
34	C₃²×C₉	—	3	2	no	—
35	C₃²×C₉	—	3	24	no	—
36	C₃²×C₉	—	3	12	no	—
37	C₃²×C₉	C₃×C₉	1	8	yes	—
38	C₃²×C₉	H(3,𝔽₃)	1	8	yes	—
39	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
40	C₃²×C₉	3₋¹⁺²	1	2	yes	—
41	C₃²×C₉	3₋¹⁺²	1	16	yes	—
42	C₃²×C₉	C₃³	1	4	yes	—
43	C₂₇⋊C₃	—	3	48	no	—
44	C₂₇⋊C₃	C₃³	1	18	yes	complemented
45	C₃×C₂₇	—	3	12	no	—
46	C₃×C₂₇	—	3	24	no	—
47	C₃×C₂₇	—	3	24	no	—
48	C₃×C₂₇	C₃³	1	9	yes	complemented
49	C₃³	—	3	9	no	—
50	C₃³	—	3	12	no	—
51	C₃³	—	9	9	no	—
52	C₃³	C₃²⋊C₉	1	8	yes	—
53	C₃³	C₃²⋊C₉	1	1	yes	characteristic
54	C₃³	C₃²×C₉	1	4	yes	—
55	3₋¹⁺²	—	3	48	no	—
56	3₋¹⁺²	—	3	18	no	—
57	3₋¹⁺²	—	3	144	no	—
58	3₋¹⁺²	C₃²×C₉	1	18	yes	—
59	H(3,𝔽₃)	—	3	24	no	—
60	H(3,𝔽₃)	C₃²×C₉	1	9	yes	—
61	C₃×C₉	—	3	24	no	—
62	C₃×C₉	—	3	9	no	—
63	C₃×C₉	—	3	72	no	—
64	C₃×C₉	C₃²⋊C₉	1	8	yes	—
65	C₃×C₉	C₃²⋊C₉	1	16	yes	—
66	C₃×C₉	C₃²⋊C₉	1	48	yes	—
67	C₃×C₉	C₃²×C₉	1	8	yes	—
68	C₃×C₉	C₃×H(3,𝔽₃)	1	4	yes	—
69	C₃×C₉	C₃×H(3,𝔽₃)	1	8	yes	—
70	C₃×C₉	C₃×3₋¹⁺²	1	8	yes	—
71	C₃×C₉	C₃×3₋¹⁺²	1	16	yes	—
72	C₃×C₉	C₃⁴	1	1	yes	characteristic
73	C₂₇	—	3	9	no	—
74	C₂₇	—	3	18	no	—
75	C₃²	—	3	24	no	—
76	C₃²	—	3	12	no	—
77	C₃²	—	3	9	no	—
78	C₃²	—	9	36	no	—
79	C₃²	#6873	1	9	yes	direct product factor
80	C₃²	C₃×C₃²⋊C₉	1	8	yes	—
81	C₃²	C₃×C₃²⋊C₉	1	4	yes	—
82	C₃²	C₃³×C₉	1	1	yes	characteristic
83	C₉	—	3	18	no	—
84	C₉	C₃×C₃²⋊C₉	1	8	yes	—
85	C₉	C₃×C₃²⋊C₉	1	16	yes	—
86	C₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
87	C₉	C₃²×3₋¹⁺²	1	2	yes	—
88	C₃	—	3	1	no	—
89	C₃	—	3	8	no	—
90	C₃	—	9	9	no	—
91	C₃	C₃×#6873	1	12	yes	direct product factor
92	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₉	H(3,𝔽₃)
commutator	C₃²	C₃³×C₉
Frattini, cosocle	C₃×C₉	C₃⁴
socle	C₃³	C₃²⋊C₉
prime radical	C₃²×H(3,𝔽₃)	C₉
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₉ ◁ C₃³×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. C₃³×C₂₇	C₃	semidirect
4. C₃×#6873	C₃	direct (N × H)
15. C₃×C₂₇⋊C₃	C₃²	semidirect
19. C₃²×C₂₇	C₃²	semidirect
20. #6873	C₃²	direct (N × H)
44. C₂₇⋊C₃	C₃³	semidirect
48. C₃×C₂₇	C₃³	semidirect
79. C₃²	#6873	direct (N × H)
91. C₃	C₃×#6873	direct (N × H)

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
C₃³×C₂₇	3
C₃×#6873	3

Subgroups — C32×#6873 #14554133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#6873 #14554133