Subgroups — #14550260

Order 2187.

374 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#737	C₃	1	1	yes	maximal, characteristic
2	C₃×#6873	C₃	1	3	yes	maximal
3	C₉×C₂₇⋊C₃	C₃	1	2	yes	maximal, complemented
4	C₃×#6864	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624468	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	C₉×3₋¹⁺²	—	3	6	no	—
10	C₉×3₋¹⁺²	C₉	1	6	yes	—
11	C₉×H(3,𝔽₃)	C₉	1	3	yes	—
12	#6887	—	3	2	no	—
13	#6887	C₃²	1	2	yes	—
14	C₃×C₉²	—	3	1	no	—
15	C₃×C₉²	C₃²	1	1	yes	characteristic
16	#6873	C₉	1	27	yes	complemented
17	#6864	—	3	2	no	—
18	#6864	—	3	1	no	—
19	#6864	—	3	2	no	—
20	#6864	—	3	2	no	—
21	C₉×C₂₇	—	3	2	no	—
22	#737	—	3	6	no	—
23	#737	C₃×C₉	1	9	yes	—
24	C₃×3₋¹⁺²	—	3	2	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	—	3	3	no	—
29	C₃²×C₉	C₃×C₉	1	3	yes	—
30	C₃²×C₉	C₃³	1	1	yes	characteristic
31	C₂₇⋊C₃	—	3	12	no	—
32	C₂₇⋊C₃	C₃×C₉	1	18	yes	complemented
33	C₃×C₂₇	—	3	6	no	—
34	C₃×C₂₇	—	3	3	no	—
35	C₃×C₂₇	—	3	6	no	—
36	C₃×C₂₇	C₃×C₉	1	9	yes	complemented
37	C₉⋊C₉	—	3	12	no	—
38	C₉⋊C₉	—	3	4	no	—
39	C₃²⋊C₉	—	3	6	no	—
40	C₃²⋊C₉	C₃×C₉	1	6	yes	—
41	C₉²	—	3	6	no	—
42	C₉²	—	3	2	no	—
43	C₉²	H(3,𝔽₃)	1	2	yes	—
44	C₉²	H(3,𝔽₃)	1	1	yes	characteristic
45	C₉²	3₋¹⁺²	1	6	yes	—
46	C₃³	—	3	1	no	—
47	C₃³	C₃²×C₉	1	1	yes	characteristic
48	3₋¹⁺²	—	3	4	no	—
49	3₋¹⁺²	—	3	18	no	—
50	3₋¹⁺²	C₉²	1	6	yes	—
51	H(3,𝔽₃)	—	3	2	no	—
52	H(3,𝔽₃)	C₉²	1	3	yes	—
53	C₃×C₉	—	3	2	no	—
54	C₃×C₉	—	3	9	no	—
55	C₃×C₉	—	3	6	no	—
56	C₃×C₉	—	9	3	no	—
57	C₃×C₉	—	9	9	no	—
58	C₃×C₉	C₃²⋊C₉	1	3	yes	—
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	2	yes	—
63	C₃×C₉	#737	1	6	yes	—
64	C₂₇	—	3	9	no	—
65	C₂₇	—	3	18	no	—
66	C₃²	—	3	3	no	—
67	C₃²	—	3	3	no	—
68	C₃²	—	9	3	no	—
69	C₃²	C₃×C₉²	1	1	yes	characteristic
70	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
71	C₃²	#6887	1	2	yes	—
72	C₉	—	3	6	no	—
73	C₉	—	3	3	no	—
74	C₉	—	3	6	no	—
75	C₉	—	9	6	no	—
76	C₉	C₉×H(3,𝔽₃)	1	3	yes	—
77	C₉	C₉×3₋¹⁺²	1	6	yes	—
78	C₃	—	3	1	no	—
79	C₃	—	3	2	no	—
80	C₃	—	9	3	no	—
81	C₃	C₉×C₃²⋊C₉	1	1	yes	characteristic
82	C₃	#10624704	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃×H(3,𝔽₃)
commutator	C₃²	C₃×C₉²
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃²	C₃×C₃²⋊C₉
prime radical	C₃×H(3,𝔽₃)	C₃×C₉
LCS	G ▷ C₃² ▷ 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
3. C₉×C₂₇⋊C₃	C₃	semidirect
4. C₃×#6864	C₃	semidirect
5. #10624468	C₃	semidirect
16. #6873	C₉	semidirect
32. C₂₇⋊C₃	C₃×C₉	semidirect
36. C₃×C₂₇	C₃×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₉×#737	3
C₃×#6873	3
C₉×C₂₇⋊C₃	3
C₃×#6864	3
#10624468	3