Subgroups — #14552002

Order 2187.

145 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624837	C₃	1	3	yes	maximal, complemented
2	C₉×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624541	C₃	1	3	yes	maximal, complemented
4	#10624540	C₃	1	3	yes	maximal, complemented
5	#10624539	C₃	1	3	yes	maximal, complemented
6	#6908	—	3	3	no	—
7	C₃×C₂₇⋊C₃	—	3	1	no	—
8	C₃×C₉²	C₃²	1	1	yes	characteristic
9	#6880	C₃²	1	3	yes	—
10	#6880	C₃²	1	3	yes	—
11	#6879	C₃²	1	3	yes	—
12	#6878	—	3	3	no	—
13	#6878	—	3	3	no	—
14	#6864	—	3	1	no	—
15	#6864	C₃²	1	3	yes	—
16	C₉×C₂₇	—	3	1	no	—
17	#737	—	9	3	no	—
18	C₃²×C₉	—	3	1	no	—
19	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
20	#733	—	3	3	no	—
21	#733	—	3	3	no	—
22	#732	—	3	3	no	—
23	#731	—	9	3	no	—
24	AΣL(1,𝔽₂₇)	—	9	3	no	—
25	C₂₇⋊C₃	—	3	9	no	—
26	C₃×C₂₇	—	3	3	no	—
27	C₉²	—	3	1	no	—
28	C₉²	—	3	1	no	—
29	C₉²	H(3,𝔽₃)	1	1	yes	characteristic
30	C₉²	H(3,𝔽₃)	1	1	yes	characteristic
31	C₉²	C₃³	1	1	yes	characteristic
32	C₃³	#732	1	1	yes	characteristic
33	3₋¹⁺²	—	9	3	no	—
34	3₋¹⁺²	—	9	3	no	—
35	3₋¹⁺²	—	27	6	no	—
36	H(3,𝔽₃)	—	9	3	no	—
37	C₃×C₉	—	3	1	no	—
38	C₃×C₉	—	3	3	no	—
39	C₃×C₉	—	3	1	no	—
40	C₃×C₉	—	3	1	no	—
41	C₃×C₉	—	9	3	no	—
42	C₃×C₉	—	27	3	no	—
43	C₃×C₉	#732	1	1	yes	characteristic
44	C₃×C₉	#732	1	1	yes	characteristic
45	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
46	C₂₇	—	3	9	no	—
47	C₃²	—	3	1	no	—
48	C₃²	—	9	1	no	—
49	C₃²	—	27	3	no	—
50	C₃²	C₃×#732	1	1	yes	characteristic
51	C₉	—	3	3	no	—
52	C₉	—	3	1	no	—
53	C₉	—	3	1	no	—
54	C₉	—	9	1	no	—
55	C₉	—	9	1	no	—
56	C₉	—	27	3	no	—
57	C₉	—	27	3	no	—
58	C₉	#6879	1	3	yes	—
59	C₃	—	3	1	no	—
60	C₃	—	9	1	no	—
61	C₃	—	81	3	no	—
62	C₃	C₃×#6879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	C₃×#6879
commutator, Frattini, cosocle	C₉²	C₃³
LCS	G ▷ C₉² ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ C₃²×C₉ ◁ C₃×C₉² ◁ G
LECS	G ▷ C₉² ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #10624837	C₃	semidirect
2. C₉×C₂₇⋊C₃	C₃	semidirect
3. #10624541	C₃	semidirect
4. #10624540	C₃	semidirect
5. #10624539	C₃	semidirect

Maximal subgroups

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

subgroup	index
#10624837	3
C₉×C₂₇⋊C₃	3
#10624541	3
#10624540	3
#10624539	3