Subgroups — #14546340

Order 2187.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈₁⋊C₃	C₃	1	2	yes	maximal
2	C₃²×C₈₁	C₃	1	1	yes	maximal, characteristic
3	#10624681	C₃	1	1	yes	maximal, characteristic
4	C₃×C₂₇⋊C₃	C₉	1	2	yes	—
5	C₃²×C₂₇	C₃²	1	1	yes	characteristic
6	#6887	C₉	1	1	yes	characteristic
7	C₈₁⋊C₃	—	3	4	no	—
8	C₈₁⋊C₃	C₉	1	6	yes	complemented
9	C₃×C₈₁	—	3	1	no	—
10	C₃×C₈₁	—	3	2	no	—
11	C₃×C₈₁	—	3	2	no	—
12	C₃×C₈₁	C₉	1	3	yes	complemented
13	#6864	—	3	2	no	—
14	C₉×C₂₇	—	3	1	no	—
15	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
16	C₂₇⋊C₃	—	3	4	no	—
17	C₂₇⋊C₃	C₂₇	1	6	yes	—
18	C₃×C₂₇	—	3	2	no	—
19	C₃×C₂₇	C₃×C₉	1	1	yes	characteristic
20	C₃×C₂₇	C₃×C₉	1	2	yes	—
21	C₃×C₂₇	H(3,𝔽₃)	1	1	yes	characteristic
22	C₃×C₂₇	3₋¹⁺²	1	2	yes	—
23	C₃×C₂₇	3₋¹⁺²	1	4	yes	—
24	C₃×C₂₇	3₋¹⁺²	1	2	yes	—
25	C₉⋊C₉	—	3	2	no	—
26	C₃²⋊C₉	C₂₇	1	3	yes	—
27	C₉²	—	3	1	no	—
28	C₈₁	—	3	3	no	—
29	C₈₁	—	3	6	no	—
30	C₃³	C₃×C₂₇	1	1	yes	characteristic
31	C₃×C₉	—	3	3	no	—
32	C₃×C₉	C₉²	1	1	yes	characteristic
33	C₃×C₉	C₃²⋊C₉	1	1	yes	characteristic
34	C₃×C₉	C₉⋊C₉	1	2	yes	—
35	C₃×C₉	C₃×C₂₇	1	2	yes	—
36	C₃×C₉	C₂₇⋊C₃	1	4	yes	—
37	C₃×C₉	C₂₇⋊C₃	1	2	yes	—
38	C₂₇	—	3	6	no	—
39	C₂₇	C₃²⋊C₉	1	1	yes	characteristic
40	C₂₇	C₃²⋊C₉	1	2	yes	—
41	C₂₇	C₉⋊C₉	1	2	yes	—
42	C₂₇	C₉⋊C₉	1	4	yes	—
43	C₃²	—	3	3	no	—
44	C₃²	C₉×C₂₇	1	1	yes	characteristic
45	C₃²	C₃²⋊C₂₇	1	1	yes	characteristic
46	C₃²	C₉⋊C₂₇	1	2	yes	—
47	C₉	—	9	3	no	—
48	C₉	#6855	1	1	yes	characteristic
49	C₉	#6864	1	2	yes	—
50	C₉	C₃²⋊C₂₇	1	2	yes	—
51	C₉	C₉⋊C₂₇	1	4	yes	—
52	C₃	—	3	1	no	—
53	C₃	—	3	2	no	—
54	C₃	#10624463	1	1	yes	characteristic
55	C₃	#10624503	1	3	yes	—

	subgroup	G/H
center	C₃×C₂₇	H(3,𝔽₃)
commutator	C₃²	C₉×C₂₇
Frattini, cosocle	C₃²×C₂₇	C₃²
socle	C₃²	C₃²⋊C₂₇
prime radical	C₃³	C₃×C₂₇
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₂₇ ◁ C₃²×C₂₇ ◁ G
LECS	G ▷ C₃²×C₂₇ ▷ C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. C₈₁⋊C₃	C₉	semidirect
12. C₃×C₈₁	C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₈₁⋊C₃	3
C₃²×C₈₁	3
#10624681	3