Subgroups — #14550270

Order 2187.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉⋊C₂₇	C₃	1	2	yes	maximal
2	C₃×#6864	C₃	1	3	yes	maximal
3	C₃×C₉×C₂₇	C₃	1	1	yes	maximal, characteristic
4	C₉×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
5	#10624463	C₃	1	6	yes	maximal
6	C₃²×C₂₇	C₃²	1	2	yes	—
7	C₃²×C₂₇	C₃²	1	6	yes	—
8	C₃²×C₂₇	C₃²	1	1	yes	characteristic
9	C₃×C₉⋊C₉	C₉	1	3	yes	—
10	C₃×C₉²	C₃²	1	3	yes	—
11	C₃×C₉²	C₃²	1	1	yes	characteristic
12	C₉⋊C₂₇	C₉	1	18	yes	complemented
13	#6864	—	3	9	no	—
14	C₉×C₂₇	C₉	1	9	yes	complemented
15	#6855	C₉	1	6	yes	—
16	C₃²×C₉	C₃×C₉	1	9	yes	—
17	C₃²×C₉	C₃×C₉	1	2	yes	—
18	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
19	C₃²×C₉	C₃³	1	1	yes	characteristic
20	C₃×C₂₇	—	3	6	no	—
21	C₃×C₂₇	—	3	6	no	—
22	C₃×C₂₇	—	3	12	no	—
23	C₃×C₂₇	C₃×C₉	1	6	yes	—
24	C₃×C₂₇	C₃×C₉	1	18	yes	—
25	C₃×C₂₇	C₃×C₉	1	3	yes	—
26	C₃×C₂₇	3₋¹⁺²	1	9	yes	—
27	C₉⋊C₉	—	3	9	no	—
28	C₉²	—	3	3	no	—
29	C₉²	—	3	6	no	—
30	C₉²	C₃×C₉	1	3	yes	—
31	C₉²	C₃×C₉	1	6	yes	—
32	C₃³	C₃²×C₉	1	1	yes	characteristic
33	C₃×C₉	—	3	6	no	—
34	C₃×C₉	—	3	9	no	—
35	C₃×C₉	—	3	9	no	—
36	C₃×C₉	—	3	18	no	—
37	C₃×C₉	C₉²	1	6	yes	—
38	C₃×C₉	C₉²	1	3	yes	—
39	C₃×C₉	C₂₇⋊C₃	1	9	yes	—
40	C₃×C₉	C₃²×C₉	1	1	yes	characteristic
41	C₃×C₉	C₃²×C₉	1	2	yes	—
42	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
43	C₃×C₉	#737	1	6	yes	—
44	C₂₇	—	3	6	no	—
45	C₂₇	—	3	18	no	—
46	C₂₇	C₉⋊C₉	1	9	yes	complemented
47	C₃²	C₃×C₉²	1	1	yes	characteristic
48	C₃²	C₉×3₋¹⁺²	1	3	yes	—
49	C₃²	C₃×C₂₇⋊C₃	1	3	yes	—
50	C₃²	#6903	1	6	yes	—
51	C₉	—	3	6	no	—
52	C₉	—	9	9	no	—
53	C₉	#6864	1	9	yes	complemented
54	C₉	C₃×C₉⋊C₉	1	1	yes	characteristic
55	C₉	#6887	1	2	yes	—
56	C₉	C₉×3₋¹⁺²	1	6	yes	—
57	C₃	C₉×C₉⋊C₉	1	1	yes	characteristic
58	C₃	C₃×#6864	1	1	yes	characteristic
59	C₃	#10624681	1	2	yes	—
60	C₃	#10624729	1	9	yes	—

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

Complements

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

N	H ≅ G/N	type
12. C₉⋊C₂₇	C₉	semidirect
14. C₉×C₂₇	C₉	semidirect
46. C₂₇	C₉⋊C₉	semidirect
53. C₉	#6864	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉⋊C₂₇	3
C₃×#6864	3
C₃×C₉×C₂₇	3
C₉×C₉⋊C₉	3
#10624463	3