Subgroups — #14550361

Order 2187.

178 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624709	C₃	1	3	yes	maximal
2	#10624681	C₃	1	1	yes	maximal, characteristic
3	#10624471	C₃	1	2	yes	maximal
4	#10624470	C₃	1	1	yes	maximal, characteristic
5	#10624469	C₃	1	3	yes	maximal
6	#10624465	C₃	1	3	yes	maximal
7	C₃×C₂₇⋊C₃	C₃²	1	6	yes	—
8	C₃×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
10	C₃²×C₂₇	C₃²	1	1	yes	characteristic
11	#6887	C₃²	1	3	yes	—
12	#6887	C₃²	1	1	yes	characteristic
13	#6875	—	3	9	no	—
14	#6864	C₉	1	6	yes	complemented
15	C₉×C₂₇	C₉	1	3	yes	complemented
16	C₃²×C₉	C₃³	1	1	yes	characteristic
17	C₂₇⋊C₃	—	3	3	no	—
18	C₂₇⋊C₃	—	3	6	no	—
19	C₂₇⋊C₃	—	3	6	no	—
20	C₂₇⋊C₃	—	3	6	no	—
21	C₂₇⋊C₃	3₋¹⁺²	1	9	yes	—
22	C₃×C₂₇	—	3	3	no	—
23	C₃×C₂₇	—	3	6	no	—
24	C₃×C₂₇	C₃×C₉	1	1	yes	characteristic
25	C₃×C₂₇	C₃×C₉	1	3	yes	—
26	C₃×C₂₇	C₃×C₉	1	2	yes	—
27	C₃×C₂₇	C₃×C₉	1	3	yes	—
28	C₉⋊C₉	—	3	2	no	—
29	C₉⋊C₉	—	3	3	no	—
30	C₉⋊C₉	—	3	3	no	—
31	C₃²⋊C₉	—	3	1	no	—
32	C₃²⋊C₉	—	3	3	no	—
33	C₉²	—	3	3	no	—
34	C₉²	C₃×C₉	1	3	yes	—
35	C₃³	#737	1	1	yes	characteristic
36	C₃×C₉	—	3	3	no	—
37	C₃×C₉	—	9	3	no	—
38	C₃×C₉	C₃²×C₉	1	1	yes	characteristic
39	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
40	C₃×C₉	#737	1	3	yes	—
41	C₃×C₉	#737	1	3	yes	—
42	C₃×C₉	#737	1	2	yes	—
43	C₂₇	—	3	3	no	—
44	C₂₇	—	3	1	no	—
45	C₂₇	—	3	2	no	—
46	C₂₇	—	9	6	no	—
47	C₂₇	C₉⋊C₉	1	9	yes	—
48	C₃²	—	9	1	no	—
49	C₃²	#6887	1	1	yes	characteristic
50	C₃²	#6899	1	3	yes	—
51	C₉	—	9	3	no	—
52	C₉	—	27	3	no	—
53	C₉	C₃×C₉⋊C₉	1	1	yes	characteristic
54	C₉	#6887	1	1	yes	characteristic
55	C₉	#6887	1	1	yes	characteristic
56	C₉	#6899	1	6	yes	—
57	C₃	—	3	1	no	—
58	C₃	—	3	3	no	—
59	C₃	#10624553	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#6887
commutator	C₃×C₉	C₃²×C₉
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃	#10624553
prime radical	C₃³	#737
LCS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉ ◁ C₃×C₂₇⋊C₃ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. #6864	C₉	semidirect
15. C₉×C₂₇	C₉	semidirect

Maximal subgroups

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

subgroup	index
#10624709	3
#10624681	3
#10624471	3
#10624470	3
#10624469	3
#10624465	3