Subgroups — #14554232

Order 2187.

606 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	#10624727	C₃	1	27	yes	maximal, complemented
3	#10624711	C₃	1	9	yes	maximal, complemented
4	C₃×C₃²⋊C₂₇	C₃	1	3	yes	maximal
5	C₃²×3₋¹⁺²	C₉	1	1	yes	characteristic
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃²×C₂₇	C₃²	1	9	yes	—
8	C₉×3₋¹⁺²	C₉	1	18	yes	—
9	C₉×3₋¹⁺²	C₃²	1	9	yes	—
10	C₃×C₉⋊C₉	C₉	1	6	yes	—
11	C₃×C₃²⋊C₉	C₉	1	2	yes	—
12	C₃×C₉²	C₃²	1	3	yes	—
13	C₉⋊C₂₇	C₃²	1	54	yes	complemented
14	C₃²⋊C₂₇	C₃²	1	9	yes	—
15	C₃²⋊C₂₇	C₃²	1	18	yes	—
16	#6864	C₃²	1	27	yes	complemented
17	C₃⁴	C₃×C₉	1	1	yes	characteristic
18	C₃×3₋¹⁺²	—	3	9	no	—
19	C₃×3₋¹⁺²	C₃×C₉	1	9	yes	—
20	C₃²×C₉	—	9	3	no	—
21	C₃²×C₉	C₃×C₉	1	3	yes	—
22	C₃²×C₉	C₃×C₉	1	2	yes	—
23	C₃²×C₉	C₃×C₉	1	6	yes	—
24	C₃²×C₉	C₃×C₉	1	6	yes	—
25	C₃²×C₉	C₃³	1	1	yes	characteristic
26	C₃²×C₉	C₃³	1	3	yes	—
27	C₃×C₂₇	—	9	9	no	—
28	C₃×C₂₇	C₃³	1	27	yes	—
29	C₉⋊C₉	C₃×C₉	1	54	yes	—
30	C₃²⋊C₉	C₃×C₉	1	18	yes	—
31	C₉²	C₃×C₉	1	18	yes	—
32	C₉²	C₃³	1	9	yes	—
33	C₃³	—	9	1	no	—
34	C₃³	—	9	2	no	—
35	C₃³	C₂₇⋊C₃	1	9	yes	—
36	C₃³	C₃²×C₉	1	3	yes	—
37	C₃³	C₃²×C₉	1	1	yes	characteristic
38	3₋¹⁺²	—	3	27	no	—
39	C₃×C₉	—	3	9	no	—
40	C₃×C₉	—	3	3	no	—
41	C₃×C₉	—	3	3	no	—
42	C₃×C₉	—	3	6	no	—
43	C₃×C₉	—	9	2	no	—
44	C₃×C₉	—	9	6	no	—
45	C₃×C₉	—	9	6	no	—
46	C₃×C₉	C₃²×C₉	1	2	yes	—
47	C₃×C₉	C₃²×C₉	1	18	yes	—
48	C₃×C₉	C₃²×C₉	1	9	yes	—
49	C₃×C₉	C₃²×C₉	1	6	yes	—
50	C₃×C₉	C₃⁴	1	1	yes	characteristic
51	C₂₇	—	3	27	no	—
52	C₃²	—	3	3	no	—
53	C₃²	—	3	1	no	—
54	C₃²	—	3	2	no	—
55	C₃²	—	3	6	no	—
56	C₃²	—	9	9	no	—
57	C₃²	C₃×C₂₇⋊C₃	1	3	yes	—
58	C₃²	C₃×C₂₇⋊C₃	1	9	yes	—
59	C₃²	C₃³×C₉	1	1	yes	characteristic
60	C₉	—	3	6	no	—
61	C₉	—	3	2	no	—
62	C₉	—	3	9	no	—
63	C₉	—	3	18	no	—
64	C₉	3₋¹⁺⁴	1	3	yes	—
65	C₃	—	3	3	no	—
66	C₃	—	3	9	no	—
67	C₃	#10624863	1	1	yes	characteristic
68	C₃	C₃²×C₂₇⋊C₃	1	1	yes	characteristic
69	C₃	#10624918	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₃×C₉	C₃⁴
commutator, socle	C₃²	C₃³×C₉
prime radical	C₃⁴	C₃×C₉
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ G
LECS	G ▷ C₃×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624727	C₃	semidirect
3. #10624711	C₃	semidirect
13. C₉⋊C₂₇	C₃²	semidirect
16. #6864	C₃²	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×3₋¹⁺²	3
#10624727	3
#10624711	3
C₃×C₃²⋊C₂₇	3