Subgroups — #14550533

Order 2187.

356 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	#10624872	C₃	1	1	yes	maximal, characteristic
3	#10624683	C₃	1	2	yes	maximal
4	#10624469	C₃	1	9	yes	maximal, complemented
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₃×C₂₇⋊C₃	—	3	3	no	—
7	C₃×C₂₇⋊C₃	—	3	9	no	—
8	C₃×C₂₇⋊C₃	C₃²	1	6	yes	—
9	C₃²×C₂₇	C₃²	1	3	yes	—
10	#6895	C₉	1	6	yes	—
11	#6894	—	3	6	no	—
12	#6891	C₉	1	3	yes	—
13	C₉×3₋¹⁺²	—	3	3	no	—
14	#6887	C₃²	1	3	yes	—
15	C₃²⋊C₂₇	—	3	6	no	—
16	C₃⁴	C₃×C₉	1	1	yes	characteristic
17	C₃×3₋¹⁺²	—	3	3	no	—
18	C₃²×C₉	—	3	3	no	—
19	C₃²×C₉	—	3	6	no	—
20	C₃²×C₉	C₃×C₉	1	2	yes	—
21	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
22	C₃²×C₉	3₋¹⁺²	1	6	yes	—
23	C₃²×C₉	C₃³	1	1	yes	characteristic
24	C₂₇⋊C₃	—	3	27	no	—
25	C₂₇⋊C₃	—	3	18	no	—
26	C₃×C₂₇	—	3	3	no	—
27	C₃×C₂₇	—	3	6	no	—
28	C₃×C₂₇	—	3	6	no	—
29	C₃×C₂₇	3₋¹⁺²	1	9	yes	complemented
30	C₉⋊C₉	—	3	18	no	—
31	C₉⋊C₉	—	3	6	no	—
32	C₃²⋊C₉	—	3	6	no	—
33	C₃²⋊C₉	C₃×C₉	1	9	yes	—
34	C₉²	—	3	3	no	—
35	C₃³	—	3	3	no	—
36	C₃³	—	9	3	no	—
37	C₃³	C₃²⋊C₉	1	3	yes	—
38	C₃³	C₃²×C₉	1	1	yes	characteristic
39	3₋¹⁺²	—	9	9	no	—
40	C₃×C₉	—	3	6	no	—
41	C₃×C₉	—	3	3	no	—
42	C₃×C₉	—	3	6	no	—
43	C₃×C₉	—	3	18	no	—
44	C₃×C₉	—	3	9	no	—
45	C₃×C₉	C₃²⋊C₉	1	6	yes	—
46	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
48	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
49	C₃×C₉	#737	1	6	yes	—
50	C₂₇	—	3	9	no	—
51	C₂₇	—	9	6	no	—
52	C₃²	—	3	3	no	—
53	C₃²	—	3	3	no	—
54	C₃²	—	3	3	no	—
55	C₃²	—	3	6	no	—
56	C₃²	—	9	9	no	—
57	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
58	C₃²	C₉×3₋¹⁺²	1	3	yes	—
59	C₉	—	3	6	no	—
60	C₉	—	9	9	no	—
61	C₉	#6891	1	1	yes	characteristic
62	C₉	#6893	1	2	yes	—
63	C₉	#6894	1	6	yes	—
64	C₃	—	3	3	no	—
65	C₃	—	9	3	no	—
66	C₃	#10624558	1	1	yes	characteristic
67	C₃	#10624702	1	3	yes	—

	subgroup	G/H
center	C₃×C₉	C₃×H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃²	C₃×C₃²⋊C₉
prime radical	C₃⁴	C₃×C₉
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₉ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #10624469	C₃	semidirect
29. C₃×C₂₇	3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂₇⋊C₃	3
#10624872	3
#10624683	3
#10624469	3