Subgroups — #14550873

Order 2187.

205 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#732	C₃	1	3	yes	maximal, complemented
2	#10624550	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624494	C₃	1	3	yes	maximal, complemented
4	#10624489	C₃	1	3	yes	maximal, complemented
5	#10624480	C₃	1	3	yes	maximal, complemented
6	C₃×#732	C₃²	1	3	yes	—
7	C₉×H(3,𝔽₃)	—	3	3	no	—
8	C₃×C₉⋊C₉	—	3	2	no	—
9	C₃×C₉²	C₃²	1	1	yes	characteristic
10	H(3,𝔽₃)⋊C₉	—	3	6	no	—
11	#6868	C₃²	1	3	yes	—
12	#6868	C₃²	1	3	yes	—
13	#6855	—	3	1	no	—
14	#6855	C₃²	1	3	yes	—
15	C₃×H(3,𝔽₃)	—	3	3	no	—
16	C₃²×C₉	—	3	3	no	—
17	C₃²×C₉	—	9	3	no	—
18	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
19	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃²×C₉	C₃³	1	1	yes	characteristic
21	#732	—	3	9	no	—
22	C₉⋊C₉	—	9	6	no	—
23	C₃²⋊C₉	—	3	3	no	—
24	C₃²⋊C₉	—	3	3	no	—
25	C₃²⋊C₉	—	9	6	no	—
26	C₉²	—	3	3	no	—
27	C₃³	—	9	3	no	—
28	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
29	H(3,𝔽₃)	—	3	9	no	—
30	H(3,𝔽₃)	—	9	6	no	—
31	C₃×C₉	—	3	2	no	—
32	C₃×C₉	—	3	6	no	—
33	C₃×C₉	—	3	1	no	—
34	C₃×C₉	—	3	3	no	—
35	C₃×C₉	—	9	1	no	—
36	C₃×C₉	—	9	6	no	—
37	C₃×C₉	—	9	3	no	—
38	C₃×C₉	—	9	3	no	—
39	C₃×C₉	—	9	3	no	—
40	C₃×C₉	—	9	3	no	—
41	C₃×C₉	—	27	9	no	—
42	C₃×C₉	#732	1	3	yes	—
43	C₃²	—	3	1	no	—
44	C₃²	—	3	2	no	—
45	C₃²	—	9	9	no	—
46	C₃²	—	27	3	no	—
47	C₃²	—	27	6	no	—
48	C₃²	C₃×#732	1	1	yes	characteristic
49	C₃²	#6908	1	3	yes	—
50	C₉	—	3	3	no	—
51	C₉	—	3	6	no	—
52	C₉	—	9	1	no	—
53	C₉	—	27	3	no	—
54	C₉	—	27	3	no	—
55	C₉	—	27	3	no	—
56	C₃	—	3	3	no	—
57	C₃	—	27	9	no	—
58	C₃	#10624585	1	1	yes	characteristic
59	C₃	#10624839	1	1	yes	characteristic
60	C₃	#10624850	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃×#732
commutator, Frattini, cosocle	C₃²×C₉	C₃³
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ C₃×C₉² ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₉×#732	C₃	semidirect
2. #10624550	C₃	semidirect
3. #10624494	C₃	semidirect
4. #10624489	C₃	semidirect
5. #10624480	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₉×#732	3
#10624550	3
#10624494	3
#10624489	3
#10624480	3