Subgroups — #14550847

Order 2187.

362 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#732	C₃	1	1	yes	maximal, characteristic
2	C₃×#6868	C₃	1	2	yes	maximal, complemented
3	#10624549	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624489	C₃	1	9	yes	maximal, complemented
5	C₃²×H(3,𝔽₃)	—	3	1	no	—
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃×#732	—	3	9	no	—
8	C₃×#732	—	3	2	no	—
9	C₃×#732	C₃²	1	3	yes	—
10	C₃×C₃²⋊C₉	—	3	2	no	—
11	C₃×C₃²⋊C₉	—	3	1	no	—
12	#6868	—	3	4	no	—
13	#6868	C₃²	1	6	yes	complemented
14	#6855	—	3	2	no	—
15	#6855	C₃²	1	3	yes	complemented
16	C₃⁴	—	9	1	no	—
17	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃×H(3,𝔽₃)	—	3	9	no	—
19	C₃×H(3,𝔽₃)	—	3	3	no	—
20	C₃×H(3,𝔽₃)	—	9	6	no	—
21	C₃×H(3,𝔽₃)	—	9	2	no	—
22	C₃²×C₉	—	3	3	no	—
23	C₃²×C₉	—	3	6	no	—
24	C₃²×C₉	—	3	1	no	—
25	C₃²×C₉	—	3	2	no	—
26	C₃²×C₉	—	3	3	no	—
27	C₃²×C₉	—	9	2	no	—
28	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
29	C₃²×C₉	C₃³	1	1	yes	characteristic
30	#732	—	3	9	no	—
31	#732	—	3	18	no	—
32	C₃²⋊C₉	—	3	6	no	—
33	C₃²⋊C₉	—	9	3	no	—
34	C₃²⋊C₉	—	9	2	no	—
35	C₃²⋊C₉	—	9	2	no	—
36	C₃³	—	9	1	no	—
37	C₃³	—	9	2	no	—
38	C₃³	—	9	9	no	—
39	C₃³	—	9	3	no	—
40	C₃³	—	27	3	no	—
41	C₃³	—	27	6	no	—
42	C₃³	AΣL(1,𝔽₂₇)	1	3	yes	—
43	C₃³	#731	1	6	yes	—
44	C₃³	#732	1	1	yes	characteristic
45	C₃³	#732	1	2	yes	—
46	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
47	H(3,𝔽₃)	—	3	9	no	—
48	H(3,𝔽₃)	—	9	18	no	—
49	H(3,𝔽₃)	—	9	6	no	—
50	C₃×C₉	—	3	2	no	—
51	C₃×C₉	—	3	1	no	—
52	C₃×C₉	—	3	4	no	—
53	C₃×C₉	—	3	2	no	—
54	C₃×C₉	—	3	2	no	—
55	C₃×C₉	—	3	1	no	—
56	C₃×C₉	—	9	6	no	—
57	C₃×C₉	—	9	3	no	—
58	C₃×C₉	—	9	3	no	—
59	C₃×C₉	—	9	6	no	—
60	C₃×C₉	—	9	3	no	—
61	C₃×C₉	—	9	6	no	—
62	C₃×C₉	—	27	6	no	—
63	C₃²	—	3	1	no	—
64	C₃²	—	3	2	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	3	2	no	—
67	C₃²	—	3	2	no	—
68	C₃²	—	3	4	no	—
69	C₃²	—	9	3	no	—
70	C₃²	—	9	6	no	—
71	C₃²	—	9	9	no	—
72	C₃²	—	27	27	no	—
73	C₃²	—	27	3	no	—
74	C₃²	—	27	6	no	—
75	C₃²	#6859	1	6	yes	—
76	C₃²	#6859	1	3	yes	—
77	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
78	C₃²	C₃×#731	1	2	yes	—
79	C₃²	C₃×#732	1	1	yes	characteristic
80	C₉	—	9	2	no	—
81	C₉	—	9	1	no	—
82	C₉	—	27	3	no	—
83	C₉	—	27	6	no	—
84	C₃	—	3	1	no	—
85	C₃	—	3	3	no	—
86	C₃	—	3	6	no	—
87	C₃	—	3	2	no	—
88	C₃	—	27	9	no	—
89	C₃	C₃×#6859	1	1	yes	characteristic
90	C₃	#10624839	1	1	yes	characteristic
91	C₃	#10624848	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃×#732
commutator, Frattini, cosocle	C₃²×C₉	C₃³
prime radical	C₃²×#732	C₃
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃×#6868	C₃	semidirect
3. #10624549	C₃	semidirect
4. #10624489	C₃	semidirect
13. #6868	C₃²	semidirect
15. #6855	C₃²	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#732	3
C₃×#6868	3
#10624549	3
#10624489	3