Subgroups — #14550857

Order 2187.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6868	C₃	1	1	yes	maximal, characteristic
2	C₃×#6868	C₃	1	2	yes	maximal
3	#10624549	C₃	1	1	yes	maximal, characteristic
4	#10624493	C₃	1	9	yes	maximal, complemented
5	C₃³×C₉	C₃²	1	1	yes	characteristic
6	C₃×C₃²⋊C₉	—	3	1	no	—
7	C₃×C₃²⋊C₉	—	3	2	no	—
8	C₃×C₃²⋊C₉	—	3	1	no	—
9	#6868	—	3	2	no	—
10	#6868	—	3	4	no	—
11	#6868	C₃²	1	3	yes	—
12	#6868	C₃²	1	6	yes	—
13	#6855	—	3	2	no	—
14	#6855	C₃²	1	3	yes	—
15	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃²×C₉	—	3	3	no	—
17	C₃²×C₉	—	3	1	no	—
18	C₃²×C₉	—	3	2	no	—
19	C₃²×C₉	—	3	6	no	—
20	C₃²×C₉	—	3	3	no	—
21	C₃²×C₉	—	9	1	no	—
22	C₃²×C₉	—	9	2	no	—
23	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
24	C₃²×C₉	C₃³	1	1	yes	characteristic
25	C₃²⋊C₉	—	3	3	no	—
26	C₃²⋊C₉	—	3	6	no	—
27	C₃²⋊C₉	—	9	3	no	—
28	C₃²⋊C₉	—	9	2	no	—
29	C₃²⋊C₉	—	9	2	no	—
30	C₃²⋊C₉	—	9	2	no	—
31	C₃³	—	9	1	no	—
32	C₃³	—	9	1	no	—
33	C₃³	—	9	1	no	—
34	C₃³	#731	1	3	yes	—
35	C₃³	#732	1	1	yes	characteristic
36	C₃³	#732	1	2	yes	—
37	C₃³	#733	1	6	yes	—
38	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₃×C₉	—	3	2	no	—
40	C₃×C₉	—	3	1	no	—
41	C₃×C₉	—	3	2	no	—
42	C₃×C₉	—	3	1	no	—
43	C₃×C₉	—	3	4	no	—
44	C₃×C₉	—	3	2	no	—
45	C₃×C₉	—	9	3	no	—
46	C₃×C₉	—	9	6	no	—
47	C₃×C₉	—	9	3	no	—
48	C₃×C₉	—	9	3	no	—
49	C₃×C₉	—	9	3	no	—
50	C₃×C₉	—	9	3	no	—
51	C₃×C₉	—	9	3	no	—
52	C₃×C₉	—	9	6	no	—
53	C₃×C₉	—	27	3	no	—
54	C₃×C₉	—	27	6	no	—
55	C₃²	—	3	1	no	—
56	C₃²	—	3	1	no	—
57	C₃²	—	3	2	no	—
58	C₃²	—	3	1	no	—
59	C₃²	—	3	2	no	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	1	no	—
62	C₃²	—	3	1	no	—
63	C₃²	—	3	2	no	—
64	C₃²	—	9	3	no	—
65	C₃²	—	9	6	no	—
66	C₃²	#6861	1	3	yes	—
67	C₃²	#6861	1	6	yes	—
68	C₃²	C₃×#731	1	1	yes	characteristic
69	C₃²	C₃×#732	1	1	yes	characteristic
70	C₃²	C₃×#733	1	2	yes	—
71	C₉	—	9	2	no	—
72	C₉	—	9	1	no	—
73	C₉	—	27	3	no	—
74	C₉	—	27	3	no	—
75	C₉	—	27	6	no	—
76	C₃	—	3	1	no	—
77	C₃	—	3	3	no	—
78	C₃	—	3	6	no	—
79	C₃	—	3	2	no	—
80	C₃	C₃×#6861	1	1	yes	characteristic
81	C₃	#10624838	1	1	yes	characteristic
82	C₃	#10624846	1	1	yes	characteristic
83	C₃	#10624847	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃×#732
commutator, Frattini, cosocle	C₃²×C₉	C₃³
prime radical	C₃⁴	H(3,𝔽₃)
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
4. #10624493	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6868	3
#10624549	3
#10624493	3