Subgroups — #14547214

Order 2187.

368 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6894	C₃	1	1	yes	maximal, characteristic
2	#10624768	C₃	1	2	yes	maximal
3	#10624744	C₃	1	3	yes	maximal
4	C₃×#6868	C₃	1	1	yes	maximal, characteristic
5	#10624600	C₃	1	6	yes	maximal
6	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
7	C₃³×C₉	C₃²	1	1	yes	characteristic
8	C₃×#733	—	3	9	no	—
9	#6894	—	3	3	no	—
10	#6894	—	3	6	no	—
11	#6891	C₃²	1	6	yes	—
12	C₃×C₉⋊C₉	—	3	2	no	—
13	C₃×C₉⋊C₉	—	3	1	no	—
14	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
15	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
16	#6868	—	3	4	no	—
17	#6868	—	3	2	no	—
18	#6868	—	3	2	no	—
19	#6868	—	3	1	no	—
20	#6861	—	3	18	no	—
21	C₃⁴	C₃³	1	1	yes	characteristic
22	C₃×3₋¹⁺²	—	3	2	no	—
23	C₃×3₋¹⁺²	—	3	6	no	—
24	C₃×3₋¹⁺²	—	9	3	no	—
25	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
26	C₃²×C₉	—	3	2	no	—
27	C₃²×C₉	—	3	1	no	—
28	C₃²×C₉	—	3	1	no	—
29	C₃²×C₉	—	3	2	no	—
30	C₃²×C₉	—	3	3	no	—
31	C₃²×C₉	—	3	3	no	—
32	C₃²×C₉	—	9	2	no	—
33	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
34	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
35	#733	—	9	27	no	—
36	C₉⋊C₉	—	3	9	no	—
37	C₉⋊C₉	—	3	18	no	—
38	C₃²⋊C₉	—	3	2	no	—
39	C₃²⋊C₉	—	3	2	no	—
40	C₃²⋊C₉	—	3	6	no	—
41	C₃²⋊C₉	—	3	12	no	—
42	C₃²⋊C₉	—	3	3	no	—
43	C₃²⋊C₉	—	3	6	no	—
44	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
45	C₃³	—	3	2	no	—
46	C₃³	—	3	1	no	—
47	C₃³	—	3	3	no	—
48	C₃³	—	9	2	no	—
49	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
50	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
51	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	3₋¹⁺²	—	3	9	no	—
53	3₋¹⁺²	—	9	6	no	—
54	3₋¹⁺²	—	9	18	no	—
55	C₃×C₉	—	3	2	no	—
56	C₃×C₉	—	3	1	no	—
57	C₃×C₉	—	3	4	no	—
58	C₃×C₉	—	3	6	no	—
59	C₃×C₉	—	3	2	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	3	2	no	—
62	C₃×C₉	—	9	3	no	—
63	C₃×C₉	—	9	6	no	—
64	C₃×C₉	—	9	6	no	—
65	C₃×C₉	—	9	3	no	—
66	C₃×C₉	—	9	3	no	—
67	C₃×C₉	—	9	6	no	—
68	C₃×C₉	—	9	9	no	—
69	C₃×C₉	#733	1	3	yes	complemented
70	C₃²	—	3	1	no	—
71	C₃²	—	3	1	no	—
72	C₃²	—	3	2	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	3	2	no	—
76	C₃²	—	3	4	no	—
77	C₃²	—	9	6	no	—
78	C₃²	—	9	3	no	—
79	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
80	C₃²	C₃×#733	1	1	yes	characteristic
81	C₃²	#6911	1	3	yes	—
82	C₃²	#6912	1	2	yes	—
83	C₉	—	3	3	no	—
84	C₉	—	9	3	no	—
85	C₉	—	9	2	no	—
86	C₉	—	9	6	no	—
87	C₉	—	27	3	no	—
88	C₉	—	27	6	no	—
89	C₃	—	3	2	no	—
90	C₃	—	3	1	no	—
91	C₃	—	3	3	no	—
92	C₃	—	9	2	no	—
93	C₃	#10624737	1	1	yes	characteristic
94	C₃	#10624744	1	1	yes	characteristic
95	C₃	#10624758	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
69. C₃×C₉	#733	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6894	3
#10624768	3
#10624744	3
C₃×#6868	3
#10624600	3