Subgroups — #14550312

Order 2187.

509 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₉²	C₃	1	1	yes	maximal, characteristic
2	#10624706	C₃	1	3	yes	maximal, complemented
3	C₃×#6869	C₃	1	3	yes	maximal
4	#10624467	C₃	1	6	yes	maximal, complemented
5	C₃³×C₉	C₉	1	3	yes	—
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃×C₂₇⋊C₃	C₃²	1	9	yes	—
8	C₃×C₉²	—	3	3	no	—
9	C₃×C₉²	—	3	2	no	—
10	C₃×C₉²	—	3	3	no	—
11	C₃×C₉²	—	3	1	no	—
12	C₃×C₉²	C₉	1	6	yes	—
13	C₃×C₉²	C₃²	1	2	yes	—
14	C₃×C₉²	C₃²	1	1	yes	characteristic
15	C₉⋊C₂₇	—	3	9	no	—
16	#6869	—	3	9	no	—
17	C₃⁴	C₃×C₉	1	1	yes	characteristic
18	C₃²×C₉	—	3	6	no	—
19	C₃²×C₉	—	3	3	no	—
20	C₃²×C₉	—	3	1	no	—
21	C₃²×C₉	—	3	1	no	—
22	C₃²×C₉	—	3	1	no	—
23	C₃²×C₉	—	3	6	no	—
24	C₃²×C₉	—	3	27	no	—
25	C₃²×C₉	C₃×C₉	1	6	yes	—
26	C₃²×C₉	C₃×C₉	1	3	yes	—
27	C₃²×C₉	C₃×C₉	1	2	yes	—
28	C₃²×C₉	3₋¹⁺²	1	9	yes	—
29	C₃²×C₉	C₃³	1	1	yes	characteristic
30	C₂₇⋊C₃	—	3	9	no	—
31	C₂₇⋊C₃	—	3	18	no	—
32	C₃×C₂₇	—	3	9	no	—
33	C₉²	—	3	12	no	—
34	C₉²	—	3	6	no	—
35	C₉²	—	3	2	no	—
36	C₉²	—	3	2	no	—
37	C₉²	—	3	2	no	—
38	C₉²	H(3,𝔽₃)	1	3	yes	—
39	C₉²	3₋¹⁺²	1	2	yes	—
40	C₉²	3₋¹⁺²	1	3	yes	—
41	C₉²	3₋¹⁺²	1	1	yes	characteristic
42	C₃³	—	3	9	no	—
43	C₃³	—	3	1	no	—
44	C₃³	—	3	2	no	—
45	C₃³	C₉⋊C₉	1	3	yes	—
46	C₃³	C₃²×C₉	1	1	yes	characteristic
47	C₃×C₉	—	3	54	no	—
48	C₃×C₉	—	3	2	no	—
49	C₃×C₉	—	3	6	no	—
50	C₃×C₉	—	3	4	no	—
51	C₃×C₉	—	3	27	no	—
52	C₃×C₉	—	3	9	no	—
53	C₃×C₉	—	3	9	no	—
54	C₃×C₉	—	3	9	no	—
55	C₃×C₉	—	3	9	no	—
56	C₃×C₉	—	3	18	no	—
57	C₃×C₉	C₃²⋊C₉	1	6	yes	—
58	C₃×C₉	C₃²⋊C₉	1	3	yes	—
59	C₃×C₉	C₉⋊C₉	1	6	yes	—
60	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
62	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
63	C₃×C₉	C₃×3₋¹⁺²	1	1	yes	characteristic
64	C₃×C₉	#737	1	6	yes	—
65	C₂₇	—	9	9	no	—
66	C₃²	—	3	9	no	—
67	C₃²	—	3	18	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	9	no	—
70	C₃²	—	3	3	no	—
71	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
72	C₃²	C₃×C₉⋊C₉	1	1	yes	characteristic
73	C₃²	#6887	1	2	yes	—
74	C₉	—	3	6	no	—
75	C₉	—	3	18	no	—
76	C₉	—	3	6	no	—
77	C₉	#6869	1	9	yes	complemented
78	C₉	C₉⋊H(3,𝔽₃)	1	3	yes	—
79	C₉	#6894	1	3	yes	—
80	C₉	#6894	1	3	yes	—
81	C₃	—	3	9	no	—
82	C₃	—	3	1	no	—
83	C₃	—	3	2	no	—
84	C₃	#10624548	1	1	yes	characteristic
85	C₃	C₃×#6869	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 ◁ C₃×C₉² ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624706	C₃	semidirect
4. #10624467	C₃	semidirect
77. C₉	#6869	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₉²	3
#10624706	3
C₃×#6869	3
#10624467	3