Subgroups — #14550323

Order 2187.

530 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	C₃×C₃²⋊C₂₇	C₃	1	3	yes	maximal
3	#10624463	C₃	1	9	yes	maximal, complemented
4	C₃³×C₉	C₉	1	2	yes	—
5	C₃³×C₉	C₉	1	1	yes	characteristic
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃²×C₂₇	C₃²	1	9	yes	—
8	C₃×C₉²	—	3	3	no	—
9	C₃×C₉²	—	3	6	no	—
10	C₃×C₉²	C₉	1	6	yes	—
11	C₃×C₉²	C₃²	1	3	yes	—
12	C₃²⋊C₂₇	—	3	9	no	—
13	C₃⁴	C₃×C₉	1	1	yes	characteristic
14	C₃²×C₉	—	3	6	no	—
15	C₃²×C₉	—	3	3	no	—
16	C₃²×C₉	—	3	6	no	—
17	C₃²×C₉	—	3	18	no	—
18	C₃²×C₉	—	3	9	no	—
19	C₃²×C₉	C₂₇	1	9	yes	complemented
20	C₃²×C₉	C₃×C₉	1	2	yes	—
21	C₃²×C₉	C₃×C₉	1	3	yes	—
22	C₃²×C₉	C₃×C₉	1	6	yes	—
23	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
24	C₃²×C₉	3₋¹⁺²	1	6	yes	—
25	C₃²×C₉	C₃³	1	1	yes	characteristic
26	C₃×C₂₇	—	3	9	no	—
27	C₃×C₂₇	—	3	27	no	—
28	C₉²	—	3	18	no	—
29	C₉²	—	3	3	no	—
30	C₉²	—	3	6	no	—
31	C₃³	—	3	3	no	—
32	C₃³	—	3	6	no	—
33	C₃³	C₃²⋊C₉	1	3	yes	—
34	C₃³	C₃×C₂₇	1	3	yes	—
35	C₃³	C₂₇⋊C₃	1	6	yes	—
36	C₃³	C₃²×C₉	1	1	yes	characteristic
37	C₃×C₉	—	3	54	no	—
38	C₃×C₉	—	3	27	no	—
39	C₃×C₉	—	3	3	no	—
40	C₃×C₉	—	3	6	no	—
41	C₃×C₉	—	3	6	no	—
42	C₃×C₉	—	3	6	no	—
43	C₃×C₉	—	3	12	no	—
44	C₃×C₉	—	3	9	no	—
45	C₃×C₉	—	3	18	no	—
46	C₃×C₉	C₃²⋊C₉	1	6	yes	—
47	C₃×C₉	C₃×C₂₇	1	9	yes	—
48	C₃×C₉	C₂₇⋊C₃	1	18	yes	—
49	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
50	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
51	C₃×C₉	#737	1	9	yes	—
52	C₂₇	—	9	9	no	—
53	C₃²	—	3	3	no	—
54	C₃²	—	3	6	no	—
55	C₃²	—	3	9	no	—
56	C₃²	—	3	18	no	—
57	C₃²	C₃²⋊C₂₇	1	9	yes	—
58	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
59	C₃²	#6887	1	3	yes	—
60	C₃²	C₃²×C₂₇	1	1	yes	characteristic
61	C₃²	C₃×C₂₇⋊C₃	1	2	yes	—
62	C₃²	C₃×C₂₇⋊C₃	1	6	yes	—
63	C₉	—	3	6	no	—
64	C₉	—	3	9	no	—
65	C₉	—	3	18	no	—
66	C₉	#6893	1	3	yes	—
67	C₉	#6895	1	6	yes	—
68	C₃	—	3	9	no	—
69	C₃	#10624549	1	1	yes	characteristic
70	C₃	C₃×C₃²⋊C₂₇	1	1	yes	characteristic
71	C₃	#10624683	1	2	yes	—
72	C₃	#10624710	1	3	yes	—
73	C₃	#10624711	1	6	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₃²×C₉	C₃³
commutator	C₃²	C₃²×C₂₇
socle	C₃³	C₃²×C₉
prime radical	C₃⁴	C₃×C₉
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #10624463	C₃	semidirect
19. C₃²×C₉	C₂₇	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
C₃×C₃²⋊C₂₇	3
#10624463	3