Subgroups — #14550522

Order 2187.

440 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃²⋊C₂₇	C₃	1	2	yes	maximal
4	#10624463	C₃	1	9	yes	maximal, complemented
5	C₃²×3₋¹⁺²	C₉	1	1	yes	characteristic
6	C₃³×C₉	C₃²	1	1	yes	characteristic
7	C₃×C₂₇⋊C₃	—	3	3	no	—
8	C₃×C₂₇⋊C₃	—	3	9	no	—
9	C₃²×C₂₇	C₃²	1	6	yes	—
10	C₃²×C₂₇	C₃²	1	3	yes	—
11	C₉×3₋¹⁺²	—	3	3	no	—
12	C₉×3₋¹⁺²	—	3	6	no	—
13	C₃×C₉⋊C₉	C₉	1	6	yes	—
14	C₃×C₃²⋊C₉	C₉	1	2	yes	—
15	C₃×C₉²	C₃²	1	3	yes	—
16	C₃²⋊C₂₇	—	3	6	no	—
17	C₃⁴	C₃×C₉	1	1	yes	characteristic
18	C₃×3₋¹⁺²	—	3	3	no	—
19	C₃×3₋¹⁺²	—	3	9	no	—
20	C₃²×C₉	—	3	3	no	—
21	C₃²×C₉	—	3	6	no	—
22	C₃²×C₉	C₃×C₉	1	6	yes	—
23	C₃²×C₉	C₃×C₉	1	2	yes	—
24	C₃²×C₉	C₃×C₉	1	3	yes	—
25	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
26	C₃²×C₉	3₋¹⁺²	1	6	yes	—
27	C₃²×C₉	C₃³	1	1	yes	characteristic
28	C₂₇⋊C₃	—	3	27	no	—
29	C₃×C₂₇	—	3	3	no	—
30	C₃×C₂₇	—	3	6	no	—
31	C₃×C₂₇	—	3	6	no	—
32	C₃×C₂₇	—	3	18	no	—
33	C₃×C₂₇	3₋¹⁺²	1	9	yes	complemented
34	C₉⋊C₉	—	3	18	no	—
35	C₃²⋊C₉	—	3	6	no	—
36	C₉²	—	3	3	no	—
37	C₉²	—	3	6	no	—
38	C₃³	—	3	3	no	—
39	C₃³	—	3	3	no	—
40	C₃³	—	3	6	no	—
41	C₃³	C₃²⋊C₉	1	3	yes	—
42	C₃³	C₃²×C₉	1	1	yes	characteristic
43	3₋¹⁺²	—	3	27	no	—
44	C₃×C₉	—	3	3	no	—
45	C₃×C₉	—	3	6	no	—
46	C₃×C₉	—	3	6	no	—
47	C₃×C₉	—	3	3	no	—
48	C₃×C₉	—	3	6	no	—
49	C₃×C₉	—	3	18	no	—
50	C₃×C₉	—	3	6	no	—
51	C₃×C₉	—	3	18	no	—
52	C₃×C₉	C₃²⋊C₉	1	6	yes	—
53	C₃×C₉	C₂₇⋊C₃	1	9	yes	complemented
54	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
55	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
56	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
57	C₃×C₉	#737	1	6	yes	—
58	C₂₇	—	3	9	no	—
59	C₂₇	—	9	6	no	—
60	C₃²	—	3	3	no	—
61	C₃²	—	3	3	no	—
62	C₃²	—	3	6	no	—
63	C₃²	—	9	9	no	—
64	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
65	C₃²	C₉×3₋¹⁺²	1	3	yes	—
66	C₃²	C₃×C₂₇⋊C₃	1	3	yes	—
67	C₃²	#6903	1	6	yes	—
68	C₉	—	3	6	no	—
69	C₉	—	3	9	no	—
70	C₉	—	9	6	no	—
71	C₉	#6891	1	1	yes	characteristic
72	C₉	#6893	1	2	yes	—
73	C₉	#6894	1	6	yes	—
74	C₃	—	9	3	no	—
75	C₃	#10624558	1	1	yes	characteristic
76	C₃	#10624683	1	1	yes	characteristic
77	C₃	#10624684	1	2	yes	—
78	C₃	#10624729	1	9	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₃²×C₉	C₃³
commutator, socle	C₃³	C₃²×C₉
prime radical	C₃⁴	C₃×C₉
LCS	G ▷ C₃³ ▷ 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
4. #10624463	C₃	semidirect
33. C₃×C₂₇	3₋¹⁺²	semidirect
53. C₃×C₉	C₂₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₂₇⋊C₃	3
C₃×C₉×3₋¹⁺²	3
C₃×C₃²⋊C₂₇	3
#10624463	3