Subgroups — #10624797

Order 729.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic
2	#6894	C₃	1	2	yes	maximal, complemented
3	C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	3₋¹⁺²⋊C₉	C₃	1	2	yes	maximal, complemented
6	3₋¹⁺²⋊C₉	C₃	1	2	yes	maximal, complemented
7	H(3,𝔽₃)⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	H(3,𝔽₃)⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
9	#6866	C₃	1	2	yes	maximal, complemented
10	C₃⁴	C₃²	1	1	yes	characteristic
11	C₃×3₋¹⁺²	C₃²	1	2	yes	—
12	C₃×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃²×C₉	—	3	1	no	—
14	C₃²×C₉	C₃²	1	1	yes	characteristic
15	C₃²×C₉	C₃²	1	1	yes	characteristic
16	C₃²×C₉	C₃²	1	1	yes	characteristic
17	AΣL(1,𝔽₂₇)	—	3	3	no	—
18	C₉⋊C₉	—	3	4	no	—
19	C₉⋊C₉	—	3	2	no	—
20	C₃²⋊C₉	—	3	2	no	—
21	C₃²⋊C₉	—	3	1	no	—
22	C₃²⋊C₉	—	3	2	no	—
23	C₃²⋊C₉	C₃²	1	2	yes	—
24	C₃²⋊C₉	C₃²	1	2	yes	—
25	C₃²⋊C₉	C₃²	1	2	yes	—
26	C₃³	—	3	3	no	—
27	C₃³	—	3	1	no	—
28	C₃³	—	3	1	no	—
29	C₃³	—	3	1	no	—
30	C₃³	—	3	2	no	—
31	C₃³	—	9	1	no	—
32	C₃³	—	9	1	no	—
33	C₃³	C₃³	1	1	yes	characteristic
34	3₋¹⁺²	—	3	2	no	—
35	3₋¹⁺²	—	3	4	no	—
36	H(3,𝔽₃)	—	3	2	no	—
37	H(3,𝔽₃)	3₋¹⁺²	1	3	yes	—
38	C₃×C₉	—	3	1	no	—
39	C₃×C₉	—	3	1	no	—
40	C₃×C₉	—	3	2	no	—
41	C₃×C₉	—	3	2	no	—
42	C₃×C₉	—	3	2	no	—
43	C₃×C₉	—	3	2	no	—
44	C₃×C₉	—	9	1	no	—
45	C₃×C₉	—	9	3	no	—
46	C₃×C₉	—	9	1	no	—
47	C₃×C₉	—	9	1	no	—
48	C₃×C₉	H(3,𝔽₃)	1	2	yes	—
49	C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
50	C₃²	—	3	1	no	—
51	C₃²	—	3	1	no	—
52	C₃²	—	3	1	no	—
53	C₃²	—	3	3	no	—
54	C₃²	—	3	3	no	—
55	C₃²	—	9	3	no	—
56	C₃²	—	9	6	no	—
57	C₃²	—	9	1	no	—
58	C₃²	—	9	1	no	—
59	C₃²	—	9	1	no	—
60	C₃²	—	9	1	no	—
61	C₃²	—	9	1	no	—
62	C₃²	—	9	1	no	—
63	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
64	C₃²	C₃×3₋¹⁺²	1	1	yes	characteristic
65	C₃²	#737	1	2	yes	—
66	C₉	—	3	1	no	—
67	C₉	—	3	2	no	—
68	C₉	—	9	2	no	—
69	C₉	—	9	2	no	—
70	C₉	—	9	2	no	—
71	C₉	—	9	1	no	—
72	C₉	—	9	1	no	—
73	C₉	—	9	2	no	—
74	C₃	—	3	1	no	—
75	C₃	—	3	2	no	—
76	C₃	—	9	3	no	—
77	C₃	—	9	3	no	—
78	C₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
79	C₃	#6908	1	1	yes	characteristic
80	C₃	#6909	1	1	yes	characteristic
81	C₃	#6913	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³
prime radical	C₃×AΣL(1,𝔽₂₇)	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #6894	C₃	semidirect
3. C₉×H(3,𝔽₃)	C₃	semidirect
4. C₃×C₃²⋊C₉	C₃	semidirect
5. 3₋¹⁺²⋊C₉	C₃	semidirect
6. 3₋¹⁺²⋊C₉	C₃	semidirect
7. H(3,𝔽₃)⋊C₉	C₃	semidirect
8. H(3,𝔽₃)⋊C₉	C₃	semidirect
9. #6866	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×AΣL(1,𝔽₂₇)	3
#6894	3
C₉×H(3,𝔽₃)	3
C₃×C₃²⋊C₉	3
3₋¹⁺²⋊C₉	3
H(3,𝔽₃)⋊C₉	3
#6866	3