Subgroups — #14550989

Order 2187.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624730	C₃	1	2	yes	maximal, complemented
2	C₃×#6866	C₃	1	2	yes	maximal, complemented
3	#10624476	C₃	1	4	yes	maximal, complemented
4	#10624475	C₃	1	4	yes	maximal, complemented
5	#10624474	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃⁵	C₃²	1	1	yes	characteristic
7	C₃×AΣL(1,𝔽₂₇)	—	3	2	no	—
8	C₃×AΣL(1,𝔽₂₇)	—	3	4	no	—
9	#6891	C₃²	1	4	yes	—
10	C₃²⋊H(3,𝔽₃)	C₃²	1	2	yes	—
11	C₃×C₃²⋊C₉	—	3	2	no	—
12	#6866	—	3	2	no	—
13	#6866	—	3	2	no	—
14	#6866	C₃²	1	2	yes	—
15	#6866	C₃²	1	4	yes	—
16	C₃⁴	—	3	2	no	—
17	C₃⁴	—	3	4	no	—
18	C₃⁴	—	9	2	no	—
19	C₃⁴	—	9	9	no	—
20	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
21	C₃⁴	H(3,𝔽₃)	1	2	yes	—
22	C₃⁴	C₃³	1	1	yes	characteristic
23	C₃×3₋¹⁺²	—	3	4	no	—
24	C₃×H(3,𝔽₃)	—	3	2	no	—
25	C₃×H(3,𝔽₃)	—	9	2	no	—
26	C₃²×C₉	—	9	2	no	—
27	AΣL(1,𝔽₂₇)	—	9	6	no	—
28	AΣL(1,𝔽₂₇)	—	9	4	no	—
29	AΣL(1,𝔽₂₇)	—	9	8	no	—
30	C₃²⋊C₉	—	3	2	no	—
31	C₃²⋊C₉	—	3	4	no	—
32	C₃²⋊C₉	—	9	4	no	—
33	C₃²⋊C₉	—	9	4	no	—
34	C₃³	—	3	2	no	—
35	C₃³	—	3	2	no	—
36	C₃³	—	3	4	no	—
37	C₃³	—	3	2	no	—
38	C₃³	—	3	2	no	—
39	C₃³	—	3	2	no	—
40	C₃³	—	9	3	no	—
41	C₃³	—	9	1	no	—
42	C₃³	—	9	4	no	—
43	C₃³	—	9	4	no	—
44	C₃³	—	9	6	no	—
45	C₃³	—	9	18	no	—
46	C₃³	—	9	18	no	—
47	C₃³	—	9	3	no	—
48	C₃³	—	9	36	no	—
49	C₃³	—	9	36	no	—
50	C₃³	—	9	2	no	—
51	C₃³	#732	1	2	yes	—
52	C₃³	#732	1	4	yes	—
53	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	3₋¹⁺²	—	9	8	no	—
55	3₋¹⁺²	—	9	4	no	—
56	H(3,𝔽₃)	—	9	4	no	—
57	H(3,𝔽₃)	—	9	2	no	—
58	H(3,𝔽₃)	—	27	6	no	—
59	C₃×C₉	—	9	2	no	—
60	C₃×C₉	—	9	4	no	—
61	C₃×C₉	—	9	4	no	—
62	C₃×C₉	—	27	6	no	—
63	C₃²	—	3	6	no	—
64	C₃²	—	3	2	no	—
65	C₃²	—	3	6	no	—
66	C₃²	—	3	4	no	—
67	C₃²	—	9	1	no	—
68	C₃²	—	9	6	no	—
69	C₃²	—	9	12	no	—
70	C₃²	—	9	9	no	—
71	C₃²	—	9	4	no	—
72	C₃²	—	9	6	no	—
73	C₃²	—	9	6	no	—
74	C₃²	—	9	6	no	—
75	C₃²	—	9	4	no	—
76	C₃²	—	9	36	no	—
77	C₃²	—	9	36	no	—
78	C₃²	—	9	1	no	—
79	C₃²	—	9	1	no	—
80	C₃²	—	27	6	no	—
81	C₃²	—	27	2	no	—
82	C₃²	C₃×#732	1	2	yes	—
83	C₃²	#6909	1	2	yes	—
84	C₉	—	27	2	no	—
85	C₉	—	27	4	no	—
86	C₉	—	27	4	no	—
87	C₃	—	3	2	no	—
88	C₃	—	3	6	no	—
89	C₃	—	3	2	no	—
90	C₃	—	9	3	no	—
91	C₃	—	9	4	no	—
92	C₃	—	9	3	no	—
93	C₃	—	81	2	no	—
94	C₃	#10624599	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	#10624599
commutator, Frattini, cosocle	C₃⁴	C₃³
LCS	G ▷ C₃⁴ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁵ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #10624730	C₃	semidirect
2. C₃×#6866	C₃	semidirect
3. #10624476	C₃	semidirect
4. #10624475	C₃	semidirect
5. #10624474	C₃	semidirect

Maximal subgroups

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

subgroup	index
#10624730	3
C₃×#6866	3
#10624476	3
#10624475	3
#10624474	3