Subgroups — #14552003

Order 2187.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624841	C₃	1	2	yes	maximal, complemented
2	#10624839	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624713	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624540	C₃	1	2	yes	maximal, complemented
5	#10624540	C₃	1	2	yes	maximal, complemented
6	#10624539	C₃	1	2	yes	maximal, complemented
7	#10624538	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624537	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624535	C₃	1	1	yes	maximal, characteristic, complemented
10	#6908	—	3	2	no	—
11	C₃×#732	—	3	1	no	—
12	#6887	C₃²	1	1	yes	characteristic
13	#6880	C₃²	1	2	yes	complemented
14	#6880	C₃²	1	2	yes	complemented
15	#6879	—	3	2	no	—
16	#6879	C₃²	1	1	yes	characteristic
17	#6879	C₃²	1	1	yes	characteristic
18	#6879	C₃²	1	1	yes	characteristic
19	#6879	C₃²	1	2	yes	complemented
20	#6878	—	3	4	no	—
21	C₉⋊C₂₇	—	3	2	no	—
22	C₃²⋊C₂₇	—	3	1	no	—
23	#6864	C₃²	1	2	yes	complemented
24	C₉×C₂₇	C₃²	1	1	yes	characteristic, complemented
25	#737	—	9	2	no	—
26	C₃×H(3,𝔽₃)	—	9	1	no	—
27	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
28	#733	—	3	2	no	—
29	#733	—	3	2	no	—
30	#732	—	3	2	no	—
31	#732	—	3	1	no	—
32	#732	—	3	1	no	—
33	#732	—	3	1	no	—
34	#732	—	9	2	no	—
35	#731	—	9	2	no	—
36	AΣL(1,𝔽₂₇)	—	9	2	no	—
37	C₃×C₂₇	—	3	1	no	—
38	C₃×C₂₇	—	3	2	no	—
39	C₉⋊C₉	—	3	2	no	—
40	C₃²⋊C₉	—	3	1	no	—
41	C₉²	H(3,𝔽₃)	1	2	yes	—
42	C₉²	C₃³	1	1	yes	characteristic
43	C₃³	—	27	1	no	—
44	C₃³	#732	1	1	yes	characteristic
45	3₋¹⁺²	—	9	2	no	—
46	3₋¹⁺²	—	9	2	no	—
47	3₋¹⁺²	—	27	4	no	—
48	H(3,𝔽₃)	—	9	2	no	—
49	H(3,𝔽₃)	—	9	1	no	—
50	H(3,𝔽₃)	—	9	1	no	—
51	H(3,𝔽₃)	—	9	1	no	—
52	H(3,𝔽₃)	—	27	2	no	—
53	C₃×C₉	—	3	2	no	—
54	C₃×C₉	—	3	1	no	—
55	C₃×C₉	—	9	1	no	—
56	C₃×C₉	—	27	2	no	—
57	C₃×C₉	#732	1	2	yes	—
58	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
59	C₂₇	—	9	1	no	—
60	C₂₇	—	9	2	no	—
61	C₃²	—	3	1	no	—
62	C₃²	—	9	1	no	—
63	C₃²	—	27	2	no	—
64	C₃²	—	27	1	no	—
65	C₃²	—	27	1	no	—
66	C₃²	—	27	1	no	—
67	C₃²	—	81	3	no	—
68	C₃²	C₃×#732	1	1	yes	characteristic
69	C₉	—	3	1	no	—
70	C₉	—	3	2	no	—
71	C₉	—	9	2	no	—
72	C₉	—	9	1	no	—
73	C₉	—	27	2	no	—
74	C₉	—	27	2	no	—
75	C₃	—	3	1	no	—
76	C₃	—	9	1	no	—
77	C₃	—	81	2	no	—
78	C₃	—	81	1	no	—
79	C₃	—	81	1	no	—
80	C₃	—	81	1	no	—
81	C₃	#10624839	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃	#10624839
commutator, Frattini, cosocle	C₉²	C₃³
LCS	G ▷ C₉² ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃²×C₉ ◁ #6887 ◁ G
LECS	G ▷ C₉² ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #10624841	C₃	semidirect
2. #10624839	C₃	semidirect
3. #10624713	C₃	semidirect
4. #10624540	C₃	semidirect
5. #10624540	C₃	semidirect
6. #10624539	C₃	semidirect
7. #10624538	C₃	semidirect
8. #10624537	C₃	semidirect
9. #10624535	C₃	semidirect
13. #6880	C₃²	semidirect
14. #6880	C₃²	semidirect
19. #6879	C₃²	semidirect
23. #6864	C₃²	semidirect
24. C₉×C₂₇	C₃²	semidirect

Maximal subgroups

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

subgroup	index
#10624841	3
#10624839	3
#10624713	3
#10624540	3
#10624539	3
#10624538	3
#10624537	3
#10624535	3