Subgroups — #14552057

Order 2187.

1394 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₉×H(3,𝔽₃)	C₃	1	3	yes	maximal
2	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	#10624556	C₃	1	24	yes	maximal, complemented
4	#10624548	C₃	1	8	yes	maximal, complemented
5	C₃×#6855	C₃	1	4	yes	maximal, complemented
6	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
7	C₃³×C₉	C₃²	1	12	yes	—
8	C₃³×C₉	C₃²	1	4	yes	—
9	C₉×H(3,𝔽₃)	—	3	27	no	—
10	C₉×H(3,𝔽₃)	C₉	1	27	yes	complemented
11	C₃×C₉⋊C₉	C₃²	1	24	yes	—
12	C₃×C₃²⋊C₉	C₃²	1	24	yes	—
13	C₃×C₃²⋊C₉	C₃²	1	8	yes	—
14	#6855	C₃²	1	1	yes	characteristic
15	#6855	C₃²	1	8	yes	—
16	#6855	C₃²	1	24	yes	—
17	#6855	C₃²	1	24	yes	—
18	C₃⁴	C₃³	1	4	yes	—
19	C₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	—
20	C₃×H(3,𝔽₃)	3₋¹⁺²	1	27	yes	—
21	C₃²×C₉	—	3	36	no	—
22	C₃²×C₉	—	3	36	no	—
23	C₃²×C₉	—	9	4	no	—
24	C₃²×C₉	—	9	12	no	—
25	C₃²×C₉	—	9	8	no	—
26	C₃²×C₉	—	9	12	no	—
27	C₃²×C₉	C₃×C₉	1	36	yes	complemented
28	C₃²×C₉	C₃³	1	24	yes	—
29	C₃²×C₉	C₃³	1	1	yes	characteristic
30	C₃²×C₉	C₃³	1	8	yes	—
31	C₃²×C₉	C₃³	1	3	yes	—
32	C₉⋊C₉	—	9	24	no	—
33	C₃²⋊C₉	—	3	72	no	—
34	C₃²⋊C₉	C₃×C₉	1	72	yes	complemented
35	C₃³	—	3	12	no	—
36	C₃³	—	3	4	no	—
37	C₃³	—	3	8	no	—
38	C₃³	—	3	12	no	—
39	C₃³	C₃²×C₉	1	12	yes	—
40	C₃³	C₃×3₋¹⁺²	1	36	yes	—
41	C₃³	C₃⁴	1	1	yes	characteristic
42	H(3,𝔽₃)	C₉⋊C₉	1	81	yes	complemented
43	C₃×C₉	—	3	3	no	—
44	C₃×C₉	—	3	8	no	—
45	C₃×C₉	—	3	1	no	—
46	C₃×C₉	—	3	3	no	—
47	C₃×C₉	—	3	2	no	—
48	C₃×C₉	—	3	3	no	—
49	C₃×C₉	—	3	16	no	—
50	C₃×C₉	—	3	24	no	—
51	C₃×C₉	—	3	24	no	—
52	C₃×C₉	—	3	24	no	—
53	C₃×C₉	—	3	24	no	—
54	C₃×C₉	—	9	36	no	—
55	C₃×C₉	—	9	108	no	—
56	C₃×C₉	C₃²×C₉	1	24	yes	—
57	C₃×C₉	C₃²×C₉	1	3	yes	—
58	C₃×C₉	#737	1	9	yes	—
59	C₃²	—	3	12	no	—
60	C₃²	—	3	12	no	—
61	C₃²	—	3	12	no	—
62	C₃²	—	3	36	no	—
63	C₃²	—	3	72	no	—
64	C₃²	C₃×C₉⋊C₉	1	36	yes	—
65	C₃²	C₃³×C₉	1	1	yes	characteristic
66	C₃²	C₃²×3₋¹⁺²	1	3	yes	—
67	C₃²	C₃×#737	1	3	yes	—
68	C₃²	H(5,𝔽₃)	1	1	yes	characteristic
69	C₃²	3₋¹⁺⁴	1	2	yes	—
70	C₃²	3₋¹⁺⁴	1	3	yes	—
71	C₉	—	3	3	no	—
72	C₉	—	3	24	no	—
73	C₉	—	3	9	no	—
74	C₉	—	9	24	no	—
75	C₃	—	3	36	no	—
76	C₃	C₃²×C₉⋊C₉	1	1	yes	characteristic
77	C₃	C₉×#737	1	1	yes	characteristic
78	C₃	#10624862	1	1	yes	characteristic
79	C₃	#10624864	1	1	yes	characteristic
80	C₃	#10624897	1	3	yes	—
81	C₃	#10624900	1	6	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₃³	C₃⁴
commutator	C₃²	C₃³×C₉
prime radical	C₃²×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃³ ◁ G
LECS	G ▷ C₃³ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #10624556	C₃	semidirect
4. #10624548	C₃	semidirect
5. C₃×#6855	C₃	semidirect
10. C₉×H(3,𝔽₃)	C₉	semidirect
27. C₃²×C₉	C₃×C₉	semidirect
34. C₃²⋊C₉	C₃×C₉	semidirect
42. H(3,𝔽₃)	C₉⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉×H(3,𝔽₃)	3
#10624556	3
#10624548	3
C₃×#6855	3