Subgroups — #14552613

Order 2187.

3221 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×C₉⋊H(3,𝔽₃)	C₃	1	12	yes	maximal, complemented
3	C₃×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	#10624555	C₃	1	24	yes	maximal, complemented
5	#10624547	C₃	1	2	yes	maximal, complemented
6	C₃⁵	C₃²	1	1	yes	characteristic
7	C₃²×3₋¹⁺²	C₃²	1	9	yes	complemented
8	C₃²×H(3,𝔽₃)	C₃²	1	12	yes	—
9	C₃³×C₉	—	3	36	no	—
10	C₃³×C₉	C₃²	1	8	yes	—
11	C₃³×C₉	C₃²	1	4	yes	—
12	C₉⋊H(3,𝔽₃)	—	3	108	no	—
13	C₃²⋊H(3,𝔽₃)	—	3	9	no	—
14	C₃×C₃²⋊C₉	C₃²	1	24	yes	—
15	C₃×C₃²⋊C₉	C₃²	1	72	yes	complemented
16	C₃⁴	—	3	12	no	—
17	C₃⁴	—	3	24	no	—
18	C₃⁴	3₋¹⁺²	1	9	yes	complemented
19	C₃⁴	C₃³	1	9	yes	—
20	C₃⁴	C₃³	1	4	yes	—
21	C₃×3₋¹⁺²	—	3	108	no	—
22	C₃×H(3,𝔽₃)	—	3	36	no	—
23	C₃×H(3,𝔽₃)	—	3	36	no	—
24	C₃×H(3,𝔽₃)	—	3	72	no	—
25	C₃²×C₉	—	3	243	no	—
26	C₃²×C₉	—	3	24	no	—
27	C₃²×C₉	—	3	72	no	—
28	C₃²×C₉	—	3	36	no	—
29	C₃²×C₉	H(3,𝔽₃)	1	36	yes	complemented
30	C₃²×C₉	C₃³	1	18	yes	—
31	C₃²×C₉	C₃³	1	8	yes	—
32	C₃²×C₉	C₃³	1	1	yes	characteristic
33	C₃²⋊C₉	—	3	72	no	—
34	C₃²⋊C₉	—	3	216	no	—
35	C₃³	—	3	27	no	—
36	C₃³	—	3	216	no	—
37	C₃³	—	3	12	no	—
38	C₃³	—	3	24	no	—
39	C₃³	—	3	108	no	—
40	C₃³	—	3	36	no	—
41	C₃³	—	3	9	no	—
42	C₃³	—	3	72	no	—
43	C₃³	C₃×H(3,𝔽₃)	1	12	yes	—
44	C₃³	C₃×3₋¹⁺²	1	12	yes	—
45	C₃³	#737	1	24	yes	—
46	C₃³	C₃⁴	1	1	yes	characteristic
47	3₋¹⁺²	—	9	81	no	—
48	H(3,𝔽₃)	—	9	108	no	—
49	C₃×C₉	—	3	216	no	—
50	C₃×C₉	—	3	24	no	—
51	C₃×C₉	—	3	108	no	—
52	C₃×C₉	—	3	72	no	—
53	C₃×C₉	C₃×H(3,𝔽₃)	1	24	yes	—
54	C₃×C₉	C₃×H(3,𝔽₃)	1	12	yes	—
55	C₃²	—	3	243	no	—
56	C₃²	—	3	12	no	—
57	C₃²	—	3	36	no	—
58	C₃²	—	3	24	no	—
59	C₃²	—	3	72	no	—
60	C₃²	—	9	9	no	—
61	C₃²	—	9	36	no	—
62	C₃²	—	9	72	no	—
63	C₃²	C₉⋊H(3,𝔽₃)	1	36	yes	complemented
64	C₃²	C₃²×H(3,𝔽₃)	1	4	yes	—
65	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
66	C₃²	C₃×#737	1	8	yes	—
67	C₉	—	3	24	no	—
68	C₉	—	9	18	no	—
69	C₉	C₃²⋊H(3,𝔽₃)	1	9	yes	complemented
70	C₃	—	3	36	no	—
71	C₃	—	27	9	no	—
72	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
73	C₃	C₃×C₉⋊H(3,𝔽₃)	1	4	yes	—
74	C₃	#10624869	1	8	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃×C₃²⋊H(3,𝔽₃)	C₃
LCS	G ▷ C₃³ ▷ 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
1. C₃⁴×C₉	C₃	semidirect
2. C₃×C₉⋊H(3,𝔽₃)	C₃	semidirect
4. #10624555	C₃	semidirect
5. #10624547	C₃	semidirect
7. C₃²×3₋¹⁺²	C₃²	semidirect
15. C₃×C₃²⋊C₉	C₃²	semidirect
18. C₃⁴	3₋¹⁺²	semidirect
29. C₃²×C₉	H(3,𝔽₃)	semidirect
63. C₃²	C₉⋊H(3,𝔽₃)	semidirect
69. C₉	C₃²⋊H(3,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃⁴×C₉	3
C₃×C₉⋊H(3,𝔽₃)	3
C₃×C₃²⋊H(3,𝔽₃)	3
#10624555	3
#10624547	3