Subgroups — C₃²⋊H(5,𝔽₃) #14555251

Order 2187.

9305 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×H(5,𝔽₃)	C₃	1	3	yes	maximal, complemented
2	C₃³×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	H(3,𝔽₃)²	C₃	1	81	yes	maximal, complemented
4	#10624880	C₃	1	24	yes	maximal, complemented
5	C₃×C₃²⋊H(3,𝔽₃)	C₃	1	12	yes	maximal, complemented
6	C₃⁵	C₃²	1	4	yes	complemented
7	H(5,𝔽₃)	—	3	81	no	—
8	C₃²×H(3,𝔽₃)	—	3	27	no	—
9	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
10	C₃²×H(3,𝔽₃)	C₃²	1	8	yes	complemented
11	C₃²×H(3,𝔽₃)	C₃²	1	27	yes	complemented
12	C₃²×H(3,𝔽₃)	C₃²	1	36	yes	complemented
13	C₃²×H(3,𝔽₃)	C₃²	1	81	yes	complemented
14	C₃²×H(3,𝔽₃)	C₃²	1	108	yes	complemented
15	C₃²⋊H(3,𝔽₃)	C₃²	1	9	yes	complemented
16	C₃²⋊H(3,𝔽₃)	C₃²	1	72	yes	complemented
17	C₃²⋊H(3,𝔽₃)	C₃²	1	216	yes	complemented
18	C₃²⋊H(3,𝔽₃)	C₃²	1	648	yes	complemented
19	C₃⁴	—	3	36	no	—
20	C₃⁴	—	9	36	no	—
21	C₃⁴	C₃³	1	1	yes	characteristic
22	C₃⁴	C₃³	1	36	yes	—
23	C₃⁴	C₃³	1	8	yes	—
24	C₃⁴	C₃³	1	4	yes	—
25	C₃⁴	C₃³	1	108	yes	complemented
26	C₃×H(3,𝔽₃)	—	3	72	no	—
27	C₃×H(3,𝔽₃)	—	3	324	no	—
28	C₃×H(3,𝔽₃)	—	3	243	no	—
29	C₃×H(3,𝔽₃)	—	3	729	no	—
30	C₃×H(3,𝔽₃)	—	9	324	no	—
31	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	27	yes	complemented
32	C₃×H(3,𝔽₃)	C₃³	1	72	yes	—
33	C₃×H(3,𝔽₃)	C₃³	1	27	yes	—
34	C₃×H(3,𝔽₃)	C₃³	1	81	yes	—
35	C₃×H(3,𝔽₃)	C₃³	1	9	yes	—
36	C₃×H(3,𝔽₃)	C₃³	1	216	yes	complemented
37	C₃×H(3,𝔽₃)	C₃³	1	648	yes	complemented
38	C₃³	—	3	3	no	—
39	C₃³	—	3	9	no	—
40	C₃³	—	3	108	no	—
41	C₃³	—	3	24	no	—
42	C₃³	—	3	108	no	—
43	C₃³	—	3	324	no	—
44	C₃³	—	9	12	no	—
45	C₃³	—	9	72	no	—
46	C₃³	—	9	24	no	—
47	C₃³	—	9	324	no	—
48	C₃³	—	9	324	no	—
49	C₃³	C₃×H(3,𝔽₃)	1	36	yes	complemented
50	C₃³	C₃⁴	1	24	yes	—
51	C₃³	C₃⁴	1	81	yes	—
52	C₃³	C₃⁴	1	1	yes	characteristic
53	C₃³	C₃⁴	1	12	yes	—
54	C₃³	C₃⁴	1	3	yes	—
55	H(3,𝔽₃)	—	3	81	no	—
56	H(3,𝔽₃)	—	3	216	no	—
57	H(3,𝔽₃)	—	3	648	no	—
58	H(3,𝔽₃)	—	3	243	no	—
59	H(3,𝔽₃)	—	9	648	no	—
60	H(3,𝔽₃)	C₃×H(3,𝔽₃)	1	81	yes	complemented
61	C₃²	—	3	27	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	6	no	—
65	C₃²	—	3	36	no	—
66	C₃²	—	3	24	no	—
67	C₃²	—	3	81	no	—
68	C₃²	—	3	243	no	—
69	C₃²	—	3	72	no	—
70	C₃²	—	9	36	no	—
71	C₃²	—	9	324	no	—
72	C₃²	—	9	72	no	—
73	C₃²	—	27	324	no	—
74	C₃²	C₃²×H(3,𝔽₃)	1	3	yes	—
75	C₃²	C₃²×H(3,𝔽₃)	1	36	yes	—
76	C₃²	H(5,𝔽₃)	1	9	yes	complemented
77	C₃²	C₃⁵	1	1	yes	characteristic
78	C₃	—	3	3	no	—
79	C₃	—	3	9	no	—
80	C₃	—	3	36	no	—
81	C₃	—	9	24	no	—
82	C₃	—	9	81	no	—
83	C₃	C₃³×H(3,𝔽₃)	1	1	yes	characteristic
84	C₃	C₃×H(5,𝔽₃)	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃²	C₃⁵
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ G
LECS	G ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃×H(5,𝔽₃)	C₃	semidirect
2. C₃³×H(3,𝔽₃)	C₃	semidirect
3. H(3,𝔽₃)²	C₃	semidirect
4. #10624880	C₃	semidirect
5. C₃×C₃²⋊H(3,𝔽₃)	C₃	semidirect
6. C₃⁵	C₃²	semidirect
10. C₃²×H(3,𝔽₃)	C₃²	semidirect
11. C₃²×H(3,𝔽₃)	C₃²	semidirect
12. C₃²×H(3,𝔽₃)	C₃²	semidirect
13. C₃²×H(3,𝔽₃)	C₃²	semidirect
14. C₃²×H(3,𝔽₃)	C₃²	semidirect
15. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
16. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
17. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
18. C₃²⋊H(3,𝔽₃)	C₃²	semidirect
25. C₃⁴	C₃³	semidirect
31. C₃×H(3,𝔽₃)	H(3,𝔽₃)	semidirect
36. C₃×H(3,𝔽₃)	C₃³	semidirect
37. C₃×H(3,𝔽₃)	C₃³	semidirect
49. C₃³	C₃×H(3,𝔽₃)	semidirect
60. H(3,𝔽₃)	C₃×H(3,𝔽₃)	semidirect
76. C₃²	H(5,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×H(5,𝔽₃)	3
C₃³×H(3,𝔽₃)	3
H(3,𝔽₃)²	3
#10624880	3
C₃×C₃²⋊H(3,𝔽₃)	3

Subgroups — C32⋊H(5,𝔽3) #14555251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²⋊H(5,𝔽₃) #14555251