Subgroups — #13091517

Order 1458.

422 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624897	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃²×#272	C₃	1	1	yes	maximal, characteristic
3	C₉×#272	C₃	1	3	yes	maximal, complemented
4	D₃×C₃²⋊C₉	—	3	4	no	maximal
5	C₃²×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
6	C₉⋊H(3,𝔽₃)	—	2	12	no	—
7	#6891	—	2	4	no	—
8	C₉×H(3,𝔽₃)	C₂×C₃	1	3	yes	complemented
9	C₃×C₃²⋊C₉	D₃	1	4	yes	complemented
10	C₃³×D₃	—	3	4	no	—
11	C₃×#272	—	3	1	no	—
12	C₃×#272	C₃²	1	1	yes	characteristic
13	C₃×D₃×C₉	—	3	12	no	—
14	C₂×C₃²⋊C₉	—	9	1	no	—
15	C₃⁴	C₃×D₃	1	4	yes	—
16	C₃×3₋¹⁺²	—	2	12	no	—
17	C₃×H(3,𝔽₃)	—	2	4	no	—
18	C₃×H(3,𝔽₃)	—	3	1	no	—
19	C₃×H(3,𝔽₃)	—	6	4	no	—
20	C₃×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
21	C₃²×C₉	C₃×D₃	1	12	yes	complemented
22	C₃²⋊C₉	—	2	12	no	—
23	C₃²⋊C₉	—	2	12	no	—
24	C₃²⋊C₉	—	2	4	no	—
25	C₃²⋊C₉	—	2	12	no	—
26	C₃²⋊C₉	C₃⋊D₃	1	1	yes	characteristic
27	C₂×C₃³	—	9	1	no	—
28	C₃²×D₃	—	3	4	no	—
29	C₃²×D₃	—	3	12	no	—
30	C₃²×D₃	—	9	4	no	—
31	C₃²×D₃	—	9	4	no	—
32	C₃²×D₃	—	9	4	no	—
33	C₂×C₃×C₉	—	9	3	no	—
34	#272	3₋¹⁺²	1	1	yes	characteristic, complemented
35	D₃×C₉	—	9	12	no	—
36	C₃³	—	2	4	no	—
37	C₃³	—	3	4	no	—
38	C₃³	—	3	12	no	—
39	C₃³	—	6	4	no	—
40	C₃³	—	9	4	no	—
41	C₃³	—	9	4	no	—
42	C₃³	C₃²×D₃	1	4	yes	—
43	C₃³	C₃×C₃⋊D₃	1	1	yes	characteristic
44	3₋¹⁺²	—	18	12	no	—
45	H(3,𝔽₃)	—	2	4	no	—
46	H(3,𝔽₃)	—	6	8	no	—
47	H(3,𝔽₃)	—	6	4	no	—
48	H(3,𝔽₃)	C₂×3₋¹⁺²	1	1	yes	characteristic
49	C₃×C₉	—	2	12	no	—
50	C₃×C₉	—	9	12	no	—
51	C₃×C₉	C₃×C₃⋊D₃	1	3	yes	—
52	C₂×C₃²	—	9	1	no	—
53	C₂×C₃²	—	9	3	no	—
54	C₂×C₃²	—	27	1	no	—
55	C₂×C₃²	—	27	1	no	—
56	C₂×C₃²	—	27	1	no	—
57	C₃×D₃	—	3	4	no	—
58	C₃×D₃	—	3	4	no	—
59	C₃×D₃	—	3	4	no	—
60	C₃×D₃	—	3	4	no	—
61	C₃×D₃	—	9	12	no	—
62	C₂×C₉	—	27	3	no	—
63	C₃²	—	2	4	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	3	4	no	—
68	C₃²	—	3	4	no	—
69	C₃²	—	3	4	no	—
70	C₃²	—	6	4	no	—
71	C₃²	—	6	4	no	—
72	C₃²	—	6	4	no	—
73	C₃²	—	6	4	no	—
74	C₃²	—	9	12	no	—
75	C₃²	—	18	12	no	—
76	C₃²	D₃×3₋¹⁺²	1	4	yes	—
77	C₃²	#4208	1	3	yes	—
78	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
79	C₉	—	3	3	no	—
80	C₉	—	6	12	no	—
81	C₂×C₃	—	9	1	no	—
82	C₂×C₃	—	9	1	no	—
83	C₂×C₃	—	9	1	no	—
84	C₂×C₃	—	9	1	no	—
85	C₂×C₃	—	27	3	no	—
86	D₃	—	3	4	no	—
87	C₃	—	3	4	no	—
88	C₃	—	3	3	no	—
89	C₃	—	6	4	no	—
90	C₃	—	18	4	no	—
91	C₃	C₃⋊D₃×3₋¹⁺²	1	1	yes	characteristic
92	C₃	C₃×#4208	1	1	yes	characteristic
93	C₃	#92392	1	1	yes	characteristic
94	C₃	#92394	1	1	yes	characteristic
95	C₂	—	9	1	no	Sylow

	subgroup	G/H
center, Frattini, socle	C₃²	C₃²×C₃⋊D₃
commutator, cosocle	C₃×H(3,𝔽₃)	C₂×C₃²
Fitting	#10624897	C₂
prime radical	C₃²×#272	C₃
hypercenter	C₃²⋊C₉	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×3₋¹⁺²
LCS	G ▷ C₃×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
1. #10624897	C₂	semidirect
3. C₉×#272	C₃	semidirect
8. C₉×H(3,𝔽₃)	C₂×C₃	semidirect
9. C₃×C₃²⋊C₉	D₃	semidirect
21. C₃²×C₉	C₃×D₃	semidirect
34. #272	3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
#10624897	2
C₃²×#272	3
C₉×#272	3
D₃×C₃²⋊C₉	3