Subgroups — #13091523

Order 1458.

322 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10624900	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃×#4208	C₃	1	1	yes	maximal, characteristic
3	C₉×#272	C₃	1	1	yes	maximal, characteristic
4	C₉×#272	C₃	1	2	yes	maximal
5	D₃×C₉⋊C₉	—	3	4	no	maximal
6	C₃×#737	C₂×C₃	1	1	yes	characteristic
7	#6895	—	2	4	no	—
8	#6894	—	2	4	no	—
9	#6894	—	2	8	no	—
10	C₉×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
11	C₉×H(3,𝔽₃)	C₂×C₃	1	2	yes	—
12	C₃×C₉⋊C₉	D₃	1	4	yes	complemented
13	#4208	—	3	1	no	—
14	C₃×#272	C₃²	1	1	yes	characteristic
15	C₃×D₃×C₉	—	3	4	no	—
16	C₃×D₃×C₉	—	3	8	no	—
17	C₃×D₃×C₉	—	3	4	no	—
18	C₂×C₉⋊C₉	—	9	1	no	—
19	#737	—	3	1	no	—
20	#737	—	6	4	no	—
21	C₃×3₋¹⁺²	—	2	4	no	—
22	C₃×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
23	C₃²×C₉	C₃×D₃	1	4	yes	—
24	C₃²×C₉	C₃×D₃	1	4	yes	—
25	C₃²×C₉	C₃×D₃	1	8	yes	—
26	C₉⋊C₉	—	2	24	no	—
27	C₉⋊C₉	—	2	8	no	—
28	C₉⋊C₉	—	2	4	no	—
29	C₉⋊C₉	—	2	4	no	—
30	C₉⋊C₉	C₃⋊D₃	1	1	yes	characteristic
31	C₃²⋊C₉	—	2	4	no	—
32	C₃²⋊C₉	—	2	8	no	—
33	C₃²×D₃	—	3	4	no	—
34	C₂×C₃×C₉	—	9	1	no	—
35	C₂×C₃×C₉	—	9	2	no	—
36	C₂×C₃×C₉	—	9	1	no	—
37	#272	3₋¹⁺²	1	1	yes	characteristic
38	D₃×C₉	—	3	12	no	—
39	D₃×C₉	—	9	4	no	—
40	D₃×C₉	—	9	8	no	—
41	C₃³	C₃²×D₃	1	4	yes	—
42	3₋¹⁺²	—	6	8	no	—
43	3₋¹⁺²	—	6	4	no	—
44	H(3,𝔽₃)	—	2	4	no	—
45	H(3,𝔽₃)	C₂×3₋¹⁺²	1	1	yes	characteristic
46	C₃×C₉	—	2	4	no	—
47	C₃×C₉	—	2	8	no	—
48	C₃×C₉	—	2	4	no	—
49	C₃×C₉	—	3	4	no	—
50	C₃×C₉	—	3	12	no	—
51	C₃×C₉	—	6	4	no	—
52	C₃×C₉	—	9	8	no	—
53	C₃×C₉	C₃×C₃⋊D₃	1	1	yes	characteristic
54	C₃×C₉	C₃×C₃⋊D₃	1	2	yes	—
55	C₃×C₉	C₃×C₃⋊D₃	1	1	yes	characteristic
56	C₂×C₃²	—	9	1	no	—
57	C₃×D₃	—	3	4	no	—
58	C₃×D₃	—	3	8	no	—
59	C₃×D₃	—	3	4	no	—
60	C₂×C₉	—	9	3	no	—
61	C₂×C₉	—	27	1	no	—
62	C₂×C₉	—	27	2	no	—
63	C₃²	—	2	4	no	—
64	C₃²	—	3	4	no	—
65	C₃²	—	3	8	no	—
66	C₃²	D₃×3₋¹⁺²	1	4	yes	—
67	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
68	C₉	—	3	1	no	—
69	C₉	—	3	2	no	—
70	C₉	—	6	4	no	—
71	C₉	—	6	4	no	—
72	C₉	—	6	8	no	—
73	C₉	#4208	1	3	yes	complemented
74	C₂×C₃	—	9	1	no	—
75	C₂×C₃	—	9	1	no	—
76	C₂×C₃	—	9	2	no	—
77	D₃	—	3	4	no	—
78	C₃	—	3	4	no	—
79	C₃	—	6	4	no	—
80	C₃	C₃⋊D₃×3₋¹⁺²	1	1	yes	characteristic
81	C₃	C₃×#4208	1	1	yes	characteristic
82	C₃	#92394	1	2	yes	—
83	C₂	—	9	1	no	Sylow

	subgroup	G/H
center, Frattini, socle	C₃²	C₃²×C₃⋊D₃
commutator, cosocle	C₃×H(3,𝔽₃)	C₂×C₃²
Fitting	#10624900	C₂
prime radical	C₃×#272	C₃²
hypercenter	C₉⋊C₉	C₃⋊D₃
hypocenter	H(3,𝔽₃)	C₂×3₋¹⁺²
LCS	G ▷ C₃×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

Complements

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

N	H ≅ G/N	type
1. #10624900	C₂	semidirect
12. C₃×C₉⋊C₉	D₃	semidirect
73. C₉	#4208	semidirect

Maximal subgroups

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

subgroup	index
#10624900	2
C₃×#4208	3
C₉×#272	3
D₃×C₉⋊C₉	3