Subgroups — #41007645

Order 26973.

1124 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#11758608	C₃	1	3	yes	maximal, complemented
2	#22582562	C₃	1	27	yes	maximal, complemented
3	#22582554	C₃	1	9	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	—	3	27	no	—
6	C₃×#11758608	C₃²	1	9	yes	—
7	C₃×#11758608	C₃²	1	18	yes	complemented
8	C₃²×#66524	C₃²	1	9	yes	complemented
9	C₃²⋊#66524	C₃²	1	81	yes	complemented
10	C₃⁴×C₃₇	C₃²	1	3	yes	—
11	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
13	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	—
14	C₃³×C₃₇	—	3	9	no	—
15	C₃³×C₃₇	—	9	3	no	—
16	C₃³×C₃₇	—	9	9	no	—
17	C₃³×C₃₇	H(3,𝔽₃)	1	9	yes	—
18	C₃³×C₃₇	C₃³	1	9	yes	—
19	C₃³×C₃₇	C₃³	1	3	yes	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	H(3,𝔽₃)×C₃₇	—	3	27	no	—
22	#11758608	—	3	27	no	—
23	#11758608	—	3	54	no	—
24	C₃×#66524	—	3	27	no	—
25	C₃×#66524	C₃³	1	27	yes	—
26	#10624881	—	37	1	no	maximal, Sylow
27	C₃²×C₃₇	—	3	3	no	—
28	C₃²×C₃₇	—	3	9	no	—
29	C₃²×C₃₇	—	3	9	no	—
30	C₃²×C₃₇	—	3	27	no	—
31	C₃²×C₃₇	—	9	9	no	—
32	C₃²×C₃₇	—	9	27	no	—
33	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
34	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
35	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
36	#66524	—	3	27	no	—
37	C₃²×3₋¹⁺²	—	37	3	no	—
38	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
39	#6893	—	37	27	no	—
40	#6891	—	37	9	no	—
41	C₃×C₃₇	—	3	3	no	—
42	C₃×C₃₇	—	3	9	no	—
43	C₃×C₃₇	—	9	9	no	—
44	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
45	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
46	C₃⁴	C₃×C₃₇⋊C₃	1	3	yes	—
47	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
48	C₃×3₋¹⁺²	—	37	9	no	—
49	C₃×3₋¹⁺²	—	37	18	no	—
50	C₃×3₋¹⁺²	—	111	27	no	—
51	C₃×H(3,𝔽₃)	—	3	9	no	—
52	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	9	yes	—
53	C₃²×C₉	—	37	9	no	—
54	C₃²⋊C₉	—	37	81	no	—
55	C₃₇	#10624881	1	1	yes	characteristic, complemented, Sylow
56	C₃³	—	3	9	no	—
57	C₃³	—	9	3	no	—
58	C₃³	—	9	9	no	—
59	C₃³	C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
60	C₃³	C₃²×C₃₇⋊C₃	1	9	yes	—
61	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
62	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
63	3₋¹⁺²	—	111	54	no	—
64	3₋¹⁺²	—	111	27	no	—
65	H(3,𝔽₃)	—	3	27	no	—
66	C₃×C₉	—	37	27	no	—
67	C₃×C₉	—	111	27	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	9	no	—
70	C₃²	—	3	9	no	—
71	C₃²	—	3	27	no	—
72	C₃²	—	9	9	no	—
73	C₃²	—	9	27	no	—
74	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	9	yes	—
75	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
76	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
77	C₉	—	111	27	no	—
78	C₃	—	3	3	no	—
79	C₃	—	3	9	no	—
80	C₃	—	9	9	no	—
81	C₃	C₃²×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
82	C₃	#22582680	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624881
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×#11758608	C₃	semidirect
2. #22582562	C₃	semidirect
3. #22582554	C₃	semidirect
7. C₃×#11758608	C₃²	semidirect
8. C₃²×#66524	C₃²	semidirect
9. C₃²⋊#66524	C₃²	semidirect
55. C₃₇	#10624881	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#11758608	3
#22582562	3
#22582554	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624881	37