Subgroups — #41007658

Order 26973.

926 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582556	C₃	1	27	yes	maximal, complemented
2	#22582554	C₃	1	9	yes	maximal, complemented
3	C₃×C₃²⋊#66524	C₃	1	3	yes	maximal, complemented
4	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
5	C₃×#11758608	C₃²	1	27	yes	complemented
6	C₃²×#66524	C₃²	1	9	yes	complemented
7	C₃²⋊#66524	C₃²	1	9	yes	—
8	C₃²⋊#66524	C₃²	1	18	yes	complemented
9	C₃²⋊#66524	C₃²	1	54	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	1	no	—
16	C₃³×C₃₇	—	9	3	no	—
17	C₃³×C₃₇	—	9	2	no	—
18	C₃³×C₃₇	—	9	6	no	—
19	C₃³×C₃₇	3₋¹⁺²	1	9	yes	complemented
20	C₃³×C₃₇	C₃³	1	9	yes	—
21	C₃³×C₃₇	C₃³	1	3	yes	—
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	H(3,𝔽₃)×C₃₇	—	3	27	no	—
24	#11758608	—	9	27	no	—
25	C₃×#66524	—	9	9	no	—
26	C₃×#66524	C₃³	1	27	yes	—
27	#10624883	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	3	2	no	—
31	C₃²×C₃₇	—	3	6	no	—
32	C₃²×C₃₇	—	3	9	no	—
33	C₃²×C₃₇	—	3	9	no	—
34	C₃²×C₃₇	—	3	18	no	—
35	C₃²×C₃₇	—	9	9	no	—
36	C₃²×C₃₇	—	9	27	no	—
37	C₃²×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
38	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
39	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
40	#66524	—	3	27	no	—
41	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
42	C₉⋊H(3,𝔽₃)	—	37	27	no	—
43	#6891	—	37	9	no	—
44	C₃×C₃²⋊C₉	—	37	3	no	—
45	C₃×C₃₇	—	3	3	no	—
46	C₃×C₃₇	—	3	9	no	—
47	C₃×C₃₇	—	9	9	no	—
48	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
49	C₃×C₃₇	H(5,𝔽₃)	1	1	yes	characteristic
50	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
51	C₃⁴	C₃×C₃₇⋊C₃	1	3	yes	—
52	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
53	C₃×3₋¹⁺²	—	37	27	no	—
54	C₃×H(3,𝔽₃)	—	3	9	no	—
55	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	9	yes	—
56	C₃²×C₉	—	37	9	no	—
57	C₃²⋊C₉	—	37	9	no	—
58	C₃²⋊C₉	—	37	18	no	—
59	C₃²⋊C₉	—	37	54	no	—
60	C₃₇	#10624883	1	1	yes	characteristic, complemented, Sylow
61	C₃³	—	3	9	no	—
62	C₃³	—	9	1	no	—
63	C₃³	—	9	3	no	—
64	C₃³	—	9	2	no	—
65	C₃³	—	9	6	no	—
66	C₃³	#11758608	1	9	yes	complemented
67	C₃³	C₃²×C₃₇⋊C₃	1	9	yes	—
68	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
69	C₃³	C₃²×C₃₇⋊C₃	1	3	yes	—
70	3₋¹⁺²	—	333	27	no	—
71	H(3,𝔽₃)	—	3	27	no	—
72	C₃×C₉	—	37	27	no	—
73	C₃×C₉	—	333	9	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	6	no	—
78	C₃²	—	3	9	no	—
79	C₃²	—	3	9	no	—
80	C₃²	—	3	18	no	—
81	C₃²	—	9	9	no	—
82	C₃²	—	9	27	no	—
83	C₃²	C₃×#11758608	1	9	yes	—
84	C₃²	C₃×#11758608	1	3	yes	—
85	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
86	C₉	—	111	27	no	—
87	C₃	—	3	3	no	—
88	C₃	—	3	9	no	—
89	C₃	—	9	9	no	—
90	C₃	C₃²×#11758608	1	1	yes	characteristic
91	C₃	C₃₇⋊H(5,𝔽₃)	1	1	yes	characteristic
92	C₃	#22582680	1	2	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	#10624883
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. #22582556	C₃	semidirect
2. #22582554	C₃	semidirect
3. C₃×C₃²⋊#66524	C₃	semidirect
5. C₃×#11758608	C₃²	semidirect
6. C₃²×#66524	C₃²	semidirect
8. C₃²⋊#66524	C₃²	semidirect
9. C₃²⋊#66524	C₃²	semidirect
19. C₃³×C₃₇	3₋¹⁺²	semidirect
60. C₃₇	#10624883	semidirect
66. C₃³	#11758608	semidirect

Maximal subgroups

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

subgroup	index
#22582556	3
#22582554	3
C₃×C₃²⋊#66524	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624883	37