Subgroups — C₃₇×#10624883 #41004672

Order 26973.

926 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	27	yes	maximal, complemented
3	C₃₇×#6891	C₃	1	9	yes	maximal, complemented
4	C₃×C₃₇×C₃²⋊C₉	C₃	1	3	yes	maximal, complemented
5	C₃⁴×C₃₇	C₃²	1	3	yes	—
6	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
7	C₃×3₋¹⁺²×C₃₇	C₃²	1	27	yes	complemented
8	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
9	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	—
10	C₃²×C₉×C₃₇	C₃²	1	9	yes	complemented
11	C₃₇×C₃²⋊C₉	C₃²	1	9	yes	—
12	C₃₇×C₃²⋊C₉	C₃²	1	18	yes	complemented
13	C₃₇×C₃²⋊C₉	C₃²	1	54	yes	complemented
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	1	yes	characteristic
22	C₃³×C₃₇	C₃³	1	3	yes	—
23	3₋¹⁺²×C₃₇	—	9	27	no	—
24	H(3,𝔽₃)×C₃₇	—	3	27	no	—
25	C₃×C₉×C₃₇	—	9	9	no	—
26	C₃×C₉×C₃₇	C₃³	1	27	yes	—
27	#10624883	C₃₇	1	1	yes	maximal, characteristic, direct product factor, 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	C₉×C₃₇	—	3	27	no	—
41	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
42	C₉⋊H(3,𝔽₃)	C₃×C₃₇	1	27	yes	complemented
43	#6891	C₃×C₃₇	1	9	yes	complemented
44	C₃×C₃²⋊C₉	C₃×C₃₇	1	3	yes	complemented
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	C₃²×C₃₇	1	3	yes	—
52	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
53	C₃×3₋¹⁺²	C₃²×C₃₇	1	27	yes	complemented
54	C₃×H(3,𝔽₃)	—	3	9	no	—
55	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	9	yes	—
56	C₃²×C₉	C₃²×C₃₇	1	9	yes	complemented
57	C₃²⋊C₉	C₃²×C₃₇	1	9	yes	—
58	C₃²⋊C₉	C₃²×C₃₇	1	18	yes	complemented
59	C₃²⋊C₉	C₃²×C₃₇	1	54	yes	complemented
60	C₃₇	#10624883	1	1	yes	characteristic, direct product factor, 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	3₋¹⁺²×C₃₇	1	9	yes	complemented
67	C₃³	C₃³×C₃₇	1	9	yes	—
68	C₃³	C₃³×C₃₇	1	1	yes	characteristic
69	C₃³	C₃³×C₃₇	1	3	yes	—
70	3₋¹⁺²	—	9	27	no	—
71	H(3,𝔽₃)	—	3	27	no	—
72	C₃×C₉	—	9	9	no	—
73	C₃×C₉	C₃³×C₃₇	1	27	yes	—
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	C₃×3₋¹⁺²×C₃₇	1	9	yes	—
84	C₃²	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
85	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
86	C₉	—	3	27	no	—
87	C₃	—	3	3	no	—
88	C₃	—	3	9	no	—
89	C₃	—	9	9	no	—
90	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
91	C₃	C₃₇×H(5,𝔽₃)	1	1	yes	characteristic
92	C₃	C₃₇×3₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×C₉⋊H(3,𝔽₃)	C₃	semidirect
3. C₃₇×#6891	C₃	semidirect
4. C₃×C₃₇×C₃²⋊C₉	C₃	semidirect
7. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
10. C₃²×C₉×C₃₇	C₃²	semidirect
12. C₃₇×C₃²⋊C₉	C₃²	semidirect
13. C₃₇×C₃²⋊C₉	C₃²	semidirect
19. C₃³×C₃₇	3₋¹⁺²	semidirect
27. #10624883	C₃₇	direct (N × H)
42. C₉⋊H(3,𝔽₃)	C₃×C₃₇	semidirect
43. #6891	C₃×C₃₇	semidirect
44. C₃×C₃²⋊C₉	C₃×C₃₇	semidirect
53. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
56. C₃²×C₉	C₃²×C₃₇	semidirect
58. C₃²⋊C₉	C₃²×C₃₇	semidirect
59. C₃²⋊C₉	C₃²×C₃₇	semidirect
60. C₃₇	#10624883	direct (N × H)
66. C₃³	3₋¹⁺²×C₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₃²×H(3,𝔽₃)×C₃₇	3
C₃₇×C₉⋊H(3,𝔽₃)	3
C₃₇×#6891	3
C₃×C₃₇×C₃²⋊C₉	3
#10624883	37

Subgroups — C37×#10624883 #41004672

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624883 #41004672