Subgroups — C₃×C₃₇×#6893 #41004657

Order 26973.

1232 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₃₇	C₃	1	2	yes	maximal, complemented
2	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#6893	C₃	1	27	yes	maximal, direct product factor
5	C₃×C₃₇×C₃²⋊C₉	C₃	1	9	yes	maximal, complemented
6	C₃⁴×C₃₇	C₃²	1	3	yes	—
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₃₇	—	3	18	no	—
9	C₃×3₋¹⁺²×C₃₇	C₃²	1	18	yes	complemented
10	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	—
12	C₃²×C₉×C₃₇	—	3	9	no	—
13	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
14	C₃²×C₉×C₃₇	C₃²	1	6	yes	—
15	C₃²×C₉×C₃₇	C₃²	1	9	yes	complemented
16	C₃₇×C₃²⋊C₉	C₃²	1	81	yes	complemented
17	C₃³×C₃₇	—	3	9	no	—
18	C₃³×C₃₇	—	3	9	no	—
19	C₃³×C₃₇	—	3	27	no	—
20	C₃³×C₃₇	H(3,𝔽₃)	1	9	yes	—
21	C₃³×C₃₇	C₃³	1	9	yes	—
22	C₃³×C₃₇	C₃³	1	3	yes	—
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	3₋¹⁺²×C₃₇	—	3	54	no	—
25	H(3,𝔽₃)×C₃₇	—	3	27	no	—
26	C₃×C₉×C₃₇	—	3	9	no	—
27	C₃×C₉×C₃₇	—	3	27	no	—
28	C₃×C₉×C₃₇	—	3	18	no	—
29	C₃×C₉×C₃₇	C₃³	1	9	yes	—
30	C₃×C₉×C₃₇	C₃³	1	18	yes	—
31	C₃×#6893	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₃²×C₃₇	—	3	9	no	—
33	C₃²×C₃₇	—	3	27	no	—
34	C₃²×C₃₇	—	3	9	no	—
35	C₃²×C₃₇	—	3	27	no	—
36	C₃²×C₃₇	—	9	27	no	—
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	9	yes	—
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
39	C₃²×C₃₇	#737	1	9	yes	—
40	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
41	C₉×C₃₇	—	3	9	no	—
42	C₉×C₃₇	—	3	18	no	—
43	C₃²×3₋¹⁺²	C₃×C₃₇	1	2	yes	complemented
44	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
45	C₃³×C₉	C₃×C₃₇	1	1	yes	characteristic, complemented
46	#6893	C₃×C₃₇	1	27	yes	direct product factor
47	C₃×C₃²⋊C₉	C₃×C₃₇	1	9	yes	complemented
48	C₃×C₃₇	—	3	9	no	—
49	C₃×C₃₇	—	9	9	no	—
50	C₃×C₃₇	#6893	1	9	yes	direct product factor
51	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃×C₃₇	C₃×#737	1	3	yes	—
53	C₃⁴	C₃²×C₃₇	1	3	yes	—
54	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
55	C₃×3₋¹⁺²	—	3	18	no	—
56	C₃×3₋¹⁺²	C₃²×C₃₇	1	18	yes	complemented
57	C₃×H(3,𝔽₃)	—	3	9	no	—
58	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	9	yes	—
59	C₃²×C₉	—	3	9	no	—
60	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
61	C₃²×C₉	C₃²×C₃₇	1	6	yes	—
62	C₃²×C₉	C₃²×C₃₇	1	9	yes	complemented
63	C₃²⋊C₉	C₃²×C₃₇	1	81	yes	complemented
64	C₃₇	C₃×#6893	1	1	yes	characteristic, direct product factor, Sylow
65	C₃³	—	3	9	no	—
66	C₃³	—	3	9	no	—
67	C₃³	—	3	27	no	—
68	C₃³	H(3,𝔽₃)×C₃₇	1	9	yes	—
69	C₃³	C₃³×C₃₇	1	9	yes	—
70	C₃³	C₃³×C₃₇	1	3	yes	—
71	C₃³	C₃³×C₃₇	1	1	yes	characteristic
72	3₋¹⁺²	—	3	54	no	—
73	H(3,𝔽₃)	—	3	27	no	—
74	C₃×C₉	—	3	9	no	—
75	C₃×C₉	—	3	27	no	—
76	C₃×C₉	—	3	18	no	—
77	C₃×C₉	C₃³×C₃₇	1	9	yes	—
78	C₃×C₉	C₃³×C₃₇	1	18	yes	—
79	C₃²	—	3	27	no	—
80	C₃²	—	3	9	no	—
81	C₃²	—	3	9	no	—
82	C₃²	—	3	27	no	—
83	C₃²	—	9	27	no	—
84	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	9	yes	—
85	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	3	yes	—
86	C₃²	C₃₇×#737	1	9	yes	—
87	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
88	C₉	—	3	9	no	—
89	C₉	—	3	18	no	—
90	C₃	—	3	9	no	—
91	C₃	—	9	9	no	—
92	C₃	C₃₇×#6893	1	9	yes	direct product factor
93	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
94	C₃	C₃×C₃₇×#737	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃²×3₋¹⁺²×C₃₇	C₃	semidirect
3. C₃³×C₉×C₃₇	C₃	semidirect
4. C₃₇×#6893	C₃	direct (N × H)
5. C₃×C₃₇×C₃²⋊C₉	C₃	semidirect
9. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
15. C₃²×C₉×C₃₇	C₃²	semidirect
16. C₃₇×C₃²⋊C₉	C₃²	semidirect
31. C₃×#6893	C₃₇	direct (N × H)
43. C₃²×3₋¹⁺²	C₃×C₃₇	semidirect
45. C₃³×C₉	C₃×C₃₇	semidirect
46. #6893	C₃×C₃₇	direct (N × H)
47. C₃×C₃²⋊C₉	C₃×C₃₇	semidirect
50. C₃×C₃₇	#6893	direct (N × H)
56. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
62. C₃²×C₉	C₃²×C₃₇	semidirect
63. C₃²⋊C₉	C₃²×C₃₇	semidirect
64. C₃₇	C₃×#6893	direct (N × H)
92. C₃	C₃₇×#6893	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×3₋¹⁺²×C₃₇	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃³×C₉×C₃₇	3
C₃₇×#6893	3
C₃×C₃₇×C₃²⋊C₉	3
C₃×#6893	37

Subgroups — C3×C37×#6893 #41004657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×#6893 #41004657