Subgroups — C₃×C₃₇×#6896 #41004662

Order 26973.

884 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
2	C₃₇×#6896	C₃	1	27	yes	maximal, direct product factor
3	C₃×C₃₇×C₉⋊C₉	C₃	1	4	yes	maximal, complemented
4	C₃×C₃₇×C₉⋊C₉	C₃	1	4	yes	maximal, complemented
5	C₃×C₃₇×C₃²⋊C₉	C₃	1	2	yes	maximal
6	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
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	—
10	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
11	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
13	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
14	C₃₇×C₉⋊C₉	C₃²	1	36	yes	complemented
15	C₃₇×C₉⋊C₉	C₃²	1	36	yes	complemented
16	C₃₇×C₃²⋊C₉	C₃²	1	18	yes	—
17	C₉²×C₃₇	C₃²	1	9	yes	complemented
18	C₃³×C₃₇	—	3	6	no	—
19	C₃³×C₃₇	—	3	6	no	—
20	C₃³×C₃₇	C₃³	1	3	yes	—
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	3₋¹⁺²×C₃₇	—	3	54	no	—
23	C₃×C₉×C₃₇	—	3	12	no	—
24	C₃×C₉×C₃₇	—	3	6	no	—
25	C₃×C₉×C₃₇	—	3	12	no	—
26	C₃×C₉×C₃₇	3₋¹⁺²	1	18	yes	complemented
27	C₃×C₉×C₃₇	C₃³	1	6	yes	—
28	C₃×C₉×C₃₇	C₃³	1	12	yes	—
29	C₃×C₉×C₃₇	C₃³	1	12	yes	—
30	C₃×C₉×C₃₇	C₃³	1	6	yes	—
31	C₃×#6896	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₃²×C₃₇	—	3	6	no	—
33	C₃²×C₃₇	—	3	6	no	—
34	C₃²×C₃₇	—	9	9	no	—
35	C₃²×C₃₇	C₃×3₋¹⁺²	1	6	yes	—
36	C₃²×C₃₇	#737	1	6	yes	—
37	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
38	C₉×C₃₇	—	3	12	no	—
39	C₉×C₃₇	—	3	6	no	—
40	C₉×C₃₇	—	3	12	no	—
41	C₉×C₃₇	C₃×3₋¹⁺²	1	18	yes	complemented
42	C₃²×3₋¹⁺²	C₃×C₃₇	1	2	yes	—
43	#6896	C₃×C₃₇	1	27	yes	direct product factor
44	C₃×C₉⋊C₉	C₃×C₃₇	1	4	yes	complemented
45	C₃×C₉⋊C₉	C₃×C₃₇	1	4	yes	complemented
46	C₃×C₃²⋊C₉	C₃×C₃₇	1	2	yes	—
47	C₃×C₉²	C₃×C₃₇	1	1	yes	characteristic, complemented
48	C₃×C₃₇	—	9	3	no	—
49	C₃×C₃₇	#6896	1	9	yes	direct product factor
50	C₃×C₃₇	C₃²×3₋¹⁺²	1	2	yes	—
51	C₃×C₃₇	C₃×#737	1	2	yes	—
52	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
53	C₃×3₋¹⁺²	—	3	18	no	—
54	C₃×3₋¹⁺²	C₃²×C₃₇	1	18	yes	—
55	C₃²×C₉	C₃²×C₃₇	1	4	yes	—
56	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
57	C₃²×C₉	C₃²×C₃₇	1	4	yes	—
58	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
59	C₉⋊C₉	C₃²×C₃₇	1	36	yes	complemented
60	C₉⋊C₉	C₃²×C₃₇	1	36	yes	complemented
61	C₃²⋊C₉	C₃²×C₃₇	1	18	yes	—
62	C₉²	C₃²×C₃₇	1	9	yes	complemented
63	C₃₇	C₃×#6896	1	1	yes	characteristic, direct product factor, Sylow
64	C₃³	—	3	6	no	—
65	C₃³	—	3	6	no	—
66	C₃³	C₃³×C₃₇	1	3	yes	—
67	C₃³	C₃³×C₃₇	1	1	yes	characteristic
68	3₋¹⁺²	—	3	54	no	—
69	C₃×C₉	—	3	12	no	—
70	C₃×C₉	—	3	6	no	—
71	C₃×C₉	—	3	12	no	—
72	C₃×C₉	3₋¹⁺²×C₃₇	1	18	yes	complemented
73	C₃×C₉	C₃³×C₃₇	1	6	yes	—
74	C₃×C₉	C₃³×C₃₇	1	12	yes	—
75	C₃×C₉	C₃³×C₃₇	1	12	yes	—
76	C₃×C₉	C₃³×C₃₇	1	6	yes	—
77	C₃²	—	3	6	no	—
78	C₃²	—	3	6	no	—
79	C₃²	—	9	9	no	—
80	C₃²	C₃×3₋¹⁺²×C₃₇	1	6	yes	—
81	C₃²	C₃₇×#737	1	6	yes	—
82	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
83	C₉	—	3	12	no	—
84	C₉	—	3	6	no	—
85	C₉	—	3	12	no	—
86	C₉	C₃×3₋¹⁺²×C₃₇	1	18	yes	complemented
87	C₃	—	9	3	no	—
88	C₃	C₃₇×#6896	1	9	yes	direct product factor
89	C₃	C₃²×3₋¹⁺²×C₃₇	1	2	yes	—
90	C₃	C₃×C₃₇×#737	1	2	yes	—

	subgroup	G/H
center, socle	C₃³×C₃₇	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
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
2. C₃₇×#6896	C₃	direct (N × H)
3. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
4. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
6. C₃×C₉²×C₃₇	C₃	semidirect
14. C₃₇×C₉⋊C₉	C₃²	semidirect
15. C₃₇×C₉⋊C₉	C₃²	semidirect
17. C₉²×C₃₇	C₃²	semidirect
26. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
31. C₃×#6896	C₃₇	direct (N × H)
41. C₉×C₃₇	C₃×3₋¹⁺²	semidirect
43. #6896	C₃×C₃₇	direct (N × H)
44. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
45. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
47. C₃×C₉²	C₃×C₃₇	semidirect
49. C₃×C₃₇	#6896	direct (N × H)
59. C₉⋊C₉	C₃²×C₃₇	semidirect
60. C₉⋊C₉	C₃²×C₃₇	semidirect
62. C₉²	C₃²×C₃₇	semidirect
63. C₃₇	C₃×#6896	direct (N × H)
72. C₃×C₉	3₋¹⁺²×C₃₇	semidirect
86. C₉	C₃×3₋¹⁺²×C₃₇	semidirect
88. C₃	C₃₇×#6896	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
C₃₇×#6896	3
C₃×C₃₇×C₉⋊C₉	3
C₃×C₃₇×C₃²⋊C₉	3
C₃×C₉²×C₃₇	3
C₃×#6896	37

Subgroups — C3×C37×#6896 #41004662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×#6896 #41004662