Subgroups — C₃₇×#10624885 #41004674

Order 26973.

716 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃₇×#6896	C₃	1	27	yes	maximal, complemented
3	C₃₇×#6895	C₃	1	6	yes	maximal, complemented
4	C₃₇×#6891	C₃	1	3	yes	maximal
5	C₃₇×#6887	C₃	1	3	yes	maximal, complemented
6	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
7	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
8	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
9	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
10	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
11	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
12	C₃₇×C₉⋊C₉	C₃²	1	18	yes	complemented
13	C₃₇×C₉⋊C₉	C₃²	1	54	yes	complemented
14	C₃₇×C₃²⋊C₉	C₃²	1	9	yes	—
15	C₃₇×C₃²⋊C₉	C₃²	1	18	yes	—
16	C₉²×C₃₇	C₃²	1	9	yes	complemented
17	C₃³×C₃₇	—	3	3	no	—
18	C₃³×C₃₇	—	9	3	no	—
19	C₃³×C₃₇	C₃³	1	3	yes	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	3₋¹⁺²×C₃₇	—	3	27	no	—
22	3₋¹⁺²×C₃₇	—	9	9	no	—
23	C₃×C₉×C₃₇	—	3	6	no	—
24	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
25	C₃×C₉×C₃₇	C₃³	1	27	yes	—
26	C₃×C₉×C₃₇	C₃³	1	6	yes	—
27	C₃×C₉×C₃₇	C₃³	1	3	yes	—
28	#10624885	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃²×C₃₇	—	3	3	no	—
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	3	6	no	—
33	C₃²×C₃₇	—	9	9	no	—
34	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
35	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
36	C₉×C₃₇	—	3	6	no	—
37	C₉×C₃₇	—	3	27	no	—
38	C₉×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
39	C₃²×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
40	#6896	C₃×C₃₇	1	27	yes	complemented
41	#6895	C₃×C₃₇	1	6	yes	complemented
42	#6891	C₃×C₃₇	1	3	yes	—
43	#6887	C₃×C₃₇	1	3	yes	complemented
44	C₃×C₃₇	—	3	3	no	—
45	C₃×C₃₇	—	9	3	no	—
46	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
47	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
48	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
49	C₃×3₋¹⁺²	—	3	9	no	—
50	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
51	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
52	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
53	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
54	C₉⋊C₉	C₃²×C₃₇	1	18	yes	complemented
55	C₉⋊C₉	C₃²×C₃₇	1	54	yes	complemented
56	C₃²⋊C₉	C₃²×C₃₇	1	9	yes	—
57	C₃²⋊C₉	C₃²×C₃₇	1	18	yes	—
58	C₉²	C₃²×C₃₇	1	9	yes	complemented
59	C₃₇	#10624885	1	1	yes	characteristic, direct product factor, Sylow
60	C₃³	—	3	3	no	—
61	C₃³	—	9	3	no	—
62	C₃³	C₃³×C₃₇	1	3	yes	—
63	C₃³	C₃³×C₃₇	1	1	yes	characteristic
64	3₋¹⁺²	—	3	27	no	—
65	3₋¹⁺²	—	9	9	no	—
66	C₃×C₉	—	3	6	no	—
67	C₃×C₉	3₋¹⁺²×C₃₇	1	9	yes	complemented
68	C₃×C₉	C₃³×C₃₇	1	27	yes	—
69	C₃×C₉	C₃³×C₃₇	1	6	yes	—
70	C₃×C₉	C₃³×C₃₇	1	3	yes	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	3	no	—
73	C₃²	—	3	3	no	—
74	C₃²	—	3	6	no	—
75	C₃²	—	9	9	no	—
76	C₃²	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
77	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
78	C₉	—	3	6	no	—
79	C₉	—	3	27	no	—
80	C₉	C₃×3₋¹⁺²×C₃₇	1	9	yes	—
81	C₃	—	3	3	no	—
82	C₃	—	9	3	no	—
83	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
84	C₃	C₃₇×3₋¹⁺⁴	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
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×#6896	C₃	semidirect
3. C₃₇×#6895	C₃	semidirect
5. C₃₇×#6887	C₃	semidirect
12. C₃₇×C₉⋊C₉	C₃²	semidirect
13. C₃₇×C₉⋊C₉	C₃²	semidirect
16. C₉²×C₃₇	C₃²	semidirect
24. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
28. #10624885	C₃₇	direct (N × H)
40. #6896	C₃×C₃₇	semidirect
41. #6895	C₃×C₃₇	semidirect
43. #6887	C₃×C₃₇	semidirect
54. C₉⋊C₉	C₃²×C₃₇	semidirect
55. C₉⋊C₉	C₃²×C₃₇	semidirect
58. C₉²	C₃²×C₃₇	semidirect
59. C₃₇	#10624885	direct (N × H)
67. C₃×C₉	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
C₃²×3₋¹⁺²×C₃₇	3
C₃₇×#6896	3
C₃₇×#6895	3
C₃₇×#6891	3
C₃₇×#6887	3
#10624885	37

Subgroups — C37×#10624885 #41004674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624885 #41004674