Subgroups — C₃₇×#10624884 #41004673

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×#6900	C₃	1	9	yes	maximal, complemented
3	C₃₇×#6899	C₃	1	18	yes	maximal, complemented
4	C₃₇×#6895	C₃	1	6	yes	maximal, complemented
5	C₃₇×#6891	C₃	1	3	yes	maximal
6	C₃₇×#6887	C₃	1	3	yes	maximal, complemented
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
9	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	—
10	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
11	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
13	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
14	C₃₇×C₉⋊C₉	C₃²	1	18	yes	complemented
15	C₃₇×C₉⋊C₉	C₃²	1	54	yes	complemented
16	C₃₇×C₃²⋊C₉	C₃²	1	9	yes	—
17	C₃₇×C₃²⋊C₉	C₃²	1	18	yes	—
18	C₉²×C₃₇	C₃²	1	9	yes	complemented
19	C₃³×C₃₇	—	3	3	no	—
20	C₃³×C₃₇	—	9	3	no	—
21	C₃³×C₃₇	C₃³	1	2	yes	—
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	C₃³×C₃₇	C₃³	1	1	yes	characteristic
24	3₋¹⁺²×C₃₇	—	3	18	no	—
25	3₋¹⁺²×C₃₇	—	3	9	no	—
26	3₋¹⁺²×C₃₇	—	9	9	no	—
27	C₃×C₉×C₃₇	—	3	6	no	—
28	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
29	C₃×C₉×C₃₇	C₃³	1	27	yes	—
30	C₃×C₉×C₃₇	C₃³	1	6	yes	—
31	C₃×C₉×C₃₇	C₃³	1	3	yes	—
32	#10624884	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₃²×C₃₇	—	3	3	no	—
34	C₃²×C₃₇	—	3	2	no	—
35	C₃²×C₃₇	—	3	1	no	—
36	C₃²×C₃₇	—	3	3	no	—
37	C₃²×C₃₇	—	3	6	no	—
38	C₃²×C₃₇	—	9	9	no	—
39	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
40	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
41	C₉×C₃₇	—	3	6	no	—
42	C₉×C₃₇	—	3	27	no	—
43	C₉×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
44	C₃²×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
45	#6900	C₃×C₃₇	1	9	yes	complemented
46	#6899	C₃×C₃₇	1	18	yes	complemented
47	#6895	C₃×C₃₇	1	6	yes	complemented
48	#6891	C₃×C₃₇	1	3	yes	—
49	#6887	C₃×C₃₇	1	3	yes	complemented
50	C₃×C₃₇	—	3	3	no	—
51	C₃×C₃₇	—	9	1	no	—
52	C₃×C₃₇	—	9	2	no	—
53	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
54	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
55	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
56	C₃×3₋¹⁺²	—	3	9	no	—
57	C₃×3₋¹⁺²	C₃²×C₃₇	1	6	yes	—
58	C₃×3₋¹⁺²	C₃²×C₃₇	1	3	yes	—
59	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
60	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
61	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
62	C₉⋊C₉	C₃²×C₃₇	1	18	yes	complemented
63	C₉⋊C₉	C₃²×C₃₇	1	54	yes	complemented
64	C₃²⋊C₉	C₃²×C₃₇	1	9	yes	—
65	C₃²⋊C₉	C₃²×C₃₇	1	18	yes	—
66	C₉²	C₃²×C₃₇	1	9	yes	complemented
67	C₃₇	#10624884	1	1	yes	characteristic, direct product factor, Sylow
68	C₃³	—	3	3	no	—
69	C₃³	—	9	3	no	—
70	C₃³	C₃³×C₃₇	1	2	yes	—
71	C₃³	C₃³×C₃₇	1	1	yes	characteristic
72	C₃³	C₃³×C₃₇	1	1	yes	characteristic
73	3₋¹⁺²	—	3	18	no	—
74	3₋¹⁺²	—	3	9	no	—
75	3₋¹⁺²	—	9	9	no	—
76	C₃×C₉	—	3	6	no	—
77	C₃×C₉	3₋¹⁺²×C₃₇	1	9	yes	complemented
78	C₃×C₉	C₃³×C₃₇	1	27	yes	—
79	C₃×C₉	C₃³×C₃₇	1	6	yes	—
80	C₃×C₉	C₃³×C₃₇	1	3	yes	—
81	C₃²	—	3	3	no	—
82	C₃²	—	3	2	no	—
83	C₃²	—	3	1	no	—
84	C₃²	—	3	3	no	—
85	C₃²	—	3	6	no	—
86	C₃²	—	9	9	no	—
87	C₃²	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
88	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
89	C₉	—	3	6	no	—
90	C₉	—	3	27	no	—
91	C₉	C₃×3₋¹⁺²×C₃₇	1	9	yes	—
92	C₃	—	3	3	no	—
93	C₃	—	9	1	no	—
94	C₃	—	9	2	no	—
95	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
96	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×#6900	C₃	semidirect
3. C₃₇×#6899	C₃	semidirect
4. C₃₇×#6895	C₃	semidirect
6. C₃₇×#6887	C₃	semidirect
14. C₃₇×C₉⋊C₉	C₃²	semidirect
15. C₃₇×C₉⋊C₉	C₃²	semidirect
18. C₉²×C₃₇	C₃²	semidirect
28. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
32. #10624884	C₃₇	direct (N × H)
45. #6900	C₃×C₃₇	semidirect
46. #6899	C₃×C₃₇	semidirect
47. #6895	C₃×C₃₇	semidirect
49. #6887	C₃×C₃₇	semidirect
62. C₉⋊C₉	C₃²×C₃₇	semidirect
63. C₉⋊C₉	C₃²×C₃₇	semidirect
66. C₉²	C₃²×C₃₇	semidirect
67. C₃₇	#10624884	direct (N × H)
77. 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×#6900	3
C₃₇×#6899	3
C₃₇×#6895	3
C₃₇×#6891	3
C₃₇×#6887	3
#10624884	37

Subgroups — C37×#10624884 #41004673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624884 #41004673