Subgroups — C₃₇×#10624872 #41004661

Order 26973.

824 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃₇×#6895	C₃	1	6	yes	maximal, complemented
3	C₃₇×#6894	C₃	1	18	yes	maximal, complemented
4	C₃₇×#6891	C₃	1	3	yes	maximal
5	C₉×3₋¹⁺²×C₃₇	C₃	1	9	yes	maximal, complemented
6	C₃₇×#6887	C₃	1	3	yes	maximal, complemented
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
9	C₃²×C₉×C₃₇	—	3	9	no	—
10	C₃²×C₉×C₃₇	C₃²	1	6	yes	—
11	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	3	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	—	3	9	no	—
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	C₃³×C₃₇	C₃³	1	3	yes	—
23	3₋¹⁺²×C₃₇	—	3	27	no	—
24	C₃×C₉×C₃₇	—	3	6	no	—
25	C₃×C₉×C₃₇	—	3	18	no	—
26	C₃×C₉×C₃₇	—	3	9	no	—
27	C₃×C₉×C₃₇	3₋¹⁺²	1	9	yes	complemented
28	C₃×C₉×C₃₇	C₃³	1	27	yes	—
29	C₃×C₉×C₃₇	C₃³	1	6	yes	—
30	C₃×C₉×C₃₇	C₃³	1	2	yes	—
31	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
32	#10624872	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₃²×C₃₇	—	3	27	no	—
34	C₃²×C₃₇	—	3	3	no	—
35	C₃²×C₃₇	—	3	3	no	—
36	C₃²×C₃₇	—	3	6	no	—
37	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
38	C₃²×C₃₇	#737	1	9	yes	—
39	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
40	C₉×C₃₇	—	3	6	no	—
41	C₉×C₃₇	—	3	27	no	—
42	C₉×C₃₇	C₃×3₋¹⁺²	1	6	yes	—
43	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
44	C₃³×C₉	C₃×C₃₇	1	1	yes	characteristic
45	#6895	C₃×C₃₇	1	6	yes	complemented
46	#6894	C₃×C₃₇	1	18	yes	complemented
47	#6891	C₃×C₃₇	1	3	yes	—
48	C₉×3₋¹⁺²	C₃×C₃₇	1	9	yes	complemented
49	#6887	C₃×C₃₇	1	3	yes	complemented
50	C₃×C₃₇	—	3	3	no	—
51	C₃×C₃₇	—	3	9	no	—
52	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
53	C₃×C₃₇	C₃×#737	1	3	yes	—
54	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
55	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
56	C₃²×C₉	—	3	9	no	—
57	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
58	C₃²×C₉	C₃²×C₃₇	1	6	yes	—
59	C₃²×C₉	C₃²×C₃₇	1	2	yes	—
60	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
61	C₉⋊C₉	C₃²×C₃₇	1	18	yes	complemented
62	C₉⋊C₉	C₃²×C₃₇	1	54	yes	complemented
63	C₃²⋊C₉	C₃²×C₃₇	1	9	yes	—
64	C₃²⋊C₉	C₃²×C₃₇	1	18	yes	—
65	C₉²	C₃²×C₃₇	1	9	yes	complemented
66	C₃₇	#10624872	1	1	yes	characteristic, direct product factor, Sylow
67	C₃³	—	3	3	no	—
68	C₃³	—	3	9	no	—
69	C₃³	C₃³×C₃₇	1	1	yes	characteristic
70	C₃³	C₃³×C₃₇	1	3	yes	—
71	3₋¹⁺²	—	3	27	no	—
72	C₃×C₉	—	3	6	no	—
73	C₃×C₉	—	3	9	no	—
74	C₃×C₉	—	3	18	no	—
75	C₃×C₉	3₋¹⁺²×C₃₇	1	9	yes	complemented
76	C₃×C₉	C₃³×C₃₇	1	27	yes	—
77	C₃×C₉	C₃³×C₃₇	1	6	yes	—
78	C₃×C₉	C₃³×C₃₇	1	2	yes	—
79	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
80	C₃²	—	3	27	no	—
81	C₃²	—	3	3	no	—
82	C₃²	—	3	3	no	—
83	C₃²	—	3	6	no	—
84	C₃²	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
85	C₃²	C₃₇×#737	1	9	yes	—
86	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
87	C₉	—	3	6	no	—
88	C₉	—	3	27	no	—
89	C₉	C₃×3₋¹⁺²×C₃₇	1	6	yes	—
90	C₉	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
91	C₃	—	3	3	no	—
92	C₃	—	3	9	no	—
93	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
94	C₃	C₃×C₃₇×#737	1	3	yes	—

	subgroup	G/H
center	C₃×C₉×C₃₇	C₃³
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
socle	C₃²×C₃₇	C₃⁴
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#6895	C₃	semidirect
3. C₃₇×#6894	C₃	semidirect
5. C₉×3₋¹⁺²×C₃₇	C₃	semidirect
6. C₃₇×#6887	C₃	semidirect
14. C₃₇×C₉⋊C₉	C₃²	semidirect
15. C₃₇×C₉⋊C₉	C₃²	semidirect
18. C₉²×C₃₇	C₃²	semidirect
27. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
32. #10624872	C₃₇	direct (N × H)
45. #6895	C₃×C₃₇	semidirect
46. #6894	C₃×C₃₇	semidirect
48. C₉×3₋¹⁺²	C₃×C₃₇	semidirect
49. #6887	C₃×C₃₇	semidirect
61. C₉⋊C₉	C₃²×C₃₇	semidirect
62. C₉⋊C₉	C₃²×C₃₇	semidirect
65. C₉²	C₃²×C₃₇	semidirect
66. C₃₇	#10624872	direct (N × H)
75. 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
C₃³×C₉×C₃₇	3
C₃₇×#6895	3
C₃₇×#6894	3
C₃₇×#6891	3
C₉×3₋¹⁺²×C₃₇	3
C₃₇×#6887	3
#10624872	37

Subgroups — C37×#10624872 #41004661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624872 #41004661