Subgroups — C₃₇×#10624910 #41004699

Order 26973.

740 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#6893	C₃	1	9	yes	maximal, complemented
2	C₃₇×C₉⋊H(3,𝔽₃)	C₃	1	27	yes	maximal, complemented
3	C₃₇×#6891	C₃	1	3	yes	maximal, complemented
4	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₃₇	C₃²	1	9	yes	—
7	C₃×3₋¹⁺²×C₃₇	C₃²	1	18	yes	complemented
8	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	9	yes	—
9	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
10	C₃²×C₉×C₃₇	C₃²	1	9	yes	complemented
11	C₃₇×C₃²⋊C₉	C₃²	1	18	yes	—
12	C₃₇×C₃²⋊C₉	C₃²	1	9	yes	—
13	C₃₇×C₃²⋊C₉	C₃²	1	54	yes	complemented
14	C₃³×C₃₇	—	9	1	no	—
15	C₃³×C₃₇	—	9	3	no	—
16	C₃³×C₃₇	C₃³	1	1	yes	characteristic
17	C₃³×C₃₇	C₃³	1	3	yes	—
18	C₃³×C₃₇	C₃³	1	9	yes	—
19	3₋¹⁺²×C₃₇	—	9	9	no	—
20	3₋¹⁺²×C₃₇	—	9	18	no	—
21	H(3,𝔽₃)×C₃₇	—	9	3	no	—
22	H(3,𝔽₃)×C₃₇	—	9	9	no	—
23	C₃×C₉×C₃₇	—	9	9	no	—
24	C₃×C₉×C₃₇	C₃³	1	27	yes	—
25	#10624910	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃²×C₃₇	—	3	9	no	—
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	3	no	—
30	C₃²×C₃₇	—	3	27	no	—
31	C₃²×C₃₇	—	3	3	no	—
32	C₃²×C₃₇	—	3	6	no	—
33	C₃²×C₃₇	—	9	9	no	—
34	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
35	C₉×C₃₇	—	3	27	no	—
36	#6893	C₃×C₃₇	1	9	yes	complemented
37	C₉⋊H(3,𝔽₃)	C₃×C₃₇	1	27	yes	complemented
38	#6891	C₃×C₃₇	1	3	yes	complemented
39	C₃²⋊H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
40	C₃×C₃₇	—	9	9	no	—
41	C₃×C₃₇	—	9	1	no	—
42	C₃×C₃₇	—	9	3	no	—
43	C₃×C₃₇	H(5,𝔽₃)	1	1	yes	characteristic
44	C₃×C₃₇	3₋¹⁺⁴	1	3	yes	—
45	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
46	C₃×3₋¹⁺²	C₃²×C₃₇	1	9	yes	—
47	C₃×3₋¹⁺²	C₃²×C₃₇	1	18	yes	complemented
48	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	9	yes	—
49	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	3	yes	—
50	C₃²×C₉	C₃²×C₃₇	1	9	yes	complemented
51	C₃²⋊C₉	C₃²×C₃₇	1	18	yes	—
52	C₃²⋊C₉	C₃²×C₃₇	1	9	yes	—
53	C₃²⋊C₉	C₃²×C₃₇	1	54	yes	complemented
54	C₃₇	#10624910	1	1	yes	characteristic, direct product factor, Sylow
55	C₃³	—	9	1	no	—
56	C₃³	—	9	3	no	—
57	C₃³	C₃³×C₃₇	1	1	yes	characteristic
58	C₃³	C₃³×C₃₇	1	3	yes	—
59	C₃³	C₃³×C₃₇	1	9	yes	—
60	3₋¹⁺²	—	9	9	no	—
61	3₋¹⁺²	—	9	18	no	—
62	H(3,𝔽₃)	—	9	3	no	—
63	H(3,𝔽₃)	—	9	9	no	—
64	C₃×C₉	—	9	9	no	—
65	C₃×C₉	C₃³×C₃₇	1	27	yes	—
66	C₃²	—	3	9	no	—
67	C₃²	—	3	1	no	—
68	C₃²	—	3	3	no	—
69	C₃²	—	3	3	no	—
70	C₃²	—	3	27	no	—
71	C₃²	—	3	3	no	—
72	C₃²	—	3	6	no	—
73	C₃²	—	9	9	no	—
74	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
75	C₉	—	3	27	no	—
76	C₃	—	9	1	no	—
77	C₃	—	9	3	no	—
78	C₃	—	9	9	no	—
79	C₃	C₃₇×H(5,𝔽₃)	1	1	yes	characteristic
80	C₃	C₃₇×3₋¹⁺⁴	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
prime radical	C₃₇×C₃²⋊H(3,𝔽₃)	C₃
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
1. C₃₇×#6893	C₃	semidirect
2. C₃₇×C₉⋊H(3,𝔽₃)	C₃	semidirect
3. C₃₇×#6891	C₃	semidirect
7. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
10. C₃²×C₉×C₃₇	C₃²	semidirect
13. C₃₇×C₃²⋊C₉	C₃²	semidirect
25. #10624910	C₃₇	direct (N × H)
36. #6893	C₃×C₃₇	semidirect
37. C₉⋊H(3,𝔽₃)	C₃×C₃₇	semidirect
38. #6891	C₃×C₃₇	semidirect
47. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
50. C₃²×C₉	C₃²×C₃₇	semidirect
53. C₃²⋊C₉	C₃²×C₃₇	semidirect
54. C₃₇	#10624910	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×#6893	3
C₃₇×C₉⋊H(3,𝔽₃)	3
C₃₇×#6891	3
C₃₇×C₃²⋊H(3,𝔽₃)	3
#10624910	37

Subgroups — C37×#10624910 #41004699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624910 #41004699