Subgroups — C₃₇×#10624607 #41004396

Order 26973.

452 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×#6891	C₃	1	2	yes	maximal
3	C₃₇×#6860	C₃	1	9	yes	maximal, complemented
4	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
6	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
7	C₃×3₋¹⁺²×C₃₇	—	3	18	no	—
8	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
9	C₃×3₋¹⁺²×C₃₇	C₃²	1	6	yes	—
10	C₃²×C₉×C₃₇	—	3	2	no	—
11	C₃₇×C₃²⋊C₉	—	3	2	no	—
12	C₃₇×C₃²⋊C₉	—	3	2	no	—
13	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
14	C₃³×C₃₇	—	3	6	no	—
15	C₃³×C₃₇	—	9	2	no	—
16	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	3₋¹⁺²×C₃₇	—	3	18	no	—
20	3₋¹⁺²×C₃₇	—	3	18	no	—
21	3₋¹⁺²×C₃₇	—	3	18	no	—
22	3₋¹⁺²×C₃₇	—	9	6	no	—
23	C₃×C₉×C₃₇	—	3	18	no	—
24	C₃×C₉×C₃₇	—	3	6	no	—
25	C₃×C₉×C₃₇	—	3	6	no	—
26	#10624607	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
27	C₃²×C₃₇	—	3	4	no	—
28	C₃²×C₃₇	—	3	2	no	—
29	C₃²×C₃₇	—	3	2	no	—
30	C₃²×C₃₇	—	3	2	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	—	3	2	no	—
33	C₃²×C₃₇	—	9	9	no	—
34	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	6	yes	—
35	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
36	C₉×C₃₇	—	3	18	no	—
37	C₉×C₃₇	—	9	6	no	—
38	C₃²×3₋¹⁺²	C₃×C₃₇	1	2	yes	—
39	#6891	C₃×C₃₇	1	2	yes	—
40	#6860	C₃×C₃₇	1	9	yes	complemented
41	C₃×C₃₇	—	3	6	no	—
42	C₃×C₃₇	—	9	1	no	—
43	C₃×C₃₇	—	9	1	no	—
44	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
45	C₃×C₃₇	#6912	1	2	yes	—
46	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
47	C₃×3₋¹⁺²	—	3	2	no	—
48	C₃×3₋¹⁺²	—	3	2	no	—
49	C₃×3₋¹⁺²	—	3	18	no	—
50	C₃×3₋¹⁺²	—	3	2	no	—
51	C₃×3₋¹⁺²	C₃²×C₃₇	1	6	yes	—
52	C₃²×C₉	—	3	2	no	—
53	C₃²⋊C₉	—	3	2	no	—
54	C₃²⋊C₉	—	3	2	no	—
55	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
56	C₃₇	#10624607	1	1	yes	characteristic, direct product factor, Sylow
57	C₃³	—	3	6	no	—
58	C₃³	—	9	2	no	—
59	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
60	C₃³	H(3,𝔽₃)×C₃₇	1	2	yes	—
61	C₃³	C₃³×C₃₇	1	1	yes	characteristic
62	3₋¹⁺²	—	3	18	no	—
63	3₋¹⁺²	—	3	18	no	—
64	3₋¹⁺²	—	3	18	no	—
65	3₋¹⁺²	—	9	6	no	—
66	C₃×C₉	—	3	18	no	—
67	C₃×C₉	—	3	6	no	—
68	C₃×C₉	—	3	6	no	—
69	C₃²	—	3	2	no	—
70	C₃²	—	3	4	no	—
71	C₃²	—	3	2	no	—
72	C₃²	—	3	2	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	9	9	no	—
76	C₃²	C₃₇×AΣL(1,𝔽₂₇)	1	6	yes	—
77	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
78	C₉	—	3	18	no	—
79	C₉	—	9	6	no	—
80	C₃	—	3	6	no	—
81	C₃	—	9	1	no	—
82	C₃	—	9	1	no	—
83	C₃	C₃×C₃₇×AΣL(1,𝔽₂₇)	1	2	yes	—
84	C₃	C₃₇×#6912	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
LCS	G ▷ C₃³ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ C₃⁴×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₇×#6860	C₃	semidirect
26. #10624607	C₃₇	direct (N × H)
40. #6860	C₃×C₃₇	semidirect
56. C₃₇	#10624607	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×#6891	3
C₃₇×#6860	3
#10624607	37

Subgroups — C37×#10624607 #41004396

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624607 #41004396