Subgroups — C₃₇×#10624589 #41004378

Order 26973.

668 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	2	yes	maximal, complemented
2	C₃₇×#6891	C₃	1	2	yes	maximal, complemented
3	C₃₇×#6856	C₃	1	9	yes	maximal, complemented
4	C₃⁴×C₃₇	—	3	2	no	—
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
7	C₃×H(3,𝔽₃)×C₃₇	—	3	2	no	—
8	C₃×H(3,𝔽₃)×C₃₇	—	3	2	no	—
9	C₃×H(3,𝔽₃)×C₃₇	—	3	18	no	—
10	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	6	yes	complemented
11	C₃₇×C₃²⋊C₉	—	3	4	no	—
12	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	complemented
13	C₃³×C₃₇	—	3	18	no	—
14	C₃³×C₃₇	—	3	6	no	—
15	C₃³×C₃₇	—	3	6	no	—
16	C₃³×C₃₇	—	9	2	no	—
17	C₃³×C₃₇	—	9	12	no	—
18	C₃³×C₃₇	—	9	6	no	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
21	C₃³×C₃₇	C₃³	1	1	yes	characteristic
22	3₋¹⁺²×C₃₇	—	9	6	no	—
23	H(3,𝔽₃)×C₃₇	—	3	18	no	—
24	H(3,𝔽₃)×C₃₇	—	3	18	no	—
25	H(3,𝔽₃)×C₃₇	—	3	18	no	—
26	C₃×C₉×C₃₇	—	3	6	no	—
27	#10624589	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₃²×C₃₇	—	3	2	no	—
29	C₃²×C₃₇	—	3	4	no	—
30	C₃²×C₃₇	—	3	2	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	—	3	18	no	—
33	C₃²×C₃₇	—	3	2	no	—
34	C₃²×C₃₇	—	3	2	no	—
35	C₃²×C₃₇	—	9	9	no	—
36	C₃²×C₃₇	—	9	54	no	—
37	C₃²×C₃₇	—	9	6	no	—
38	C₃²×C₃₇	—	9	12	no	—
39	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	6	yes	—
40	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
41	C₉×C₃₇	—	9	6	no	—
42	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	2	yes	complemented
43	#6891	C₃×C₃₇	1	2	yes	complemented
44	#6856	C₃×C₃₇	1	9	yes	complemented
45	C₃×C₃₇	—	3	6	no	—
46	C₃×C₃₇	—	9	1	no	—
47	C₃×C₃₇	—	9	1	no	—
48	C₃×C₃₇	—	9	18	no	—
49	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
50	C₃×C₃₇	#6911	1	2	yes	—
51	C₃⁴	—	3	2	no	—
52	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
53	C₃×3₋¹⁺²	—	3	2	no	—
54	C₃×H(3,𝔽₃)	—	3	2	no	—
55	C₃×H(3,𝔽₃)	—	3	2	no	—
56	C₃×H(3,𝔽₃)	—	3	18	no	—
57	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	6	yes	complemented
58	C₃²⋊C₉	—	3	4	no	—
59	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	complemented
60	C₃₇	#10624589	1	1	yes	characteristic, direct product factor, Sylow
61	C₃³	—	3	18	no	—
62	C₃³	—	3	6	no	—
63	C₃³	—	3	6	no	—
64	C₃³	—	9	2	no	—
65	C₃³	—	9	12	no	—
66	C₃³	—	9	6	no	—
67	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
68	C₃³	H(3,𝔽₃)×C₃₇	1	2	yes	—
69	C₃³	C₃³×C₃₇	1	1	yes	characteristic
70	3₋¹⁺²	—	9	6	no	—
71	H(3,𝔽₃)	—	3	18	no	—
72	H(3,𝔽₃)	—	3	18	no	—
73	H(3,𝔽₃)	—	3	18	no	—
74	C₃×C₉	—	3	6	no	—
75	C₃²	—	3	2	no	—
76	C₃²	—	3	4	no	—
77	C₃²	—	3	2	no	—
78	C₃²	—	3	2	no	—
79	C₃²	—	3	18	no	—
80	C₃²	—	3	2	no	—
81	C₃²	—	3	2	no	—
82	C₃²	—	9	9	no	—
83	C₃²	—	9	54	no	—
84	C₃²	—	9	6	no	—
85	C₃²	—	9	12	no	—
86	C₃²	C₃₇×AΣL(1,𝔽₂₇)	1	6	yes	—
87	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
88	C₉	—	9	6	no	—
89	C₃	—	3	6	no	—
90	C₃	—	9	1	no	—
91	C₃	—	9	1	no	—
92	C₃	—	9	18	no	—
93	C₃	C₃×C₃₇×AΣL(1,𝔽₂₇)	1	2	yes	—
94	C₃	C₃₇×#6911	1	2	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃³×C₃₇
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
1. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
2. C₃₇×#6891	C₃	semidirect
3. C₃₇×#6856	C₃	semidirect
10. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
12. C₃₇×C₃²⋊C₉	C₃²	semidirect
27. #10624589	C₃₇	direct (N × H)
42. C₃²×H(3,𝔽₃)	C₃×C₃₇	semidirect
43. #6891	C₃×C₃₇	semidirect
44. #6856	C₃×C₃₇	semidirect
57. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
59. C₃²⋊C₉	C₃²×C₃₇	semidirect
60. C₃₇	#10624589	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×H(3,𝔽₃)×C₃₇	3
C₃₇×#6891	3
C₃₇×#6856	3
#10624589	37

Subgroups — C37×#10624589 #41004378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624589 #41004378