Subgroups — C₃₇×#10624761 #41004550

Order 26973.

476 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃₇×#6910	C₃	1	8	yes	maximal, complemented
3	C₃×C₃₇×AΣL(1,𝔽₂₇)	C₃	1	4	yes	maximal, complemented
4	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic, complemented
5	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	complemented
6	C₃×H(3,𝔽₃)×C₃₇	—	3	12	no	—
7	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	4	yes	—
8	C₃₇×#731	—	3	24	no	—
9	C₃₇×AΣL(1,𝔽₂₇)	—	3	12	no	—
10	C₃³×C₃₇	—	3	9	no	—
11	C₃³×C₃₇	—	3	4	no	—
12	C₃³×C₃₇	—	3	4	no	—
13	C₃³×C₃₇	—	9	3	no	—
14	C₃³×C₃₇	C₃³	1	1	yes	characteristic
15	3₋¹⁺²×C₃₇	—	3	8	no	—
16	3₋¹⁺²×C₃₇	—	3	16	no	—
17	H(3,𝔽₃)×C₃₇	—	3	18	no	—
18	H(3,𝔽₃)×C₃₇	—	3	8	no	—
19	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	12	yes	—
20	C₃×C₉×C₃₇	—	3	8	no	—
21	#10624761	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₃²×C₃₇	—	3	12	no	—
23	C₃²×C₃₇	—	9	1	no	—
24	C₃²×C₃₇	—	9	12	no	—
25	C₃²×C₃₇	—	9	1	no	—
26	C₃²×C₃₇	—	27	4	no	—
27	C₃²×C₃₇	—	27	9	no	—
28	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	4	yes	—
29	C₉×C₃₇	—	9	8	no	—
30	H(5,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic, complemented
31	#6910	C₃×C₃₇	1	8	yes	complemented
32	C₃×AΣL(1,𝔽₂₇)	C₃×C₃₇	1	4	yes	complemented
33	C₃×C₃₇	—	3	4	no	—
34	C₃×C₃₇	—	9	12	no	—
35	C₃×C₃₇	—	9	3	no	—
36	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
37	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic, complemented
38	C₃×3₋¹⁺²	C₃²×C₃₇	1	8	yes	complemented
39	C₃×H(3,𝔽₃)	—	3	12	no	—
40	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	4	yes	—
41	#731	—	3	24	no	—
42	AΣL(1,𝔽₂₇)	—	3	12	no	—
43	C₃₇	#10624761	1	1	yes	characteristic, direct product factor, Sylow
44	C₃³	—	3	9	no	—
45	C₃³	—	3	4	no	—
46	C₃³	—	3	4	no	—
47	C₃³	—	9	3	no	—
48	C₃³	C₃³×C₃₇	1	1	yes	characteristic
49	3₋¹⁺²	—	3	8	no	—
50	3₋¹⁺²	—	3	16	no	—
51	H(3,𝔽₃)	—	3	18	no	—
52	H(3,𝔽₃)	—	3	8	no	—
53	H(3,𝔽₃)	H(3,𝔽₃)×C₃₇	1	12	yes	—
54	C₃×C₉	—	3	8	no	—
55	C₃²	—	3	12	no	—
56	C₃²	—	9	1	no	—
57	C₃²	—	9	12	no	—
58	C₃²	—	9	1	no	—
59	C₃²	—	27	4	no	—
60	C₃²	—	27	9	no	—
61	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	4	yes	—
62	C₉	—	9	8	no	—
63	C₃	—	3	4	no	—
64	C₃	—	9	12	no	—
65	C₃	—	9	3	no	—
66	C₃	C₃₇×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₇×H(5,𝔽₃)	C₃	semidirect
2. C₃₇×#6910	C₃	semidirect
3. C₃×C₃₇×AΣL(1,𝔽₂₇)	C₃	semidirect
4. C₃⁴×C₃₇	C₃²	semidirect
5. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
21. #10624761	C₃₇	direct (N × H)
30. H(5,𝔽₃)	C₃×C₃₇	semidirect
31. #6910	C₃×C₃₇	semidirect
32. C₃×AΣL(1,𝔽₂₇)	C₃×C₃₇	semidirect
37. C₃⁴	C₃²×C₃₇	semidirect
38. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
43. C₃₇	#10624761	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×H(5,𝔽₃)	3
C₃₇×#6910	3
C₃×C₃₇×AΣL(1,𝔽₂₇)	3
#10624761	37

Subgroups — C37×#10624761 #41004550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624761 #41004550