Subgroups — C₃₇×#10624933 #41004722

Order 26973.

936 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×3₋¹⁺⁴	C₃	1	3	yes	maximal
2	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
3	C₃₇×#6912	C₃	1	24	yes	maximal, complemented
4	C₃×C₃₇×#733	C₃	1	12	yes	maximal, complemented
5	C₃₇×#737	—	3	9	no	—
6	C₃₇×#737	—	3	27	no	—
7	C₃×3₋¹⁺²×C₃₇	C₃²	1	36	yes	—
8	C₃×3₋¹⁺²×C₃₇	C₃²	1	8	yes	—
9	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
10	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
11	C₃₇×#733	C₃²	1	9	yes	—
12	C₃₇×#733	C₃²	1	72	yes	—
13	C₃³×C₃₇	C₃³	1	4	yes	—
14	3₋¹⁺²×C₃₇	—	3	72	no	—
15	3₋¹⁺²×C₃₇	—	3	24	no	—
16	3₋¹⁺²×C₃₇	C₃³	1	27	yes	—
17	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
18	C₃×C₉×C₃₇	—	3	12	no	—
19	C₃×C₉×C₃₇	—	3	36	no	—
20	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
21	C₃×C₉×C₃₇	C₃³	1	8	yes	—
22	#10624933	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₃²×C₃₇	—	9	4	no	—
24	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
25	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
26	C₉×C₃₇	—	3	1	no	—
27	C₉×C₃₇	—	3	27	no	—
28	C₉×C₃₇	—	3	8	no	—
29	3₋¹⁺⁴	C₃×C₃₇	1	3	yes	—
30	C₃×#737	C₃×C₃₇	1	1	yes	characteristic
31	#6912	C₃×C₃₇	1	24	yes	complemented
32	C₃×#733	C₃×C₃₇	1	12	yes	complemented
33	C₃×C₃₇	—	3	1	no	—
34	C₃×C₃₇	—	3	12	no	—
35	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
36	#737	—	3	9	no	—
37	#737	—	3	27	no	—
38	C₃×3₋¹⁺²	C₃²×C₃₇	1	36	yes	—
39	C₃×3₋¹⁺²	C₃²×C₃₇	1	8	yes	—
40	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
41	C₃²×C₉	C₃²×C₃₇	1	4	yes	—
42	#733	C₃²×C₃₇	1	72	yes	—
43	#733	C₃²×C₃₇	1	9	yes	—
44	C₃₇	#10624933	1	1	yes	characteristic, direct product factor, Sylow
45	C₃³	C₃³×C₃₇	1	4	yes	—
46	3₋¹⁺²	—	3	72	no	—
47	3₋¹⁺²	—	3	24	no	—
48	3₋¹⁺²	C₃³×C₃₇	1	27	yes	—
49	H(3,𝔽₃)	H(3,𝔽₃)×C₃₇	1	9	yes	—
50	C₃×C₉	—	3	12	no	—
51	C₃×C₉	—	3	36	no	—
52	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
53	C₃×C₉	C₃³×C₃₇	1	8	yes	—
54	C₃²	—	9	4	no	—
55	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	12	yes	—
56	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
57	C₉	—	3	1	no	—
58	C₉	—	3	27	no	—
59	C₉	—	3	8	no	—
60	C₃	—	3	1	no	—
61	C₃	—	3	12	no	—
62	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃₇×#6912	C₃	semidirect
4. C₃×C₃₇×#733	C₃	semidirect
22. #10624933	C₃₇	direct (N × H)
31. #6912	C₃×C₃₇	semidirect
32. C₃×#733	C₃×C₃₇	semidirect
44. C₃₇	#10624933	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×3₋¹⁺⁴	3
C₃×C₃₇×#737	3
C₃₇×#6912	3
C₃×C₃₇×#733	3
#10624933	37

Subgroups — C37×#10624933 #41004722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624933 #41004722