Subgroups — C₃₇×#10624880 #41004669

Order 26973.

1718 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	4	yes	maximal, complemented
2	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	36	yes	maximal, complemented
3	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
4	C₃⁴×C₃₇	C₃²	1	12	yes	complemented
5	C₃×H(3,𝔽₃)×C₃₇	—	3	36	no	—
6	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	36	yes	complemented
7	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	81	yes	complemented
8	C₃³×C₃₇	—	3	36	no	—
9	C₃³×C₃₇	—	9	3	no	—
10	C₃³×C₃₇	—	9	36	no	—
11	C₃³×C₃₇	H(3,𝔽₃)	1	9	yes	complemented
12	C₃³×C₃₇	C₃³	1	36	yes	—
13	C₃³×C₃₇	C₃³	1	4	yes	—
14	H(3,𝔽₃)×C₃₇	—	3	108	no	—
15	H(3,𝔽₃)×C₃₇	—	9	81	no	—
16	#10624880	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₃²×C₃₇	—	3	12	no	—
18	C₃²×C₃₇	—	3	36	no	—
19	C₃²×C₃₇	—	3	108	no	—
20	C₃²×C₃₇	—	9	9	no	—
21	C₃²×C₃₇	—	9	108	no	—
22	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
23	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
24	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	4	yes	complemented
25	C₃²⋊H(3,𝔽₃)	C₃×C₃₇	1	36	yes	complemented
26	C₃×C₃₇	—	3	12	no	—
27	C₃×C₃₇	—	9	36	no	—
28	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
29	C₃×C₃₇	H(5,𝔽₃)	1	3	yes	—
30	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
31	C₃⁴	C₃²×C₃₇	1	12	yes	complemented
32	C₃×H(3,𝔽₃)	—	3	36	no	—
33	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	36	yes	complemented
34	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	81	yes	complemented
35	C₃₇	#10624880	1	1	yes	characteristic, direct product factor, Sylow
36	C₃³	—	3	36	no	—
37	C₃³	—	9	3	no	—
38	C₃³	—	9	36	no	—
39	C₃³	H(3,𝔽₃)×C₃₇	1	9	yes	complemented
40	C₃³	C₃³×C₃₇	1	36	yes	—
41	C₃³	C₃³×C₃₇	1	4	yes	—
42	H(3,𝔽₃)	—	3	108	no	—
43	H(3,𝔽₃)	—	9	81	no	—
44	C₃²	—	3	12	no	—
45	C₃²	—	3	36	no	—
46	C₃²	—	3	108	no	—
47	C₃²	—	9	9	no	—
48	C₃²	—	9	108	no	—
49	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	12	yes	—
50	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
51	C₃	—	3	12	no	—
52	C₃	—	9	36	no	—
53	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
54	C₃	C₃₇×H(5,𝔽₃)	1	3	yes	—

	subgroup	G/H
center, socle	C₃²×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃²	C₃⁴×C₃₇
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. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
2. C₃₇×C₃²⋊H(3,𝔽₃)	C₃	semidirect
4. C₃⁴×C₃₇	C₃²	semidirect
6. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
7. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
11. C₃³×C₃₇	H(3,𝔽₃)	semidirect
16. #10624880	C₃₇	direct (N × H)
24. C₃²×H(3,𝔽₃)	C₃×C₃₇	semidirect
25. C₃²⋊H(3,𝔽₃)	C₃×C₃₇	semidirect
31. C₃⁴	C₃²×C₃₇	semidirect
33. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
34. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
35. C₃₇	#10624880	direct (N × H)
39. C₃³	H(3,𝔽₃)×C₃₇	semidirect

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×C₃²⋊H(3,𝔽₃)	3
#10624880	37

Subgroups — C37×#10624880 #41004669

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624880 #41004669