Subgroups — C₃₇×#10624571 #41004360

Order 26973.

716 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₃₇	C₃	1	4	yes	maximal
2	C₃₇×#6855	C₃	1	9	yes	maximal, complemented
3	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
4	C₃×3₋¹⁺²×C₃₇	—	3	12	no	—
5	C₃×3₋¹⁺²×C₃₇	—	3	36	no	—
6	C₃²×C₉×C₃₇	C₃²	1	12	yes	—
7	C₃³×C₃₇	—	3	12	no	—
8	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
9	C₃³×C₃₇	C₃³	1	1	yes	characteristic
10	3₋¹⁺²×C₃₇	—	3	108	no	—
11	C₃×C₉×C₃₇	—	3	12	no	—
12	C₃×C₉×C₃₇	—	3	24	no	—
13	C₃×C₉×C₃₇	3₋¹⁺²	1	36	yes	complemented
14	#10624571	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₃²×C₃₇	—	3	12	no	—
16	C₃²×C₃₇	—	9	9	no	—
17	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
18	C₃²×C₃₇	C₃×3₋¹⁺²	1	12	yes	—
19	C₉×C₃₇	—	3	36	no	—
20	C₃²×3₋¹⁺²	C₃×C₃₇	1	4	yes	—
21	#6855	C₃×C₃₇	1	9	yes	complemented
22	C₃×C₃₇	—	9	3	no	—
23	C₃×C₃₇	#6891	1	4	yes	—
24	C₃×C₃₇	#6900	1	9	yes	—
25	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
26	C₃×3₋¹⁺²	—	3	12	no	—
27	C₃×3₋¹⁺²	—	3	36	no	—
28	C₃²×C₉	C₃²×C₃₇	1	12	yes	—
29	C₃₇	#10624571	1	1	yes	characteristic, direct product factor, Sylow
30	C₃³	—	3	12	no	—
31	C₃³	H(3,𝔽₃)×C₃₇	1	3	yes	—
32	C₃³	C₃³×C₃₇	1	1	yes	characteristic
33	3₋¹⁺²	—	3	108	no	—
34	C₃×C₉	—	3	12	no	—
35	C₃×C₉	—	3	24	no	—
36	C₃×C₉	3₋¹⁺²×C₃₇	1	36	yes	complemented
37	C₃²	—	3	12	no	—
38	C₃²	—	9	9	no	—
39	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
40	C₃²	C₃×3₋¹⁺²×C₃₇	1	12	yes	—
41	C₉	—	3	36	no	—
42	C₃	—	9	3	no	—
43	C₃	C₃₇×#6891	1	4	yes	—
44	C₃	C₃₇×#6900	1	9	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₇×#6855	C₃	semidirect
13. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
14. #10624571	C₃₇	direct (N × H)
21. #6855	C₃×C₃₇	semidirect
29. C₃₇	#10624571	direct (N × H)
36. C₃×C₉	3₋¹⁺²×C₃₇	semidirect

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×#6855	3
#10624571	37

Subgroups — C37×#10624571 #41004360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624571 #41004360