Subgroups — C₃₇×#10624943 #41004732

Order 26973.

5490 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×3₋¹⁺⁴	C₃	1	80	yes	maximal, complemented
2	C₃₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×C₃₇×#737	C₃	1	40	yes	maximal, complemented
4	C₃₇×#737	C₃²	1	90	yes	complemented
5	C₃₇×#737	C₃²	1	720	yes	complemented
6	C₃×3₋¹⁺²×C₃₇	C₃²	1	320	yes	complemented
7	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	40	yes	—
8	C₃²×C₉×C₃₇	C₃²	1	40	yes	complemented
9	C₃³×C₃₇	C₃³	1	40	yes	—
10	3₋¹⁺²×C₃₇	C₃³	1	720	yes	—
11	H(3,𝔽₃)×C₃₇	C₃³	1	90	yes	—
12	C₃×C₉×C₃₇	C₃³	1	320	yes	—
13	C₃×C₉×C₃₇	C₃³	1	40	yes	—
14	#10624943	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₃²×C₃₇	—	9	40	no	—
16	C₃²×C₃₇	C₃⁴	1	40	yes	—
17	C₉×C₃₇	C₃⁴	1	80	yes	—
18	C₉×C₃₇	C₃⁴	1	1	yes	characteristic
19	3₋¹⁺⁴	C₃×C₃₇	1	80	yes	complemented
20	H(5,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
21	C₃×#737	C₃×C₃₇	1	40	yes	complemented
22	C₃×C₃₇	—	3	40	no	—
23	C₃×C₃₇	C₃⁵	1	1	yes	characteristic
24	#737	C₃²×C₃₇	1	90	yes	complemented
25	#737	C₃²×C₃₇	1	720	yes	complemented
26	C₃×3₋¹⁺²	C₃²×C₃₇	1	320	yes	complemented
27	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	40	yes	—
28	C₃²×C₉	C₃²×C₃₇	1	40	yes	complemented
29	C₃₇	#10624943	1	1	yes	characteristic, direct product factor, Sylow
30	C₃³	C₃³×C₃₇	1	40	yes	—
31	3₋¹⁺²	C₃³×C₃₇	1	720	yes	—
32	H(3,𝔽₃)	C₃³×C₃₇	1	90	yes	—
33	C₃×C₉	C₃³×C₃₇	1	40	yes	—
34	C₃×C₉	C₃³×C₃₇	1	320	yes	—
35	C₃²	—	9	40	no	—
36	C₃²	C₃⁴×C₃₇	1	40	yes	—
37	C₉	C₃⁴×C₃₇	1	1	yes	characteristic
38	C₉	C₃⁴×C₃₇	1	80	yes	—
39	C₃	—	3	40	no	—
40	C₃	C₃⁵×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₉×C₃₇	C₃⁴
commutator, Frattini, cosocle	C₃	C₃⁵×C₃₇
socle	C₃×C₃₇	C₃⁵
prime radical	C₃₇×H(5,𝔽₃)	C₃
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₇×3₋¹⁺⁴	C₃	semidirect
3. C₃×C₃₇×#737	C₃	semidirect
4. C₃₇×#737	C₃²	semidirect
5. C₃₇×#737	C₃²	semidirect
6. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
8. C₃²×C₉×C₃₇	C₃²	semidirect
14. #10624943	C₃₇	direct (N × H)
19. 3₋¹⁺⁴	C₃×C₃₇	semidirect
21. C₃×#737	C₃×C₃₇	semidirect
24. #737	C₃²×C₃₇	semidirect
25. #737	C₃²×C₃₇	semidirect
26. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
28. C₃²×C₉	C₃²×C₃₇	semidirect
29. C₃₇	#10624943	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
C₃₇×H(5,𝔽₃)	3
C₃×C₃₇×#737	3
#10624943	37

Subgroups — C37×#10624943 #41004732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624943 #41004732