Subgroups — C₃₇×#10624567 #41004356

Order 26973.

524 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₃₇	C₃	1	2	yes	maximal
2	C₃×C₃₇×C₃²⋊C₉	C₃	1	2	yes	maximal
3	C₃₇×#6855	C₃	1	9	yes	maximal, complemented
4	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²×C₃₇	—	3	6	no	—
6	C₃×3₋¹⁺²×C₃₇	—	3	18	no	—
7	C₃²×C₉×C₃₇	C₃²	1	6	yes	—
8	C₃²×C₉×C₃₇	C₃²	1	6	yes	—
9	C₃₇×C₃²⋊C₉	—	3	6	no	—
10	C₃³×C₃₇	—	3	6	no	—
11	C₃³×C₃₇	—	3	6	no	—
12	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
13	C₃³×C₃₇	C₃³	1	1	yes	characteristic
14	3₋¹⁺²×C₃₇	—	3	54	no	—
15	C₃×C₉×C₃₇	—	3	6	no	—
16	C₃×C₉×C₃₇	—	3	12	no	—
17	C₃×C₉×C₃₇	—	3	6	no	—
18	C₃×C₉×C₃₇	—	3	18	no	—
19	C₃×C₉×C₃₇	3₋¹⁺²	1	18	yes	complemented
20	#10624567	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₃²×C₃₇	—	3	6	no	—
22	C₃²×C₃₇	—	3	6	no	—
23	C₃²×C₃₇	—	9	9	no	—
24	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
25	C₃²×C₃₇	C₃×3₋¹⁺²	1	6	yes	—
26	C₃²×C₃₇	#737	1	6	yes	—
27	C₉×C₃₇	—	3	18	no	—
28	C₉×C₃₇	—	9	6	no	—
29	C₃²×3₋¹⁺²	C₃×C₃₇	1	2	yes	—
30	C₃×C₃²⋊C₉	C₃×C₃₇	1	2	yes	—
31	#6855	C₃×C₃₇	1	9	yes	complemented
32	C₃×C₃₇	—	9	3	no	—
33	C₃×C₃₇	#6891	1	2	yes	—
34	C₃×C₃₇	#6893	1	2	yes	—
35	C₃×C₃₇	#6896	1	9	yes	—
36	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
37	C₃×3₋¹⁺²	—	3	6	no	—
38	C₃×3₋¹⁺²	—	3	18	no	—
39	C₃²×C₉	C₃²×C₃₇	1	6	yes	—
40	C₃²×C₉	C₃²×C₃₇	1	6	yes	—
41	C₃²⋊C₉	—	3	6	no	—
42	C₃₇	#10624567	1	1	yes	characteristic, direct product factor, Sylow
43	C₃³	—	3	6	no	—
44	C₃³	—	3	6	no	—
45	C₃³	H(3,𝔽₃)×C₃₇	1	3	yes	—
46	C₃³	C₃³×C₃₇	1	1	yes	characteristic
47	3₋¹⁺²	—	3	54	no	—
48	C₃×C₉	—	3	6	no	—
49	C₃×C₉	—	3	12	no	—
50	C₃×C₉	—	3	6	no	—
51	C₃×C₉	—	3	18	no	—
52	C₃×C₉	3₋¹⁺²×C₃₇	1	18	yes	complemented
53	C₃²	—	3	6	no	—
54	C₃²	—	3	6	no	—
55	C₃²	—	9	9	no	—
56	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
57	C₃²	C₃×3₋¹⁺²×C₃₇	1	6	yes	—
58	C₃²	C₃₇×#737	1	6	yes	—
59	C₉	—	3	18	no	—
60	C₉	—	9	6	no	—
61	C₃	—	9	3	no	—
62	C₃	C₃₇×#6891	1	2	yes	—
63	C₃	C₃₇×#6893	1	2	yes	—
64	C₃	C₃₇×#6896	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
3. C₃₇×#6855	C₃	semidirect
19. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
20. #10624567	C₃₇	direct (N × H)
31. #6855	C₃×C₃₇	semidirect
42. C₃₇	#10624567	direct (N × H)
52. 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
C₃×C₃₇×C₃²⋊C₉	3
C₃₇×#6855	3
#10624567	37

Subgroups — C37×#10624567 #41004356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624567 #41004356