Subgroups — C₃₇×#10624918 #41004707

Order 26973.

828 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×#737	C₃	1	1	yes	maximal, characteristic
2	C₃₇×#6903	C₃	1	27	yes	maximal, complemented
3	C₃×C₃₇×C₂₇⋊C₃	C₃	1	12	yes	maximal, complemented
4	C₃₇×#737	C₉	1	18	yes	—
5	C₃₇×#737	C₃²	1	9	yes	—
6	C₃×3₋¹⁺²×C₃₇	C₉	1	8	yes	—
7	C₃×H(3,𝔽₃)×C₃₇	C₉	1	1	yes	characteristic
8	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
9	C₃₇×C₂₇⋊C₃	C₃²	1	9	yes	—
10	C₃₇×C₂₇⋊C₃	C₃²	1	72	yes	complemented
11	C₃×C₂₇×C₃₇	C₃²	1	36	yes	complemented
12	C₃³×C₃₇	C₃×C₉	1	4	yes	—
13	3₋¹⁺²×C₃₇	C₃×C₉	1	72	yes	—
14	H(3,𝔽₃)×C₃₇	C₃×C₉	1	9	yes	—
15	C₃×C₉×C₃₇	C₃×C₉	1	8	yes	—
16	C₃×C₉×C₃₇	C₃×C₉	1	24	yes	—
17	C₃×C₉×C₃₇	C₃³	1	12	yes	—
18	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
19	C₂₇×C₃₇	C₃³	1	27	yes	—
20	#10624918	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₃²×C₃₇	—	9	4	no	—
22	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
23	C₃²×C₃₇	C₃²×C₉	1	12	yes	—
24	C₉×C₃₇	C₃²×C₉	1	2	yes	—
25	C₉×C₃₇	C₃²×C₉	1	24	yes	—
26	C₉×C₃₇	C₃⁴	1	1	yes	characteristic
27	C₃×#737	C₃×C₃₇	1	1	yes	characteristic
28	#6903	C₃×C₃₇	1	27	yes	complemented
29	C₃×C₂₇⋊C₃	C₃×C₃₇	1	12	yes	complemented
30	C₃×C₃₇	—	3	1	no	—
31	C₃×C₃₇	—	3	12	no	—
32	C₃×C₃₇	C₃³×C₉	1	1	yes	characteristic
33	#737	C₉×C₃₇	1	18	yes	—
34	#737	C₃²×C₃₇	1	9	yes	—
35	C₃×3₋¹⁺²	C₉×C₃₇	1	8	yes	—
36	C₃×H(3,𝔽₃)	C₉×C₃₇	1	1	yes	characteristic
37	C₃²×C₉	C₃²×C₃₇	1	4	yes	—
38	C₂₇⋊C₃	C₃²×C₃₇	1	9	yes	—
39	C₂₇⋊C₃	C₃²×C₃₇	1	72	yes	complemented
40	C₃×C₂₇	C₃²×C₃₇	1	36	yes	complemented
41	C₃₇	#10624918	1	1	yes	characteristic, direct product factor, Sylow
42	C₃³	C₃×C₉×C₃₇	1	4	yes	—
43	3₋¹⁺²	C₃×C₉×C₃₇	1	72	yes	—
44	H(3,𝔽₃)	C₃×C₉×C₃₇	1	9	yes	—
45	C₃×C₉	C₃×C₉×C₃₇	1	8	yes	—
46	C₃×C₉	C₃×C₉×C₃₇	1	24	yes	—
47	C₃×C₉	C₃³×C₃₇	1	12	yes	—
48	C₃×C₉	C₃³×C₃₇	1	1	yes	characteristic
49	C₂₇	C₃³×C₃₇	1	27	yes	—
50	C₃²	—	9	4	no	—
51	C₃²	C₃²×C₉×C₃₇	1	1	yes	characteristic
52	C₃²	C₃²×C₉×C₃₇	1	12	yes	—
53	C₉	C₃²×C₉×C₃₇	1	2	yes	—
54	C₉	C₃²×C₉×C₃₇	1	24	yes	—
55	C₉	C₃⁴×C₃₇	1	1	yes	characteristic
56	C₃	—	3	1	no	—
57	C₃	—	3	12	no	—
58	C₃	C₃³×C₉×C₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₉×C₃₇	C₃⁴
commutator	C₃	C₃³×C₉×C₃₇
Frattini, cosocle	C₉	C₃⁴×C₃₇
socle	C₃×C₃₇	C₃³×C₉
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
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
2. C₃₇×#6903	C₃	semidirect
3. C₃×C₃₇×C₂₇⋊C₃	C₃	semidirect
10. C₃₇×C₂₇⋊C₃	C₃²	semidirect
11. C₃×C₂₇×C₃₇	C₃²	semidirect
20. #10624918	C₃₇	direct (N × H)
28. #6903	C₃×C₃₇	semidirect
29. C₃×C₂₇⋊C₃	C₃×C₃₇	semidirect
39. C₂₇⋊C₃	C₃²×C₃₇	semidirect
40. C₃×C₂₇	C₃²×C₃₇	semidirect
41. C₃₇	#10624918	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₃₇×#737	3
C₃₇×#6903	3
C₃×C₃₇×C₂₇⋊C₃	3
#10624918	37

Subgroups — C37×#10624918 #41004707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624918 #41004707