Subgroups — C₃×C₃₇×#6903 #41004706

Order 26973.

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

	subgroup	G/H
center	C₃×C₂₇×C₃₇	C₃²
commutator	C₃	C₃³×C₉×C₃₇
Frattini, cosocle	C₉	C₃⁴×C₃₇
socle	C₃²×C₃₇	C₃²×C₉
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₉
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₂₇×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#6903	C₃	direct (N × H)
3. C₃×C₃₇×C₂₇⋊C₃	C₃	semidirect
4. C₃²×C₂₇×C₃₇	C₃	semidirect
10. C₃₇×C₂₇⋊C₃	C₃²	semidirect
13. C₃×C₂₇×C₃₇	C₃²	semidirect
23. C₃×#6903	C₃₇	direct (N × H)
31. #6903	C₃×C₃₇	direct (N × H)
32. C₃×C₂₇⋊C₃	C₃×C₃₇	semidirect
33. C₃²×C₂₇	C₃×C₃₇	semidirect
35. C₃×C₃₇	#6903	direct (N × H)
42. C₂₇⋊C₃	C₃²×C₃₇	semidirect
45. C₃×C₂₇	C₃²×C₃₇	semidirect
46. C₃₇	C₃×#6903	direct (N × H)
63. C₃	C₃₇×#6903	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
C₃²×C₂₇×C₃₇	3
C₃×#6903	37

Subgroups — C3×C37×#6903 #41004706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×#6903 #41004706