Subgroups — C₃×C₃₇×#6895 #41004660

Order 26973.

836 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₉×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃₇×#6895	C₃	1	27	yes	maximal, direct product factor
3	C₃×C₃₇×C₉⋊C₉	C₃	1	9	yes	maximal, complemented
4	C₃×C₃₇×C₃²⋊C₉	C₃	1	3	yes	maximal
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃²×C₉×C₃₇	—	3	9	no	—
7	C₃²×C₉×C₃₇	C₃²	1	9	yes	—
8	C₃²×C₉×C₃₇	C₃²	1	9	yes	—
9	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
10	C₃₇×C₉⋊C₉	C₃²	1	81	yes	complemented
11	C₃₇×C₃²⋊C₉	C₃²	1	27	yes	—
12	C₃³×C₃₇	—	3	9	no	—
13	C₃³×C₃₇	3₋¹⁺²	1	9	yes	—
14	C₃³×C₃₇	C₃³	1	3	yes	—
15	C₃³×C₃₇	C₃³	1	1	yes	characteristic
16	C₃×C₉×C₃₇	—	3	27	no	—
17	C₃×C₉×C₃₇	—	3	9	no	—
18	C₃×C₉×C₃₇	—	3	27	no	—
19	C₃×C₉×C₃₇	C₃³	1	27	yes	—
20	C₃×C₉×C₃₇	C₃³	1	9	yes	—
21	C₃×#6895	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₃²×C₃₇	—	3	9	no	—
23	C₃²×C₃₇	—	3	27	no	—
24	C₃²×C₃₇	C₃×3₋¹⁺²	1	9	yes	—
25	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
26	C₃²×C₃₇	#737	1	9	yes	—
27	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
28	C₉×C₃₇	—	3	9	no	—
29	C₉×C₃₇	—	3	27	no	—
30	C₃³×C₉	C₃×C₃₇	1	1	yes	characteristic
31	#6895	C₃×C₃₇	1	27	yes	direct product factor
32	C₃×C₉⋊C₉	C₃×C₃₇	1	9	yes	complemented
33	C₃×C₃²⋊C₉	C₃×C₃₇	1	3	yes	—
34	C₃×C₃₇	—	3	9	no	—
35	C₃×C₃₇	#6895	1	9	yes	direct product factor
36	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
37	C₃×C₃₇	C₃×#737	1	3	yes	—
38	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
39	C₃²×C₉	—	3	9	no	—
40	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
41	C₃²×C₉	C₃²×C₃₇	1	9	yes	—
42	C₃²×C₉	C₃²×C₃₇	1	9	yes	—
43	C₉⋊C₉	C₃²×C₃₇	1	81	yes	complemented
44	C₃²⋊C₉	C₃²×C₃₇	1	27	yes	—
45	C₃₇	C₃×#6895	1	1	yes	characteristic, direct product factor, Sylow
46	C₃³	—	3	9	no	—
47	C₃³	3₋¹⁺²×C₃₇	1	9	yes	—
48	C₃³	C₃³×C₃₇	1	3	yes	—
49	C₃³	C₃³×C₃₇	1	1	yes	characteristic
50	C₃×C₉	—	3	27	no	—
51	C₃×C₉	—	3	9	no	—
52	C₃×C₉	—	3	27	no	—
53	C₃×C₉	C₃³×C₃₇	1	27	yes	—
54	C₃×C₉	C₃³×C₃₇	1	9	yes	—
55	C₃²	—	3	27	no	—
56	C₃²	—	3	9	no	—
57	C₃²	C₃×3₋¹⁺²×C₃₇	1	9	yes	—
58	C₃²	C₃×3₋¹⁺²×C₃₇	1	3	yes	—
59	C₃²	C₃₇×#737	1	9	yes	—
60	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
61	C₉	—	3	9	no	—
62	C₉	—	3	27	no	—
63	C₃	—	3	9	no	—
64	C₃	C₃₇×#6895	1	9	yes	direct product factor
65	C₃	C₃²×3₋¹⁺²×C₃₇	1	1	yes	characteristic
66	C₃	C₃×C₃₇×#737	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₃₇×#6895	C₃	direct (N × H)
3. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
10. C₃₇×C₉⋊C₉	C₃²	semidirect
21. C₃×#6895	C₃₇	direct (N × H)
31. #6895	C₃×C₃₇	direct (N × H)
32. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
35. C₃×C₃₇	#6895	direct (N × H)
43. C₉⋊C₉	C₃²×C₃₇	semidirect
45. C₃₇	C₃×#6895	direct (N × H)
64. C₃	C₃₇×#6895	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₉×C₃₇	3
C₃₇×#6895	3
C₃×C₃₇×C₉⋊C₉	3
C₃×C₃₇×C₃²⋊C₉	3
C₃×#6895	37

Subgroups — C3×C37×#6895 #41004660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×#6895 #41004660