Subgroups — C₃×C₃₇×#6897 #41004663

Order 26973.

692 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#6897	C₃	1	27	yes	maximal, direct product factor
2	C₃×C₃₇×C₉⋊C₉	C₃	1	8	yes	maximal, complemented
3	C₃×C₃₇×C₃²⋊C₉	C₃	1	4	yes	maximal
4	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃²×C₉×C₃₇	C₃²	1	4	yes	—
7	C₃²×C₉×C₃₇	C₃²	1	8	yes	—
8	C₃₇×C₉⋊C₉	C₃²	1	72	yes	complemented
9	C₃₇×C₃²⋊C₉	C₃²	1	36	yes	—
10	C₉²×C₃₇	C₃²	1	9	yes	complemented
11	C₃³×C₃₇	—	3	12	no	—
12	C₃³×C₃₇	C₃³	1	3	yes	—
13	C₃³×C₃₇	C₃³	1	1	yes	characteristic
14	C₃×C₉×C₃₇	—	3	12	no	—
15	C₃×C₉×C₃₇	—	3	24	no	—
16	C₃×C₉×C₃₇	C₃³	1	12	yes	—
17	C₃×C₉×C₃₇	C₃³	1	24	yes	—
18	C₃×#6897	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₃²×C₃₇	—	3	12	no	—
20	C₃²×C₃₇	—	9	9	no	—
21	C₃²×C₃₇	#737	1	12	yes	—
22	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
23	C₉×C₃₇	—	3	12	no	—
24	C₉×C₃₇	—	3	24	no	—
25	#6897	C₃×C₃₇	1	27	yes	direct product factor
26	C₃×C₉⋊C₉	C₃×C₃₇	1	8	yes	complemented
27	C₃×C₃²⋊C₉	C₃×C₃₇	1	4	yes	—
28	C₃×C₉²	C₃×C₃₇	1	1	yes	characteristic, complemented
29	C₃×C₃₇	—	9	3	no	—
30	C₃×C₃₇	#6897	1	9	yes	direct product factor
31	C₃×C₃₇	C₃×#737	1	4	yes	—
32	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
33	C₃²×C₉	C₃²×C₃₇	1	4	yes	—
34	C₃²×C₉	C₃²×C₃₇	1	8	yes	—
35	C₉⋊C₉	C₃²×C₃₇	1	72	yes	complemented
36	C₃²⋊C₉	C₃²×C₃₇	1	36	yes	—
37	C₉²	C₃²×C₃₇	1	9	yes	complemented
38	C₃₇	C₃×#6897	1	1	yes	characteristic, direct product factor, Sylow
39	C₃³	—	3	12	no	—
40	C₃³	C₃³×C₃₇	1	3	yes	—
41	C₃³	C₃³×C₃₇	1	1	yes	characteristic
42	C₃×C₉	—	3	12	no	—
43	C₃×C₉	—	3	24	no	—
44	C₃×C₉	C₃³×C₃₇	1	12	yes	—
45	C₃×C₉	C₃³×C₃₇	1	24	yes	—
46	C₃²	—	3	12	no	—
47	C₃²	—	9	9	no	—
48	C₃²	C₃₇×#737	1	12	yes	—
49	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
50	C₉	—	3	12	no	—
51	C₉	—	3	24	no	—
52	C₃	—	9	3	no	—
53	C₃	C₃₇×#6897	1	9	yes	direct product factor
54	C₃	C₃×C₃₇×#737	1	4	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
1. C₃₇×#6897	C₃	direct (N × H)
2. C₃×C₃₇×C₉⋊C₉	C₃	semidirect
4. C₃×C₉²×C₃₇	C₃	semidirect
8. C₃₇×C₉⋊C₉	C₃²	semidirect
10. C₉²×C₃₇	C₃²	semidirect
18. C₃×#6897	C₃₇	direct (N × H)
25. #6897	C₃×C₃₇	direct (N × H)
26. C₃×C₉⋊C₉	C₃×C₃₇	semidirect
28. C₃×C₉²	C₃×C₃₇	semidirect
30. C₃×C₃₇	#6897	direct (N × H)
35. C₉⋊C₉	C₃²×C₃₇	semidirect
37. C₉²	C₃²×C₃₇	semidirect
38. C₃₇	C₃×#6897	direct (N × H)
53. C₃	C₃₇×#6897	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×#6897	3
C₃×C₃₇×C₉⋊C₉	3
C₃×C₃₇×C₃²⋊C₉	3
C₃×C₉²×C₃₇	3
C₃×#6897	37

Subgroups — C3×C37×#6897 #41004663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×#6897 #41004663