Subgroups — C₃×C₃₇×#6856 #41004367

Order 26973.

692 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	2	yes	maximal, complemented
2	C₃×C₃₇×C₃²⋊C₉	C₃	1	2	yes	maximal, complemented
3	C₃₇×#6856	C₃	1	9	yes	maximal, direct product factor
4	C₃⁴×C₃₇	—	3	2	no	—
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×H(3,𝔽₃)×C₃₇	—	3	4	no	—
7	C₃×H(3,𝔽₃)×C₃₇	—	3	18	no	—
8	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	6	yes	complemented
9	C₃²×C₉×C₃₇	—	3	2	no	—
10	C₃₇×C₃²⋊C₉	—	3	4	no	—
11	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	complemented
12	C₃³×C₃₇	—	3	6	no	—
13	C₃³×C₃₇	—	3	18	no	—
14	C₃³×C₃₇	—	3	6	no	—
15	C₃³×C₃₇	—	3	6	no	—
16	C₃³×C₃₇	—	9	6	no	—
17	C₃³×C₃₇	—	9	12	no	—
18	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
19	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	H(3,𝔽₃)×C₃₇	—	3	18	no	—
22	H(3,𝔽₃)×C₃₇	—	3	36	no	—
23	C₃×C₉×C₃₇	—	3	6	no	—
24	C₃×C₉×C₃₇	—	9	6	no	—
25	C₃×#6856	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃²×C₃₇	—	3	4	no	—
27	C₃²×C₃₇	—	3	2	no	—
28	C₃²×C₃₇	—	3	18	no	—
29	C₃²×C₃₇	—	3	4	no	—
30	C₃²×C₃₇	—	3	2	no	—
31	C₃²×C₃₇	—	9	9	no	—
32	C₃²×C₃₇	—	9	54	no	—
33	C₃²×C₃₇	—	9	6	no	—
34	C₃²×C₃₇	—	9	12	no	—
35	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	6	yes	—
36	C₃²×C₃₇	#732	1	6	yes	—
37	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
38	C₉×C₃₇	—	9	6	no	—
39	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	2	yes	complemented
40	C₃×C₃²⋊C₉	C₃×C₃₇	1	2	yes	complemented
41	#6856	C₃×C₃₇	1	9	yes	direct product factor
42	C₃×C₃₇	—	9	2	no	—
43	C₃×C₃₇	—	9	1	no	—
44	C₃×C₃₇	—	9	18	no	—
45	C₃×C₃₇	#6856	1	9	yes	direct product factor
46	C₃×C₃₇	C₃×AΣL(1,𝔽₂₇)	1	2	yes	—
47	C₃×C₃₇	C₃×#732	1	2	yes	—
48	C₃⁴	—	3	2	no	—
49	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
50	C₃×H(3,𝔽₃)	—	3	18	no	—
51	C₃×H(3,𝔽₃)	—	3	4	no	—
52	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	6	yes	complemented
53	C₃²×C₉	—	3	2	no	—
54	C₃²⋊C₉	—	3	4	no	—
55	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	complemented
56	C₃₇	C₃×#6856	1	1	yes	characteristic, direct product factor, Sylow
57	C₃³	—	3	6	no	—
58	C₃³	—	3	18	no	—
59	C₃³	—	3	6	no	—
60	C₃³	—	3	6	no	—
61	C₃³	—	9	6	no	—
62	C₃³	—	9	12	no	—
63	C₃³	H(3,𝔽₃)×C₃₇	1	2	yes	—
64	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
65	C₃³	C₃³×C₃₇	1	1	yes	characteristic
66	H(3,𝔽₃)	—	3	18	no	—
67	H(3,𝔽₃)	—	3	36	no	—
68	C₃×C₉	—	3	6	no	—
69	C₃×C₉	—	9	6	no	—
70	C₃²	—	3	4	no	—
71	C₃²	—	3	2	no	—
72	C₃²	—	3	18	no	—
73	C₃²	—	3	4	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	9	9	no	—
76	C₃²	—	9	54	no	—
77	C₃²	—	9	6	no	—
78	C₃²	—	9	12	no	—
79	C₃²	C₃₇×AΣL(1,𝔽₂₇)	1	6	yes	—
80	C₃²	C₃₇×#732	1	6	yes	—
81	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
82	C₉	—	9	6	no	—
83	C₃	—	9	2	no	—
84	C₃	—	9	1	no	—
85	C₃	—	9	18	no	—
86	C₃	C₃₇×#6856	1	9	yes	direct product factor
87	C₃	C₃×C₃₇×AΣL(1,𝔽₂₇)	1	2	yes	—
88	C₃	C₃×C₃₇×#732	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
2. C₃×C₃₇×C₃²⋊C₉	C₃	semidirect
3. C₃₇×#6856	C₃	direct (N × H)
8. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
11. C₃₇×C₃²⋊C₉	C₃²	semidirect
25. C₃×#6856	C₃₇	direct (N × H)
39. C₃²×H(3,𝔽₃)	C₃×C₃₇	semidirect
40. C₃×C₃²⋊C₉	C₃×C₃₇	semidirect
41. #6856	C₃×C₃₇	direct (N × H)
45. C₃×C₃₇	#6856	direct (N × H)
52. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
55. C₃²⋊C₉	C₃²×C₃₇	semidirect
56. C₃₇	C₃×#6856	direct (N × H)
86. C₃	C₃₇×#6856	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×H(3,𝔽₃)×C₃₇	3
C₃×C₃₇×C₃²⋊C₉	3
C₃₇×#6856	3
C₃×#6856	37

Subgroups — C3×C37×#6856 #41004367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃₇×#6856 #41004367