Subgroups — C₃₇×#10624452 #41004241

Order 26973.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃×C₃₇×C₃²⋊C₉	C₃	1	3	yes	maximal
3	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
4	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
5	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
6	C₃×3₋¹⁺²×C₃₇	C₉	1	3	yes	complemented
7	C₃²×C₉×C₃₇	—	3	1	no	—
8	C₃²×C₉×C₃₇	—	3	3	no	—
9	C₃₇×C₃²⋊C₉	—	3	9	no	—
10	C₃³×C₃₇	—	3	3	no	—
11	C₃³×C₃₇	—	3	3	no	—
12	C₃³×C₃₇	—	3	3	no	—
13	C₃³×C₃₇	—	3	3	no	—
14	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃³×C₃₇	3₋¹⁺²	1	2	yes	—
17	3₋¹⁺²×C₃₇	—	3	18	no	—
18	3₋¹⁺²×C₃₇	—	3	9	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	—	3	9	no	—
21	C₃×C₉×C₃₇	—	3	9	no	—
22	C₃×C₉×C₃₇	—	9	3	no	—
23	C₃×C₉×C₃₇	—	9	3	no	—
24	C₃×C₉×C₃₇	—	9	3	no	—
25	#10624452	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃²×C₃₇	—	3	1	no	—
27	C₃²×C₃₇	—	3	2	no	—
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	6	no	—
30	C₃²×C₃₇	—	9	3	no	—
31	C₃²×C₃₇	—	9	3	no	—
32	C₃²×C₃₇	—	9	3	no	—
33	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
34	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
35	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
36	C₃²×C₃₇	#731	1	3	yes	—
37	C₃²×C₃₇	#733	1	3	yes	—
38	C₉×C₃₇	—	3	9	no	—
39	C₉×C₃₇	—	9	9	no	—
40	C₃²×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
41	C₃×C₃²⋊C₉	C₃×C₃₇	1	3	yes	—
42	C₃×C₃₇	—	9	1	no	—
43	C₃×C₃₇	—	9	2	no	—
44	C₃×C₃₇	#6857	1	3	yes	—
45	C₃×C₃₇	#6858	1	3	yes	—
46	C₃×C₃₇	#6860	1	3	yes	—
47	C₃×C₃₇	#6866	1	1	yes	characteristic
48	C₃×C₃₇	3₋¹⁺²⋊C₉	1	3	yes	—
49	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
50	C₃×3₋¹⁺²	—	3	2	no	—
51	C₃×3₋¹⁺²	—	3	9	no	—
52	C₃×3₋¹⁺²	C₉×C₃₇	1	3	yes	complemented
53	C₃²×C₉	—	3	1	no	—
54	C₃²×C₉	—	3	3	no	—
55	C₃²⋊C₉	—	3	9	no	—
56	C₃₇	#10624452	1	1	yes	characteristic, direct product factor, Sylow
57	C₃³	—	3	3	no	—
58	C₃³	—	3	3	no	—
59	C₃³	—	3	3	no	—
60	C₃³	—	3	3	no	—
61	C₃³	C₃×C₉×C₃₇	1	1	yes	characteristic
62	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
63	C₃³	3₋¹⁺²×C₃₇	1	2	yes	—
64	3₋¹⁺²	—	3	18	no	—
65	3₋¹⁺²	—	3	9	no	—
66	C₃×C₉	—	3	3	no	—
67	C₃×C₉	—	3	9	no	—
68	C₃×C₉	—	3	9	no	—
69	C₃×C₉	—	9	3	no	—
70	C₃×C₉	—	9	3	no	—
71	C₃×C₉	—	9	3	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	6	no	—
76	C₃²	—	9	3	no	—
77	C₃²	—	9	3	no	—
78	C₃²	—	9	3	no	—
79	C₃²	C₃₇×C₃²⋊C₉	1	1	yes	characteristic
80	C₃²	C₃₇×AΣL(1,𝔽₂₇)	1	3	yes	—
81	C₃²	C₃₇×AΣL(1,𝔽₂₇)	1	3	yes	—
82	C₃²	C₃₇×#731	1	3	yes	—
83	C₃²	C₃₇×#733	1	3	yes	—
84	C₉	—	3	9	no	—
85	C₉	—	9	9	no	—
86	C₃	—	9	1	no	—
87	C₃	—	9	2	no	—
88	C₃	C₃₇×#6857	1	3	yes	—
89	C₃	C₃₇×#6858	1	3	yes	—
90	C₃	C₃₇×#6860	1	3	yes	—
91	C₃	C₃₇×#6866	1	1	yes	characteristic
92	C₃	C₃₇×3₋¹⁺²⋊C₉	1	3	yes	—

	subgroup	G/H
center, socle	C₃³×C₃₇	H(3,𝔽₃)
commutator	C₃³	C₃×C₉×C₃₇
Frattini, cosocle	C₃⁴	C₃²×C₃₇
prime radical	C₃⁴×C₃₇	C₃²
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
6. C₃×3₋¹⁺²×C₃₇	C₉	semidirect
25. #10624452	C₃₇	direct (N × H)
52. C₃×3₋¹⁺²	C₉×C₃₇	semidirect
56. C₃₇	#10624452	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×3₋¹⁺²×C₃₇	3
C₃×C₃₇×C₃²⋊C₉	3
#10624452	37

Subgroups — C37×#10624452 #41004241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624452 #41004241