Subgroups — C₃₇×#10624691 #41004480

Order 26973.

404 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	C₃₇×#6868	C₃	1	9	yes	maximal, complemented
4	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²×C₃₇	—	3	3	no	—
6	C₃×3₋¹⁺²×C₃₇	—	3	6	no	—
7	C₃×3₋¹⁺²×C₃₇	C₉	1	9	yes	complemented
8	C₃²×C₉×C₃₇	—	3	3	no	—
9	C₃²×C₉×C₃₇	C₃²	1	3	yes	—
10	C₃₇×C₃²⋊C₉	—	3	6	no	—
11	C₃₇×C₃²⋊C₉	C₃²	1	9	yes	—
12	C₃³×C₃₇	—	9	2	no	—
13	C₃³×C₃₇	—	9	1	no	—
14	C₃³×C₃₇	C₃×C₉	1	3	yes	—
15	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
16	C₃³×C₃₇	3₋¹⁺²	1	6	yes	—
17	C₃³×C₃₇	C₃³	1	1	yes	characteristic
18	3₋¹⁺²×C₃₇	—	3	27	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	—	3	6	no	—
21	C₃×C₉×C₃₇	—	3	9	no	—
22	C₃×C₉×C₃₇	—	9	9	no	—
23	C₃×C₉×C₃₇	C₃×C₉	1	9	yes	complemented
24	#10624691	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₃²×C₃₇	—	3	6	no	—
26	C₃²×C₃₇	—	3	2	no	—
27	C₃²×C₃₇	—	3	3	no	—
28	C₃²×C₃₇	—	3	1	no	—
29	C₃²×C₃₇	—	9	3	no	—
30	C₃²×C₃₇	—	9	6	no	—
31	C₃²×C₃₇	C₃²⋊C₉	1	9	yes	—
32	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
33	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
34	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
35	C₉×C₃₇	—	3	9	no	—
36	C₉×C₃₇	—	9	9	no	—
37	C₃²×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
38	C₃×C₃²⋊C₉	C₃×C₃₇	1	3	yes	—
39	#6868	C₃×C₃₇	1	9	yes	complemented
40	C₃×C₃₇	—	3	9	no	—
41	C₃×C₃₇	—	3	1	no	—
42	C₃×C₃₇	—	3	2	no	—
43	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
44	C₃×C₃₇	#6911	1	1	yes	characteristic
45	C₃×C₃₇	#6912	1	2	yes	—
46	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
47	C₃×3₋¹⁺²	—	3	3	no	—
48	C₃×3₋¹⁺²	—	3	6	no	—
49	C₃×3₋¹⁺²	C₉×C₃₇	1	9	yes	complemented
50	C₃²×C₉	—	3	3	no	—
51	C₃²×C₉	C₃²×C₃₇	1	3	yes	—
52	C₃²⋊C₉	—	3	6	no	—
53	C₃²⋊C₉	C₃²×C₃₇	1	9	yes	—
54	C₃₇	#10624691	1	1	yes	characteristic, direct product factor, Sylow
55	C₃³	—	9	1	no	—
56	C₃³	—	9	2	no	—
57	C₃³	C₃×C₉×C₃₇	1	3	yes	—
58	C₃³	H(3,𝔽₃)×C₃₇	1	3	yes	—
59	C₃³	3₋¹⁺²×C₃₇	1	6	yes	—
60	C₃³	C₃³×C₃₇	1	1	yes	characteristic
61	3₋¹⁺²	—	3	27	no	—
62	C₃×C₉	—	3	3	no	—
63	C₃×C₉	—	3	6	no	—
64	C₃×C₉	—	3	9	no	—
65	C₃×C₉	—	9	9	no	—
66	C₃×C₉	C₃×C₉×C₃₇	1	9	yes	complemented
67	C₃²	—	3	3	no	—
68	C₃²	—	3	1	no	—
69	C₃²	—	3	6	no	—
70	C₃²	—	3	2	no	—
71	C₃²	—	9	3	no	—
72	C₃²	—	9	6	no	—
73	C₃²	C₃₇×C₃²⋊C₉	1	9	yes	—
74	C₃²	C₃²×C₉×C₃₇	1	1	yes	characteristic
75	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
76	C₃²	C₃×3₋¹⁺²×C₃₇	1	2	yes	—
77	C₉	—	3	9	no	—
78	C₉	—	9	9	no	—
79	C₃	—	3	9	no	—
80	C₃	—	3	1	no	—
81	C₃	—	3	2	no	—
82	C₃	C₃×C₃₇×C₃²⋊C₉	1	1	yes	characteristic
83	C₃	C₃₇×#6911	1	1	yes	characteristic
84	C₃	C₃₇×#6912	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₇×#6868	C₃	semidirect
7. C₃×3₋¹⁺²×C₃₇	C₉	semidirect
23. C₃×C₉×C₃₇	C₃×C₉	semidirect
24. #10624691	C₃₇	direct (N × H)
39. #6868	C₃×C₃₇	semidirect
49. C₃×3₋¹⁺²	C₉×C₃₇	semidirect
54. C₃₇	#10624691	direct (N × H)
66. C₃×C₉	C₃×C₉×C₃₇	semidirect

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
C₃₇×#6868	3
#10624691	37

Subgroups — C37×#10624691 #41004480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624691 #41004480