Subgroups — C₃₇×#10624505 #41004294

Order 26973.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
2	C₃₇×#6866	C₃	1	3	yes	maximal, complemented
3	C₃⁴×C₃₇	C₉	1	3	yes	complemented
4	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
5	C₃×H(3,𝔽₃)×C₃₇	—	3	1	no	—
6	C₃×H(3,𝔽₃)×C₃₇	—	3	2	no	—
7	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
8	C₃₇×C₃²⋊C₉	—	3	3	no	—
9	C₃³×C₃₇	—	3	9	no	—
10	C₃³×C₃₇	—	3	3	no	—
11	C₃³×C₃₇	—	3	3	no	—
12	C₃³×C₃₇	—	9	9	no	—
13	C₃³×C₃₇	—	9	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	H(3,𝔽₃)×C₃₇	—	3	18	no	—
18	H(3,𝔽₃)×C₃₇	—	3	9	no	—
19	C₃×C₉×C₃₇	—	9	3	no	—
20	#10624505	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₃²×C₃₇	—	3	3	no	—
22	C₃²×C₃₇	—	3	1	no	—
23	C₃²×C₃₇	—	3	9	no	—
24	C₃²×C₃₇	—	3	2	no	—
25	C₃²×C₃₇	—	9	1	no	—
26	C₃²×C₃₇	—	9	9	no	—
27	C₃²×C₃₇	—	9	2	no	—
28	C₃²×C₃₇	—	9	27	no	—
29	C₃²×C₃₇	—	9	9	no	—
30	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
31	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
32	C₉×C₃₇	—	27	3	no	—
33	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic
34	#6866	C₃×C₃₇	1	3	yes	complemented
35	C₃×C₃₇	—	3	3	no	—
36	C₃×C₃₇	—	3	3	no	—
37	C₃×C₃₇	—	3	1	no	—
38	C₃×C₃₇	—	9	9	no	—
39	C₃×C₃₇	—	9	2	no	—
40	C₃×C₃₇	#6866	1	1	yes	characteristic
41	C₃⁴	C₉×C₃₇	1	3	yes	complemented
42	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
43	C₃×H(3,𝔽₃)	—	3	1	no	—
44	C₃×H(3,𝔽₃)	—	3	2	no	—
45	C₃×H(3,𝔽₃)	—	3	9	no	—
46	C₃²⋊C₉	—	3	3	no	—
47	C₃₇	#10624505	1	1	yes	characteristic, direct product factor, Sylow
48	C₃³	—	3	9	no	—
49	C₃³	—	3	3	no	—
50	C₃³	—	3	3	no	—
51	C₃³	—	9	9	no	—
52	C₃³	—	9	3	no	—
53	C₃³	C₃×C₉×C₃₇	1	1	yes	characteristic
54	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
55	C₃³	3₋¹⁺²×C₃₇	1	2	yes	—
56	H(3,𝔽₃)	—	3	18	no	—
57	H(3,𝔽₃)	—	3	9	no	—
58	C₃×C₉	—	9	3	no	—
59	C₃²	—	3	3	no	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	2	no	—
62	C₃²	—	3	9	no	—
63	C₃²	—	9	1	no	—
64	C₃²	—	9	9	no	—
65	C₃²	—	9	2	no	—
66	C₃²	—	9	27	no	—
67	C₃²	—	9	9	no	—
68	C₃²	C₃₇×C₃²⋊C₉	1	1	yes	characteristic
69	C₃²	C₃₇×AΣL(1,𝔽₂₇)	1	3	yes	—
70	C₉	—	27	3	no	—
71	C₃	—	3	3	no	—
72	C₃	—	3	1	no	—
73	C₃	—	3	3	no	—
74	C₃	—	9	2	no	—
75	C₃	—	9	9	no	—
76	C₃	C₃₇×#6866	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃×C₃₇	#6866
commutator	C₃³	C₃×C₉×C₃₇
Frattini, cosocle	C₃⁴	C₃²×C₃₇
prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
LCS	G ▷ C₃³ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₃₇ ◁ C₃³×C₃₇ ◁ C₃⁴×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₇×#6866	C₃	semidirect
3. C₃⁴×C₃₇	C₉	semidirect
20. #10624505	C₃₇	direct (N × H)
34. #6866	C₃×C₃₇	semidirect
41. C₃⁴	C₉×C₃₇	semidirect
47. C₃₇	#10624505	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
C₃₇×#6866	3
#10624505	37

Subgroups — C37×#10624505 #41004294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624505 #41004294