Subgroups — C₃₇×#10624930 #41004719

Order 26973.

1118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×3₋¹⁺⁴	C₃	1	2	yes	maximal, complemented
2	C₃₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#6909	C₃	1	24	yes	maximal, complemented
5	C₃×C₃₇×AΣL(1,𝔽₂₇)	C₃	1	12	yes	maximal, complemented
6	C₃⁴×C₃₇	C₃²	1	4	yes	complemented
7	C₃₇×#737	—	3	18	no	—
8	C₃×3₋¹⁺²×C₃₇	C₃²	1	24	yes	complemented
9	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
10	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
11	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
12	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	8	yes	complemented
13	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	12	yes	complemented
14	C₃₇×AΣL(1,𝔽₂₇)	C₃²	1	9	yes	complemented
15	C₃₇×AΣL(1,𝔽₂₇)	C₃²	1	72	yes	complemented
16	C₃³×C₃₇	—	3	12	no	—
17	C₃³×C₃₇	—	3	12	no	—
18	C₃³×C₃₇	—	9	12	no	—
19	C₃³×C₃₇	C₃³	1	1	yes	characteristic
20	C₃³×C₃₇	C₃³	1	8	yes	—
21	C₃³×C₃₇	C₃³	1	4	yes	—
22	3₋¹⁺²×C₃₇	—	3	48	no	—
23	3₋¹⁺²×C₃₇	C₃³	1	18	yes	—
24	H(3,𝔽₃)×C₃₇	—	3	24	no	—
25	H(3,𝔽₃)×C₃₇	—	3	24	no	—
26	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	9	yes	—
27	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	—
28	C₃×C₉×C₃₇	—	3	24	no	—
29	#10624930	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₃²×C₃₇	—	3	3	no	—
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	3	8	no	—
33	C₃²×C₃₇	—	3	9	no	—
34	C₃²×C₃₇	—	9	4	no	—
35	C₃²×C₃₇	—	9	36	no	—
36	C₃²×C₃₇	—	9	8	no	—
37	C₃²×C₃₇	—	27	12	no	—
38	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	12	yes	—
39	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
40	C₉×C₃₇	—	3	18	no	—
41	3₋¹⁺⁴	C₃×C₃₇	1	2	yes	complemented
42	H(5,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic, complemented
43	C₃²×H(3,𝔽₃)	C₃×C₃₇	1	1	yes	characteristic, complemented
44	#6909	C₃×C₃₇	1	24	yes	complemented
45	C₃×AΣL(1,𝔽₂₇)	C₃×C₃₇	1	12	yes	complemented
46	C₃×C₃₇	—	3	3	no	—
47	C₃×C₃₇	—	3	1	no	—
48	C₃×C₃₇	—	3	12	no	—
49	C₃×C₃₇	—	9	8	no	—
50	C₃×C₃₇	—	9	9	no	—
51	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
52	C₃⁴	C₃²×C₃₇	1	4	yes	complemented
53	#737	—	3	18	no	—
54	C₃×3₋¹⁺²	C₃²×C₃₇	1	24	yes	complemented
55	C₃×H(3,𝔽₃)	—	3	9	no	—
56	C₃×H(3,𝔽₃)	—	3	9	no	—
57	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	1	yes	characteristic
58	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	8	yes	complemented
59	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	12	yes	complemented
60	AΣL(1,𝔽₂₇)	C₃²×C₃₇	1	9	yes	complemented
61	AΣL(1,𝔽₂₇)	C₃²×C₃₇	1	72	yes	complemented
62	C₃₇	#10624930	1	1	yes	characteristic, direct product factor, Sylow
63	C₃³	—	3	12	no	—
64	C₃³	—	3	12	no	—
65	C₃³	—	9	12	no	—
66	C₃³	C₃³×C₃₇	1	1	yes	characteristic
67	C₃³	C₃³×C₃₇	1	8	yes	—
68	C₃³	C₃³×C₃₇	1	4	yes	—
69	3₋¹⁺²	—	3	48	no	—
70	3₋¹⁺²	C₃³×C₃₇	1	18	yes	—
71	H(3,𝔽₃)	—	3	24	no	—
72	H(3,𝔽₃)	—	3	24	no	—
73	H(3,𝔽₃)	H(3,𝔽₃)×C₃₇	1	9	yes	—
74	H(3,𝔽₃)	C₃³×C₃₇	1	9	yes	—
75	C₃×C₉	—	3	24	no	—
76	C₃²	—	3	3	no	—
77	C₃²	—	3	1	no	—
78	C₃²	—	3	8	no	—
79	C₃²	—	3	9	no	—
80	C₃²	—	9	4	no	—
81	C₃²	—	9	36	no	—
82	C₃²	—	9	8	no	—
83	C₃²	—	27	12	no	—
84	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	12	yes	—
85	C₃²	C₃⁴×C₃₇	1	1	yes	characteristic
86	C₉	—	3	18	no	—
87	C₃	—	3	1	no	—
88	C₃	—	3	3	no	—
89	C₃	—	3	12	no	—
90	C₃	—	9	8	no	—
91	C₃	—	9	9	no	—
92	C₃	C₃²×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₇×3₋¹⁺⁴	C₃	semidirect
2. C₃₇×H(5,𝔽₃)	C₃	semidirect
3. C₃²×H(3,𝔽₃)×C₃₇	C₃	semidirect
4. C₃₇×#6909	C₃	semidirect
5. C₃×C₃₇×AΣL(1,𝔽₂₇)	C₃	semidirect
6. C₃⁴×C₃₇	C₃²	semidirect
8. C₃×3₋¹⁺²×C₃₇	C₃²	semidirect
12. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
13. C₃×H(3,𝔽₃)×C₃₇	C₃²	semidirect
14. C₃₇×AΣL(1,𝔽₂₇)	C₃²	semidirect
15. C₃₇×AΣL(1,𝔽₂₇)	C₃²	semidirect
29. #10624930	C₃₇	direct (N × H)
41. 3₋¹⁺⁴	C₃×C₃₇	semidirect
42. H(5,𝔽₃)	C₃×C₃₇	semidirect
43. C₃²×H(3,𝔽₃)	C₃×C₃₇	semidirect
44. #6909	C₃×C₃₇	semidirect
45. C₃×AΣL(1,𝔽₂₇)	C₃×C₃₇	semidirect
52. C₃⁴	C₃²×C₃₇	semidirect
54. C₃×3₋¹⁺²	C₃²×C₃₇	semidirect
58. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
59. C₃×H(3,𝔽₃)	C₃²×C₃₇	semidirect
60. AΣL(1,𝔽₂₇)	C₃²×C₃₇	semidirect
61. AΣL(1,𝔽₂₇)	C₃²×C₃₇	semidirect
62. C₃₇	#10624930	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₇×3₋¹⁺⁴	3
C₃₇×H(5,𝔽₃)	3
C₃²×H(3,𝔽₃)×C₃₇	3
C₃₇×#6909	3
C₃×C₃₇×AΣL(1,𝔽₂₇)	3
#10624930	37

Subgroups — C37×#10624930 #41004719

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624930 #41004719