Subgroups — C₃₇×#10624656 #41004445

Order 26973.

320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₇×#6891	C₃	1	2	yes	maximal, complemented
2	C₃₇×C₃²⋊H(3,𝔽₃)	C₃	1	2	yes	maximal, complemented
3	C₃₇×#6856	C₃	1	9	yes	maximal, complemented
4	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
6	C₃×H(3,𝔽₃)×C₃₇	—	3	2	no	—
7	C₃×H(3,𝔽₃)×C₃₇	—	3	2	no	—
8	C₃×H(3,𝔽₃)×C₃₇	—	3	2	no	—
9	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	6	yes	—
10	C₃₇×C₃²⋊C₉	—	3	2	no	—
11	C₃₇×C₃²⋊C₉	—	3	2	no	—
12	C₃₇×C₃²⋊C₉	C₃²	1	6	yes	—
13	C₃³×C₃₇	—	3	6	no	—
14	C₃³×C₃₇	—	9	2	no	—
15	C₃³×C₃₇	—	9	2	no	—
16	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃³×C₃₇	H(3,𝔽₃)	1	2	yes	—
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	3₋¹⁺²×C₃₇	—	9	6	no	—
20	H(3,𝔽₃)×C₃₇	—	9	6	no	—
21	H(3,𝔽₃)×C₃₇	—	9	6	no	—
22	H(3,𝔽₃)×C₃₇	—	9	6	no	—
23	H(3,𝔽₃)×C₃₇	—	9	6	no	—
24	C₃×C₉×C₃₇	—	3	6	no	—
25	#10624656	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃²×C₃₇	—	3	2	no	—
27	C₃²×C₃₇	—	3	2	no	—
28	C₃²×C₃₇	—	3	2	no	—
29	C₃²×C₃₇	—	3	2	no	—
30	C₃²×C₃₇	—	3	2	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	—	3	2	no	—
33	C₃²×C₃₇	—	3	2	no	—
34	C₃²×C₃₇	—	9	9	no	—
35	C₃²×C₃₇	—	9	6	no	—
36	C₃²×C₃₇	—	9	6	no	—
37	C₃²×C₃₇	—	9	6	no	—
38	C₃²×C₃₇	—	9	6	no	—
39	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
40	C₉×C₃₇	—	9	6	no	—
41	#6891	C₃×C₃₇	1	2	yes	complemented
42	C₃²⋊H(3,𝔽₃)	C₃×C₃₇	1	2	yes	complemented
43	#6856	C₃×C₃₇	1	9	yes	complemented
44	C₃×C₃₇	—	9	2	no	—
45	C₃×C₃₇	—	9	1	no	—
46	C₃×C₃₇	—	9	1	no	—
47	C₃×C₃₇	—	27	6	no	—
48	C₃×C₃₇	#6909	1	2	yes	—
49	C₃×C₃₇	#6911	1	2	yes	—
50	C₃⁴	C₃²×C₃₇	1	1	yes	characteristic
51	C₃×3₋¹⁺²	—	3	2	no	—
52	C₃×H(3,𝔽₃)	—	3	2	no	—
53	C₃×H(3,𝔽₃)	—	3	2	no	—
54	C₃×H(3,𝔽₃)	—	3	2	no	—
55	C₃×H(3,𝔽₃)	C₃²×C₃₇	1	6	yes	—
56	C₃²⋊C₉	—	3	2	no	—
57	C₃²⋊C₉	—	3	2	no	—
58	C₃²⋊C₉	C₃²×C₃₇	1	6	yes	—
59	C₃₇	#10624656	1	1	yes	characteristic, direct product factor, Sylow
60	C₃³	—	3	6	no	—
61	C₃³	—	9	2	no	—
62	C₃³	—	9	2	no	—
63	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
64	C₃³	H(3,𝔽₃)×C₃₇	1	2	yes	—
65	C₃³	C₃³×C₃₇	1	1	yes	characteristic
66	3₋¹⁺²	—	9	6	no	—
67	H(3,𝔽₃)	—	9	6	no	—
68	H(3,𝔽₃)	—	9	6	no	—
69	H(3,𝔽₃)	—	9	6	no	—
70	H(3,𝔽₃)	—	9	6	no	—
71	C₃×C₉	—	3	6	no	—
72	C₃²	—	3	2	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	2	no	—
75	C₃²	—	3	2	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	3	2	no	—
78	C₃²	—	3	2	no	—
79	C₃²	—	3	2	no	—
80	C₃²	—	9	9	no	—
81	C₃²	—	9	6	no	—
82	C₃²	—	9	6	no	—
83	C₃²	—	9	6	no	—
84	C₃²	—	9	6	no	—
85	C₃²	C₃×H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
86	C₉	—	9	6	no	—
87	C₃	—	9	2	no	—
88	C₃	—	9	1	no	—
89	C₃	—	9	1	no	—
90	C₃	—	27	6	no	—
91	C₃	C₃₇×#6909	1	2	yes	—
92	C₃	C₃₇×#6911	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₃₇×#6891	C₃	semidirect
2. C₃₇×C₃²⋊H(3,𝔽₃)	C₃	semidirect
3. C₃₇×#6856	C₃	semidirect
25. #10624656	C₃₇	direct (N × H)
41. #6891	C₃×C₃₇	semidirect
42. C₃²⋊H(3,𝔽₃)	C₃×C₃₇	semidirect
43. #6856	C₃×C₃₇	semidirect
59. C₃₇	#10624656	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×#6891	3
C₃₇×C₃²⋊H(3,𝔽₃)	3
C₃₇×#6856	3
#10624656	37

Subgroups — C37×#10624656 #41004445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624656 #41004445