Subgroups — C₃₇×#10624524 #41004313

Order 26973.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×#732	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉²×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃₇×#6868	C₃	1	2	yes	maximal, complemented
4	C₃×H(3,𝔽₃)×C₃₇	—	3	1	no	—
5	C₃²×C₉×C₃₇	—	3	1	no	—
6	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
7	C₃₇×#732	—	3	2	no	—
8	C₃₇×#732	C₉	1	3	yes	complemented
9	C₃₇×C₃²⋊C₉	—	3	2	no	—
10	C₉²×C₃₇	—	3	3	no	—
11	C₃³×C₃₇	—	9	1	no	—
12	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
13	H(3,𝔽₃)×C₃₇	—	3	3	no	—
14	H(3,𝔽₃)×C₃₇	—	9	1	no	—
15	H(3,𝔽₃)×C₃₇	—	9	1	no	—
16	C₃×C₉×C₃₇	—	3	6	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	—	3	1	no	—
21	C₃×C₉×C₃₇	—	3	1	no	—
22	C₃×C₉×C₃₇	—	9	2	no	—
23	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
24	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
25	#10624524	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₃²×C₃₇	—	3	2	no	—
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	—	9	3	no	—
29	C₃²×C₃₇	—	27	2	no	—
30	C₃²×C₃₇	—	27	1	no	—
31	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
32	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
33	C₃²×C₃₇	#731	1	1	yes	characteristic
34	C₃²×C₃₇	#732	1	1	yes	characteristic
35	C₉×C₃₇	—	3	9	no	—
36	C₉×C₃₇	—	3	1	no	—
37	C₉×C₃₇	—	3	2	no	—
38	C₉×C₃₇	—	27	2	no	—
39	C₃×#732	C₃×C₃₇	1	1	yes	characteristic
40	C₃×C₉²	C₃×C₃₇	1	1	yes	characteristic, complemented
41	#6868	C₃×C₃₇	1	2	yes	complemented
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	27	3	no	—
46	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
47	C₃×C₃₇	#6878	1	2	yes	—
48	C₃×C₃₇	#6879	1	1	yes	characteristic
49	C₃×H(3,𝔽₃)	—	3	1	no	—
50	C₃²×C₉	—	3	1	no	—
51	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
52	#732	—	3	2	no	—
53	#732	C₉×C₃₇	1	3	yes	complemented
54	C₃²⋊C₉	—	3	2	no	—
55	C₉²	—	3	3	no	—
56	C₃₇	#10624524	1	1	yes	characteristic, direct product factor, Sylow
57	C₃³	—	9	1	no	—
58	C₃³	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
59	H(3,𝔽₃)	—	3	3	no	—
60	H(3,𝔽₃)	—	9	1	no	—
61	H(3,𝔽₃)	—	9	1	no	—
62	C₃×C₉	—	3	6	no	—
63	C₃×C₉	—	3	3	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	3	3	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	9	2	no	—
69	C₃×C₉	C₃×C₉×C₃₇	1	1	yes	characteristic
70	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
71	C₃²	—	3	2	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	9	3	no	—
74	C₃²	—	27	2	no	—
75	C₃²	—	27	1	no	—
76	C₃²	C₃₇×C₃²⋊C₉	1	1	yes	characteristic
77	C₃²	C₃₇×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
78	C₃²	C₃₇×#731	1	1	yes	characteristic
79	C₃²	C₃₇×#732	1	1	yes	characteristic
80	C₉	—	3	9	no	—
81	C₉	—	3	1	no	—
82	C₉	—	3	2	no	—
83	C₉	—	27	2	no	—
84	C₃	—	3	1	no	—
85	C₃	—	3	1	no	—
86	C₃	—	3	1	no	—
87	C₃	—	27	3	no	—
88	C₃	C₃₇×H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
89	C₃	C₃₇×#6878	1	2	yes	—
90	C₃	C₃₇×#6879	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²×C₃₇	#732
commutator	C₃×C₉	C₃×C₉×C₃₇
Frattini, cosocle	C₃²×C₉	C₃²×C₃₇
prime radical	C₃×C₃₇×#732	C₃
LCS	G ▷ C₃×C₉ ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃²×C₃₇ ◁ C₃³×C₃₇ ◁ C₃²×C₉×C₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₉²×C₃₇	C₃	semidirect
3. C₃₇×#6868	C₃	semidirect
8. C₃₇×#732	C₉	semidirect
25. #10624524	C₃₇	direct (N × H)
40. C₃×C₉²	C₃×C₃₇	semidirect
41. #6868	C₃×C₃₇	semidirect
53. #732	C₉×C₃₇	semidirect
56. C₃₇	#10624524	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₃₇×#732	3
C₃×C₉²×C₃₇	3
C₃₇×#6868	3
#10624524	37

Subgroups — C37×#10624524 #41004313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624524 #41004313