Subgroups — C₃₇×#10624454 #41004243

Order 26973.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃₇×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic
2	C₉×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃₇×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
4	C₃₇×C₃²⋊C₂₇	C₃	1	1	yes	maximal, characteristic
5	C₃×3₋¹⁺²×C₃₇	C₉	1	1	yes	characteristic
6	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
7	C₃₇×C₂₇⋊C₃	—	3	3	no	—
8	C₃×C₂₇×C₃₇	—	3	1	no	—
9	C₃×C₂₇×C₃₇	—	3	1	no	—
10	C₃×C₂₇×C₃₇	—	3	1	no	—
11	C₃₇×C₉⋊C₉	—	3	2	no	—
12	C₃₇×C₃²⋊C₉	C₉	1	1	yes	characteristic
13	C₃₇×C₃²⋊C₉	C₉	1	1	yes	characteristic
14	C₉²×C₃₇	—	3	1	no	—
15	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
16	3₋¹⁺²×C₃₇	—	3	3	no	—
17	C₃×C₉×C₃₇	—	3	1	no	—
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×C₉×C₃₇	—	3	1	no	—
21	C₃×C₉×C₃₇	—	3	1	no	—
22	C₃×C₉×C₃₇	—	3	1	no	—
23	C₃×C₉×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
25	C₂₇×C₃₇	—	3	3	no	—
26	C₂₇×C₃₇	—	9	1	no	—
27	C₂₇×C₃₇	—	9	1	no	—
28	#10624454	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	1	no	—
32	C₃²×C₃₇	—	3	1	no	—
33	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
34	C₉×C₃₇	—	3	3	no	—
35	C₉×C₃₇	—	3	2	no	—
36	C₉×C₃₇	—	9	1	no	—
37	C₉×C₃₇	—	9	1	no	—
38	C₉×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
39	C₉×C₃₇	#731	1	1	yes	characteristic
40	C₉×C₃₇	#733	1	1	yes	characteristic
41	C₃×C₂₇⋊C₃	C₃×C₃₇	1	1	yes	characteristic
42	C₉×3₋¹⁺²	C₃×C₃₇	1	1	yes	characteristic
43	C₃²⋊C₂₇	C₃×C₃₇	1	1	yes	characteristic
44	C₃²⋊C₂₇	C₃×C₃₇	1	1	yes	characteristic
45	C₃×C₃₇	—	9	1	no	—
46	C₃×C₃₇	#6869	1	1	yes	characteristic
47	C₃×C₃₇	3₋¹⁺²⋊C₉	1	1	yes	characteristic
48	C₃×C₃₇	#6872	1	1	yes	characteristic
49	C₃×C₃₇	#6873	1	1	yes	characteristic
50	C₃×3₋¹⁺²	C₉×C₃₇	1	1	yes	characteristic
51	C₃²×C₉	C₃²×C₃₇	1	1	yes	characteristic
52	C₂₇⋊C₃	—	3	3	no	—
53	C₃×C₂₇	—	3	1	no	—
54	C₃×C₂₇	—	3	1	no	—
55	C₃×C₂₇	—	3	1	no	—
56	C₉⋊C₉	—	3	2	no	—
57	C₃²⋊C₉	C₉×C₃₇	1	1	yes	characteristic
58	C₃²⋊C₉	C₉×C₃₇	1	1	yes	characteristic
59	C₉²	—	3	1	no	—
60	C₃₇	#10624454	1	1	yes	characteristic, direct product factor, Sylow
61	C₃³	C₃×C₉×C₃₇	1	1	yes	characteristic
62	3₋¹⁺²	—	3	3	no	—
63	C₃×C₉	—	3	1	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	1	no	—
67	C₃×C₉	—	3	1	no	—
68	C₃×C₉	—	3	1	no	—
69	C₃×C₉	H(3,𝔽₃)×C₃₇	1	1	yes	characteristic
70	C₃×C₉	3₋¹⁺²×C₃₇	1	2	yes	—
71	C₂₇	—	3	3	no	—
72	C₂₇	—	9	1	no	—
73	C₂₇	—	9	1	no	—
74	C₃²	—	3	1	no	—
75	C₃²	—	3	1	no	—
76	C₃²	—	3	1	no	—
77	C₃²	—	3	1	no	—
78	C₃²	C₃₇×C₃²⋊C₉	1	1	yes	characteristic
79	C₉	—	3	3	no	—
80	C₉	—	3	2	no	—
81	C₉	—	9	1	no	—
82	C₉	—	9	1	no	—
83	C₉	C₃₇×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
84	C₉	C₃₇×#731	1	1	yes	characteristic
85	C₉	C₃₇×#733	1	1	yes	characteristic
86	C₃	—	9	1	no	—
87	C₃	C₃₇×#6869	1	1	yes	characteristic
88	C₃	C₃₇×3₋¹⁺²⋊C₉	1	1	yes	characteristic
89	C₃	C₃₇×#6872	1	1	yes	characteristic
90	C₃	C₃₇×#6873	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
28. #10624454	C₃₇	direct (N × H)
60. C₃₇	#10624454	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×C₂₇⋊C₃	3
C₉×3₋¹⁺²×C₃₇	3
C₃₇×C₃²⋊C₂₇	3
#10624454	37

Subgroups — C37×#10624454 #41004243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#10624454 #41004243