Subgroups — #41004830

Order 26973.

154 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582554	C₃	1	1	yes	maximal, characteristic
2	#22582468	C₃	1	2	yes	maximal
3	C₃₇×#6891	C₃	1	1	yes	maximal, characteristic
4	C₃×#11758608	—	3	1	no	—
5	C₃²⋊#66524	—	3	3	no	—
6	C₃²⋊#66524	—	3	2	no	—
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
9	C₃₇×C₃²⋊C₉	—	3	3	no	—
10	C₃³×C₃₇	—	3	1	no	—
11	C₃³×C₃₇	—	3	1	no	—
12	C₃³×C₃₇	—	3	2	no	—
13	C₃³×C₃₇	—	9	3	no	—
14	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
15	3₋¹⁺²×C₃₇	—	9	1	no	—
16	3₋¹⁺²×C₃₇	—	9	2	no	—
17	C₃×C₉×C₃₇	—	3	3	no	—
18	#11758608	—	9	1	no	—
19	#11758608	—	9	2	no	—
20	C₃×#66524	—	3	3	no	—
21	C₃×#66524	—	9	2	no	—
22	#10624476	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	3	no	—
24	C₃²×C₃₇	—	3	3	no	—
25	C₃²×C₃₇	—	9	1	no	—
26	C₃²×C₃₇	—	9	6	no	—
27	C₃²×C₃₇	—	9	2	no	—
28	C₃²×C₃₇	—	9	2	no	—
29	C₃²×C₃₇	—	9	1	no	—
30	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
31	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
32	C₃²×C₃₇	#732	1	2	yes	—
33	C₉×C₃₇	—	9	3	no	—
34	#66524	—	9	3	no	—
35	#66524	—	27	2	no	—
36	#6891	—	37	1	no	—
37	#6891	C₃₇⋊C₃	1	1	yes	characteristic
38	#6866	—	37	2	no	—
39	C₃×C₃₇	—	3	2	no	—
40	C₃×C₃₇	—	3	1	no	—
41	C₃×C₃₇	—	3	1	no	—
42	C₃×C₃₇	—	9	3	no	—
43	C₃×C₃₇	#6856	1	1	yes	characteristic
44	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
45	C₃×3₋¹⁺²	—	3	1	no	—
46	C₃×3₋¹⁺²	—	111	1	no	—
47	C₃²⋊C₉	—	3	3	no	—
48	C₃²⋊C₉	—	111	2	no	—
49	C₃²⋊C₉	—	111	3	no	—
50	C₃₇	#10624476	1	1	yes	characteristic, complemented, Sylow
51	C₃³	—	3	1	no	—
52	C₃³	—	3	1	no	—
53	C₃³	—	3	2	no	—
54	C₃³	—	9	3	no	—
55	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
56	3₋¹⁺²	—	9	1	no	—
57	3₋¹⁺²	—	9	2	no	—
58	3₋¹⁺²	—	333	1	no	—
59	3₋¹⁺²	—	333	2	no	—
60	C₃×C₉	—	3	3	no	—
61	C₃×C₉	—	333	2	no	—
62	C₃×C₉	—	111	3	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	3	no	—
65	C₃²	—	9	1	no	—
66	C₃²	—	9	6	no	—
67	C₃²	—	9	2	no	—
68	C₃²	—	9	2	no	—
69	C₃²	—	9	1	no	—
70	C₃²	#15688512	1	1	yes	characteristic
71	C₃²	#15688513	1	1	yes	characteristic
72	C₃²	#15688518	1	2	yes	—
73	C₉	—	9	3	no	—
74	C₉	—	333	3	no	—
75	C₉	—	999	2	no	—
76	C₃	—	3	2	no	—
77	C₃	—	3	1	no	—
78	C₃	—	3	1	no	—
79	C₃	—	9	3	no	—
80	C₃	#22582442	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582442
commutator, prime radical, cosocle	C₃⁴×C₃₇	C₃²
Frattini	C₃⁴	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6891	C₃
socle	C₃×C₃₇	#6856
hypercenter	#6891	C₃₇⋊C₃
hypocenter	C₃₇	#10624476
LCS	G ▷ C₃⁴×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃⁴ ◁ #6891

Complements

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

N	H ≅ G/N	type
50. C₃₇	#10624476	semidirect

Maximal subgroups

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

subgroup	index
#22582554	3
#22582468	3
C₃₇×#6891	3
#10624476	37