Subgroups — #41004761

Order 26973.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₃²⋊#66524	C₃	1	3	yes	maximal
2	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃²×#66524	—	3	3	no	—
4	C₃²⋊#66524	—	3	9	no	—
5	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
6	C₃×3₋¹⁺²×C₃₇	—	3	2	no	—
7	C₃×3₋¹⁺²×C₃₇	—	3	9	no	—
8	C₃×3₋¹⁺²×C₃₇	C₉	1	3	yes	complemented
9	C₃²×C₉×C₃₇	—	3	1	no	—
10	C₃³×C₃₇	—	3	3	no	—
11	C₃³×C₃₇	—	3	3	no	—
12	C₃³×C₃₇	—	3	3	no	—
13	C₃³×C₃₇	—	3	3	no	—
14	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
15	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
16	C₃³×C₃₇	3₋¹⁺²	1	2	yes	—
17	3₋¹⁺²×C₃₇	—	3	18	no	—
18	3₋¹⁺²×C₃₇	—	3	9	no	—
19	C₃×C₉×C₃₇	—	3	3	no	—
20	C₃×C₉×C₃₇	—	3	9	no	—
21	C₃×#66524	—	3	9	no	—
22	C₃×#66524	—	9	3	no	—
23	C₃×#66524	—	9	3	no	—
24	C₃×#66524	—	9	3	no	—
25	#10624452	—	37	1	no	maximal, Sylow
26	C₃²×C₃₇	—	3	1	no	—
27	C₃²×C₃₇	—	3	2	no	—
28	C₃²×C₃₇	—	3	3	no	—
29	C₃²×C₃₇	—	3	6	no	—
30	C₃²×C₃₇	—	9	3	no	—
31	C₃²×C₃₇	—	9	3	no	—
32	C₃²×C₃₇	—	9	3	no	—
33	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
34	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
35	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
36	C₃²×C₃₇	#731	1	3	yes	—
37	C₃²×C₃₇	#733	1	3	yes	—
38	C₉×C₃₇	—	3	9	no	—
39	#66524	—	9	9	no	—
40	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
41	C₃×C₃²⋊C₉	—	37	3	no	—
42	C₃×C₃₇	—	9	1	no	—
43	C₃×C₃₇	—	9	2	no	—
44	C₃×C₃₇	#6857	1	3	yes	—
45	C₃×C₃₇	#6858	1	3	yes	—
46	C₃×C₃₇	#6860	1	3	yes	—
47	C₃×C₃₇	#6866	1	1	yes	characteristic
48	C₃×C₃₇	3₋¹⁺²⋊C₉	1	3	yes	—
49	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
50	C₃×3₋¹⁺²	—	3	2	no	—
51	C₃×3₋¹⁺²	—	3	9	no	—
52	C₃×3₋¹⁺²	#66524	1	3	yes	complemented
53	C₃²×C₉	—	3	1	no	—
54	C₃²×C₉	—	111	3	no	—
55	C₃²⋊C₉	—	111	9	no	—
56	C₃₇	#10624452	1	1	yes	characteristic, complemented, Sylow
57	C₃³	—	3	3	no	—
58	C₃³	—	3	3	no	—
59	C₃³	—	3	3	no	—
60	C₃³	—	3	3	no	—
61	C₃³	C₃×#66524	1	1	yes	characteristic
62	C₃³	#11758608	1	2	yes	—
63	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
64	3₋¹⁺²	—	3	18	no	—
65	3₋¹⁺²	—	3	9	no	—
66	C₃×C₉	—	3	3	no	—
67	C₃×C₉	—	3	9	no	—
68	C₃×C₉	—	333	3	no	—
69	C₃×C₉	—	333	3	no	—
70	C₃×C₉	—	333	3	no	—
71	C₃×C₉	—	111	9	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	2	no	—
74	C₃²	—	3	3	no	—
75	C₃²	—	3	6	no	—
76	C₃²	—	9	3	no	—
77	C₃²	—	9	3	no	—
78	C₃²	—	9	3	no	—
79	C₃²	C₃²⋊#66524	1	1	yes	characteristic
80	C₃²	#15688513	1	3	yes	—
81	C₃²	#15688514	1	3	yes	—
82	C₃²	#15688517	1	3	yes	—
83	C₃²	#15688520	1	3	yes	—
84	C₉	—	3	9	no	—
85	C₉	—	333	9	no	—
86	C₃	—	9	1	no	—
87	C₃	—	9	2	no	—
88	C₃	#22582444	1	3	yes	—
89	C₃	#22582447	1	3	yes	—
90	C₃	#22582454	1	3	yes	—
91	C₃	#22582468	1	1	yes	characteristic
92	C₃	3₋¹⁺²⋊#66524	1	3	yes	—

	subgroup	G/H
center	C₃³	C₃²⋊(C₃₇⋊C₃)
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃⁴	C₃×C₃₇⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃³×C₃₇	H(3,𝔽₃)
prime radical, cosocle	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624452
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ C₃⁴ ◁ C₃²×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
8. C₃×3₋¹⁺²×C₃₇	C₉	semidirect
52. C₃×3₋¹⁺²	#66524	semidirect
56. C₃₇	#10624452	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₃²⋊#66524	3
C₃²×3₋¹⁺²×C₃₇	3
#10624452	37