Subgroups — #41004968

Order 26973.

242 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582468	C₃	1	3	yes	maximal
2	C₃²×3₋¹⁺²×C₃₇	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#66524	—	3	3	no	—
4	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
5	C₃×3₋¹⁺²×C₃₇	—	3	1	no	—
6	C₃×3₋¹⁺²×C₃₇	—	3	1	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	—	9	3	no	—
12	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
13	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
14	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
15	C₃³×C₃₇	3₋¹⁺²	1	1	yes	characteristic
16	3₋¹⁺²×C₃₇	—	3	9	no	—
17	3₋¹⁺²×C₃₇	—	3	9	no	—
18	3₋¹⁺²×C₃₇	—	3	9	no	—
19	C₃×C₉×C₃₇	—	3	9	no	—
20	C₃×C₉×C₃₇	—	3	3	no	—
21	C₃×#66524	—	9	3	no	—
22	#10624509	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	3	no	—
24	C₃²×C₃₇	—	3	1	no	—
25	C₃²×C₃₇	—	3	1	no	—
26	C₃²×C₃₇	—	3	1	no	—
27	C₃²×C₃₇	—	9	1	no	—
28	C₃²×C₃₇	—	9	9	no	—
29	C₃²×C₃₇	—	9	1	no	—
30	C₃²×C₃₇	—	9	1	no	—
31	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
32	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	3	yes	—
33	C₉×C₃₇	—	3	9	no	—
34	#66524	—	27	3	no	—
35	C₃²×3₋¹⁺²	C₃₇⋊C₃	1	1	yes	characteristic
36	#6866	—	37	3	no	—
37	C₃×C₃₇	—	3	3	no	—
38	C₃×C₃₇	—	3	3	no	—
39	C₃×C₃₇	—	3	1	no	—
40	C₃×C₃₇	—	9	1	no	—
41	C₃×C₃₇	—	9	1	no	—
42	C₃×C₃₇	#6866	1	1	yes	characteristic
43	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
44	C₃×3₋¹⁺²	—	3	1	no	—
45	C₃×3₋¹⁺²	—	3	1	no	—
46	C₃×3₋¹⁺²	—	3	9	no	—
47	C₃×3₋¹⁺²	#66524	1	3	yes	complemented
48	C₃²×C₉	—	3	1	no	—
49	C₃²⋊C₉	—	111	3	no	—
50	C₃₇	#10624509	1	1	yes	characteristic, complemented, Sylow
51	C₃³	—	3	3	no	—
52	C₃³	—	9	3	no	—
53	C₃³	C₃×#66524	1	1	yes	characteristic
54	C₃³	#11758608	1	1	yes	characteristic
55	C₃³	#11758608	1	1	yes	characteristic
56	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
57	3₋¹⁺²	—	3	9	no	—
58	3₋¹⁺²	—	3	9	no	—
59	3₋¹⁺²	—	3	9	no	—
60	C₃×C₉	—	3	9	no	—
61	C₃×C₉	—	3	3	no	—
62	C₃×C₉	—	333	3	no	—
63	C₃²	—	3	3	no	—
64	C₃²	—	3	1	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	3	1	no	—
67	C₃²	—	9	1	no	—
68	C₃²	—	9	9	no	—
69	C₃²	—	9	1	no	—
70	C₃²	—	9	1	no	—
71	C₃²	C₃²⋊#66524	1	1	yes	characteristic
72	C₃²	#15688513	1	3	yes	—
73	C₉	—	3	9	no	—
74	C₉	—	999	3	no	—
75	C₃	—	3	3	no	—
76	C₃	—	3	1	no	—
77	C₃	—	3	3	no	—
78	C₃	—	9	1	no	—
79	C₃	—	9	1	no	—
80	C₃	#22582468	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582468
commutator	C₃³×C₃₇	C₃×C₉
Frattini	C₃⁴	C₃×C₃₇⋊C₃
Fitting	C₃²×3₋¹⁺²×C₃₇	C₃
socle	C₃×C₃₇	#6866
prime radical, cosocle	C₃⁴×C₃₇	C₃²
hypercenter	C₃²×3₋¹⁺²	C₃₇⋊C₃
hypocenter	C₃₇	#10624509
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ 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
47. C₃×3₋¹⁺²	#66524	semidirect
50. C₃₇	#10624509	semidirect

Maximal subgroups

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

subgroup	index
#22582468	3
C₃²×3₋¹⁺²×C₃₇	3
#10624509	37