Subgroups — #41004839

Order 26973.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688521	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688519	C₃	1	1	yes	maximal, characteristic
3	#22582470	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃₇×#6867	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×#11758608	—	3	1	no	—
6	C₃×C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
7	#15688521	—	3	3	no	—
8	#15688519	—	3	3	no	—
9	C₃²⋊#66524	—	3	1	no	—
10	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
11	C₃₇×C₃²⋊C₉	—	3	1	no	—
12	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
13	C₃²×C₃₇⋊C₃	—	9	1	no	—
14	C₃×C₉×C₃₇	—	3	1	no	—
15	C₃×C₉×C₃₇	—	3	1	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	1	no	—
18	C₃×C₉×C₃₇	—	9	1	no	—
19	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
20	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
21	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
22	#11758608	—	3	3	no	—
23	#11758608	—	9	1	no	—
24	#11758608	—	9	1	no	—
25	C₃×#66524	—	9	1	no	—
26	C₃×#66524	—	9	1	no	—
27	#10624478	—	37	1	no	maximal, Sylow
28	C₃²×C₃₇	—	9	1	no	—
29	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
30	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
31	C₃²×C₃₇	#732	1	1	yes	characteristic
32	C₃²×C₃₇	#732	1	1	yes	characteristic
33	C₃×C₃₇⋊C₃	—	9	3	no	—
34	C₃×C₃₇⋊C₃	—	27	3	no	—
35	C₉×C₃₇	—	9	1	no	—
36	C₉×C₃₇	—	27	1	no	—
37	#66524	—	9	3	no	—
38	#66524	—	27	1	no	—
39	C₃×#733	—	37	1	no	—
40	C₃×#732	—	37	1	no	—
41	#6867	—	37	1	no	—
42	#6867	C₃₇⋊C₃	1	1	yes	characteristic, complemented
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	3	1	no	—
46	C₃×C₃₇	—	3	1	no	—
47	C₃×C₃₇	#6856	1	1	yes	characteristic
48	C₃₇⋊C₃	—	27	3	no	—
49	C₃×3₋¹⁺²	—	111	1	no	—
50	C₃×H(3,𝔽₃)	—	111	1	no	—
51	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
52	#733	—	111	3	no	—
53	#732	—	111	3	no	—
54	C₃²⋊C₉	—	3	1	no	—
55	C₃²⋊C₉	—	111	1	no	—
56	C₃₇	#10624478	1	1	yes	characteristic, complemented, Sylow
57	C₃³	—	333	1	no	—
58	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
59	3₋¹⁺²	—	333	1	no	—
60	3₋¹⁺²	—	333	1	no	—
61	3₋¹⁺²	—	111	3	no	—
62	H(3,𝔽₃)	—	333	1	no	—
63	H(3,𝔽₃)	—	333	1	no	—
64	H(3,𝔽₃)	—	111	3	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	—	9	1	no	—
70	C₃×C₉	—	333	1	no	—
71	C₃×C₉	—	333	1	no	—
72	C₃²	—	9	1	no	—
73	C₃²	—	333	3	no	—
74	C₃²	—	999	3	no	—
75	C₃²	#15688514	1	1	yes	characteristic
76	C₃²	#15688514	1	1	yes	characteristic
77	C₃²	#15688518	1	1	yes	characteristic
78	C₃²	#15688519	1	1	yes	characteristic
79	C₉	—	9	1	no	—
80	C₉	—	27	1	no	—
81	C₉	—	333	3	no	—
82	C₉	—	999	1	no	—
83	C₃	—	3	1	no	—
84	C₃	—	3	1	no	—
85	C₃	—	3	1	no	—
86	C₃	—	3	1	no	—
87	C₃	—	999	3	no	—
88	C₃	#22582443	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582443
commutator, cosocle	C₃²×C₉×C₃₇	C₃²
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting	C₃₇×#6867	C₃
socle	C₃×C₃₇	#6856
prime radical	C₃×#15688519	C₃
hypercenter	#6867	C₃₇⋊C₃
hypocenter	C₃₇	#10624478
LCS	G ▷ C₃²×C₉×C₃₇ ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃²×C₉ ◁ #6867

Complements

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

N	H ≅ G/N	type
1. C₃×#15688521	C₃	semidirect
3. #22582470	C₃	semidirect
4. C₃₇×#6867	C₃	semidirect
42. #6867	C₃₇⋊C₃	semidirect
56. C₃₇	#10624478	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688521	3
C₃×#15688519	3
#22582470	3
C₃₇×#6867	3
#10624478	37