Subgroups — #41005029

Order 26973.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688506	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582472	C₃	1	2	yes	maximal, complemented
3	C₃×C₃₇×#732	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	—	3	1	no	—
5	#15688506	—	3	3	no	—
6	C₃²⋊#66524	—	3	2	no	—
7	C₃×H(3,𝔽₃)×C₃₇	—	3	1	no	—
8	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
9	C₃₇×#732	—	3	2	no	—
10	C₃₇×#732	C₉	1	3	yes	complemented
11	C₃³×C₃₇	—	9	1	no	—
12	C₃³×C₃₇	H(3,𝔽₃)	1	1	yes	characteristic
13	H(3,𝔽₃)×C₃₇	—	3	3	no	—
14	H(3,𝔽₃)×C₃₇	—	9	1	no	—
15	H(3,𝔽₃)×C₃₇	—	9	1	no	—
16	C₃×C₉×C₃₇	—	3	1	no	—
17	C₃×C₉×C₃₇	—	3	1	no	—
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
20	C₃×C₉×C₃₇	3₋¹⁺²	1	2	yes	—
21	C₃×#66524	—	3	3	no	—
22	C₃×#66524	—	9	2	no	—
23	C₃×#66524	—	9	1	no	—
24	C₃×#66524	—	9	2	no	—
25	#10624527	—	37	1	no	maximal, Sylow
26	C₃²×C₃₇	—	3	2	no	—
27	C₃²×C₃₇	—	3	1	no	—
28	C₃²×C₃₇	—	9	3	no	—
29	C₃²×C₃₇	—	27	2	no	—
30	C₃²×C₃₇	—	27	1	no	—
31	C₃²×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
32	C₃²×C₃₇	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
33	C₃²×C₃₇	#731	1	1	yes	characteristic
34	C₃²×C₃₇	#732	1	1	yes	characteristic
35	C₉×C₃₇	—	3	1	no	—
36	C₉×C₃₇	—	3	2	no	—
37	#66524	—	9	3	no	—
38	#66524	—	27	2	no	—
39	C₃×#732	C₃₇⋊C₃	1	1	yes	characteristic
40	C₃×C₉⋊C₉	—	37	1	no	—
41	#6868	—	37	2	no	—
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	—	3	1	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	27	3	no	—
46	C₃×C₃₇	H(3,𝔽₃)⋊C₉	1	1	yes	characteristic
47	C₃×C₃₇	#6881	1	2	yes	—
48	C₃×C₃₇	#6883	1	1	yes	characteristic
49	C₃×H(3,𝔽₃)	—	3	1	no	—
50	C₃²×C₉	—	111	1	no	—
51	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
52	#732	—	3	2	no	—
53	#732	#66524	1	3	yes	complemented
54	C₉⋊C₉	—	111	3	no	—
55	C₃²⋊C₉	—	111	2	no	—
56	C₃₇	#10624527	1	1	yes	characteristic, complemented, Sylow
57	C₃³	—	9	1	no	—
58	C₃³	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
59	H(3,𝔽₃)	—	3	3	no	—
60	H(3,𝔽₃)	—	9	1	no	—
61	H(3,𝔽₃)	—	9	1	no	—
62	C₃×C₉	—	3	1	no	—
63	C₃×C₉	—	3	1	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	333	2	no	—
66	C₃×C₉	—	333	1	no	—
67	C₃×C₉	—	333	2	no	—
68	C₃×C₉	—	111	3	no	—
69	C₃×C₉	C₃×#66524	1	1	yes	characteristic
70	C₃×C₉	#11758608	1	2	yes	—
71	C₃²	—	3	2	no	—
72	C₃²	—	3	1	no	—
73	C₃²	—	9	3	no	—
74	C₃²	—	27	2	no	—
75	C₃²	—	27	1	no	—
76	C₃²	C₃²⋊#66524	1	1	yes	characteristic
77	C₃²	#15688512	1	1	yes	characteristic
78	C₃²	#15688515	1	1	yes	characteristic
79	C₃²	#15688518	1	1	yes	characteristic
80	C₉	—	3	1	no	—
81	C₉	—	3	2	no	—
82	C₉	—	333	3	no	—
83	C₉	—	999	2	no	—
84	C₃	—	3	1	no	—
85	C₃	—	3	1	no	—
86	C₃	—	3	1	no	—
87	C₃	—	27	3	no	—
88	C₃	H(3,𝔽₃)⋊#66524	1	1	yes	characteristic
89	C₃	#22582511	1	2	yes	—
90	C₃	#22582524	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	#15688518
commutator	C₃×C₉×C₃₇	C₃×C₉
Frattini	C₃²×C₉	C₃×C₃₇⋊C₃
Fitting, prime radical	C₃×C₃₇×#732	C₃
socle	C₃²×C₃₇	#732
cosocle	C₃²×C₉×C₃₇	C₃²
hypercenter	C₃×#732	C₃₇⋊C₃
hypocenter	C₃₇	#10624527
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉ ◁ C₃×#732

Complements

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

N	H ≅ G/N	type
1. C₃×#15688506	C₃	semidirect
2. #22582472	C₃	semidirect
10. C₃₇×#732	C₉	semidirect
53. #732	#66524	semidirect
56. C₃₇	#10624527	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688506	3
#22582472	3
C₃×C₃₇×#732	3
#10624527	37