Subgroups — #41006430

Order 26973.

250 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	3₋¹⁺²×#66524	C₃	1	3	yes	maximal
2	C₃×#15688506	C₃	1	1	yes	maximal, characteristic
3	3₋¹⁺²⋊#66524	C₃	1	6	yes	maximal
4	#22582472	C₃	1	2	yes	maximal
5	C₃×C₃₇×#733	C₃	1	1	yes	maximal, characteristic
6	C₃²×#66524	C₃²	1	2	yes	—
7	C₃²×#66524	C₃²	1	1	yes	characteristic
8	#15688506	—	3	3	no	—
9	#15688506	—	3	3	no	—
10	#15688505	—	3	3	no	—
11	C₃²⋊#66524	C₃²	1	6	yes	—
12	C₉×#66524	—	3	3	no	—
13	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
14	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
15	C₃₇×#733	C₉	1	9	yes	complemented
16	C₃³×C₃₇	C₃³	1	1	yes	characteristic
17	3₋¹⁺²×C₃₇	—	3	6	no	—
18	3₋¹⁺²×C₃₇	C₃×C₉	1	9	yes	—
19	C₃×C₉×C₃₇	—	3	1	no	—
20	C₃×C₉×C₃₇	—	3	3	no	—
21	C₃×C₉×C₃₇	—	3	2	no	—
22	C₃×C₉×C₃₇	C₃×C₉	1	3	yes	—
23	C₃×#66524	—	3	6	no	—
24	C₃×#66524	—	3	2	no	—
25	C₃×#66524	—	9	1	no	—
26	C₃×#66524	—	9	2	no	—
27	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
28	#10624723	—	37	1	no	maximal, Sylow
29	C₃²×C₃₇	—	3	1	no	—
30	C₃²×C₃₇	—	3	2	no	—
31	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
32	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃²×C₃₇	#737	1	2	yes	—
34	C₉×C₃₇	—	3	3	no	—
35	C₉×C₃₇	—	3	9	no	—
36	#66524	—	3	3	no	—
37	#66524	—	9	6	no	—
38	#66524	—	9	2	no	—
39	C₃×#733	C₃₇⋊C₃	1	1	yes	characteristic
40	C₉×3₋¹⁺²	—	37	3	no	—
41	C₃×C₉⋊C₉	—	37	1	no	—
42	3₋¹⁺²⋊C₉	—	37	6	no	—
43	#6868	—	37	2	no	—
44	C₃×C₃₇	—	3	1	no	—
45	C₃×C₃₇	—	3	2	no	—
46	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
47	C₃×C₃₇	#6912	1	1	yes	characteristic
48	C₃×C₃₇	#6913	1	2	yes	—
49	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
50	C₃²×C₉	—	37	1	no	—
51	C₃²×C₉	—	37	2	no	—
52	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
53	#733	#66524	1	9	yes	complemented
54	C₉⋊C₉	—	111	3	no	—
55	C₉⋊C₉	—	111	3	no	—
56	C₉⋊C₉	—	111	3	no	—
57	C₃²⋊C₉	—	37	6	no	—
58	C₉²	—	111	3	no	—
59	C₃₇	#10624723	1	1	yes	characteristic, complemented, Sylow
60	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
61	3₋¹⁺²	—	3	6	no	—
62	3₋¹⁺²	C₃×#66524	1	9	yes	—
63	C₃×C₉	—	3	3	no	—
64	C₃×C₉	—	3	1	no	—
65	C₃×C₉	—	3	2	no	—
66	C₃×C₉	—	37	3	no	—
67	C₃×C₉	—	333	2	no	—
68	C₃×C₉	—	333	1	no	—
69	C₃×C₉	—	111	6	no	—
70	C₃×C₉	—	111	2	no	—
71	C₃×C₉	C₃×#66524	1	3	yes	—
72	C₃²	—	3	1	no	—
73	C₃²	—	3	2	no	—
74	C₃²	C₃²×#66524	1	1	yes	characteristic
75	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
76	C₃²	#15688530	1	2	yes	—
77	C₉	—	3	3	no	—
78	C₉	—	3	9	no	—
79	C₉	—	333	2	no	—
80	C₉	—	333	6	no	—
81	C₉	—	111	3	no	—
82	C₃	—	3	1	no	—
83	C₃	—	3	2	no	—
84	C₃	H(3,𝔽₃)×#66524	1	1	yes	characteristic
85	C₃	#22582650	1	1	yes	characteristic
86	C₃	#22582663	1	2	yes	—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#733	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical, cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃×#733	C₃₇⋊C₃
hypocenter	C₃₇	#10624723
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#733

Complements

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

N	H ≅ G/N	type
15. C₃₇×#733	C₉	semidirect
53. #733	#66524	semidirect
59. C₃₇	#10624723	semidirect

Maximal subgroups

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

subgroup	index
3₋¹⁺²×#66524	3
C₃×#15688506	3
3₋¹⁺²⋊#66524	3
#22582472	3
C₃×C₃₇×#733	3
#10624723	37