Subgroups — #41006235

Order 26973.

512 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, complemented
2	#22582468	C₃	1	9	yes	maximal, complemented
3	C₃²×H(3,𝔽₃)×C₃₇	C₃	1	1	yes	maximal, characteristic
4	C₃²×#66524	—	3	3	no	—
5	C₃²⋊#66524	—	3	6	no	—
6	C₃²⋊#66524	C₃²	1	9	yes	—
7	C₃⁴×C₃₇	C₃²	1	3	yes	—
8	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
9	C₃×H(3,𝔽₃)×C₃₇	—	3	3	no	—
10	C₃×H(3,𝔽₃)×C₃₇	—	3	6	no	—
11	C₃×H(3,𝔽₃)×C₃₇	C₉	1	9	yes	complemented
12	C₃³×C₃₇	—	3	3	no	—
13	C₃³×C₃₇	—	3	6	no	—
14	C₃³×C₃₇	—	9	3	no	—
15	C₃³×C₃₇	—	9	9	no	—
16	C₃³×C₃₇	C₃×C₉	1	3	yes	—
17	C₃³×C₃₇	C₃×C₉	1	9	yes	complemented
18	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
19	C₃³×C₃₇	3₋¹⁺²	1	6	yes	—
20	C₃³×C₃₇	C₃³	1	1	yes	characteristic
21	H(3,𝔽₃)×C₃₇	—	3	27	no	—
22	C₃×#66524	—	3	9	no	—
23	C₃×#66524	—	9	9	no	—
24	#10624689	—	37	1	no	maximal, Sylow
25	C₃²×C₃₇	—	3	3	no	—
26	C₃²×C₃₇	—	3	3	no	—
27	C₃²×C₃₇	—	3	6	no	—
28	C₃²×C₃₇	—	3	9	no	—
29	C₃²×C₃₇	—	9	3	no	—
30	C₃²×C₃₇	—	9	6	no	—
31	C₃²×C₃₇	—	9	9	no	—
32	C₃²×C₃₇	—	9	18	no	—
33	C₃²×C₃₇	—	9	9	no	—
34	C₃²×C₃₇	C₃²⋊C₉	1	9	yes	—
35	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
36	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
37	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
38	#66524	—	9	9	no	—
39	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
40	C₃×C₃²⋊C₉	—	37	3	no	—
41	#6866	—	37	9	no	—
42	C₃×C₃₇	—	3	1	no	—
43	C₃×C₃₇	—	3	2	no	—
44	C₃×C₃₇	—	3	9	no	—
45	C₃×C₃₇	—	9	9	no	—
46	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
47	C₃×C₃₇	#6909	1	3	yes	—
48	C₃⁴	C₃×C₃₇⋊C₃	1	3	yes	—
49	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
50	C₃×H(3,𝔽₃)	—	3	6	no	—
51	C₃×H(3,𝔽₃)	—	3	3	no	—
52	C₃×H(3,𝔽₃)	#66524	1	9	yes	complemented
53	C₃²×C₉	—	111	3	no	—
54	C₃²⋊C₉	—	37	9	no	—
55	C₃²⋊C₉	—	111	6	no	—
56	C₃₇	#10624689	1	1	yes	characteristic, complemented, Sylow
57	C₃³	—	3	3	no	—
58	C₃³	—	3	6	no	—
59	C₃³	—	9	3	no	—
60	C₃³	—	9	9	no	—
61	C₃³	C₃×#66524	1	3	yes	—
62	C₃³	C₃×#66524	1	9	yes	complemented
63	C₃³	#11758608	1	6	yes	—
64	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
65	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
66	H(3,𝔽₃)	—	3	27	no	—
67	C₃×C₉	—	333	9	no	—
68	C₃×C₉	—	111	9	no	—
69	C₃²	—	3	3	no	—
70	C₃²	—	3	3	no	—
71	C₃²	—	3	6	no	—
72	C₃²	—	3	9	no	—
73	C₃²	—	9	3	no	—
74	C₃²	—	9	6	no	—
75	C₃²	—	9	9	no	—
76	C₃²	—	9	18	no	—
77	C₃²	—	9	9	no	—
78	C₃²	C₃²⋊#66524	1	9	yes	—
79	C₃²	C₃²×#66524	1	1	yes	characteristic
80	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
81	C₃²	C₃×#11758608	1	2	yes	—
82	C₉	—	333	9	no	—
83	C₃	—	3	1	no	—
84	C₃	—	3	2	no	—
85	C₃	—	3	9	no	—
86	C₃	—	9	9	no	—
87	C₃	C₃×C₃²⋊#66524	1	1	yes	characteristic
88	C₃	#22582638	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
cosocle	C₃³×C₃₇	C₃³
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624689
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ C₃²×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃×C₃²⋊#66524	C₃	semidirect
2. #22582468	C₃	semidirect
11. C₃×H(3,𝔽₃)×C₃₇	C₉	semidirect
17. C₃³×C₃₇	C₃×C₉	semidirect
52. C₃×H(3,𝔽₃)	#66524	semidirect
56. C₃₇	#10624689	semidirect
62. C₃³	C₃×#66524	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
#22582468	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624689	37