Subgroups — #41006725

Order 26973.

464 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582554	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	C₃×#11758608	—	3	3	no	—
5	C₃²⋊#66524	—	3	6	no	—
6	C₃²⋊#66524	C₃²	1	9	yes	—
7	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
8	C₃⁴×C₃₇	C₃²	1	3	yes	—
9	C₃×H(3,𝔽₃)×C₃₇	—	3	3	no	—
10	C₃×H(3,𝔽₃)×C₃₇	—	3	9	no	—
11	C₃³×C₃₇	—	3	3	no	—
12	C₃³×C₃₇	—	3	6	no	—
13	C₃³×C₃₇	—	3	3	no	—
14	C₃³×C₃₇	—	9	3	no	—
15	C₃³×C₃₇	—	9	9	no	—
16	C₃³×C₃₇	H(3,𝔽₃)	1	3	yes	—
17	C₃³×C₃₇	3₋¹⁺²	1	9	yes	complemented
18	C₃³×C₃₇	C₃³	1	1	yes	characteristic
19	H(3,𝔽₃)×C₃₇	—	3	27	no	—
20	#11758608	—	9	9	no	—
21	C₃×#66524	—	3	9	no	—
22	#10624769	—	37	1	no	maximal, Sylow
23	C₃²×C₃₇	—	3	3	no	—
24	C₃²×C₃₇	—	3	3	no	—
25	C₃²×C₃₇	—	3	9	no	—
26	C₃²×C₃₇	—	3	3	no	—
27	C₃²×C₃₇	—	3	6	no	—
28	C₃²×C₃₇	—	9	9	no	—
29	C₃²×C₃₇	—	9	9	no	—
30	C₃²×C₃₇	—	9	18	no	—
31	C₃²×C₃₇	—	9	9	no	—
32	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
33	C₃²×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
34	#66524	—	9	9	no	—
35	C₃²×H(3,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic
36	#6891	—	37	3	no	—
37	#6866	—	37	9	no	—
38	C₃×C₃₇	—	3	3	no	—
39	C₃×C₃₇	—	9	3	no	—
40	C₃×C₃₇	—	9	9	no	—
41	C₃×C₃₇	#6891	1	1	yes	characteristic
42	C₃×C₃₇	#6909	1	3	yes	—
43	C₃⁴	C₃×C₃₇⋊C₃	1	3	yes	—
44	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
45	C₃×3₋¹⁺²	—	111	3	no	—
46	C₃×H(3,𝔽₃)	—	3	9	no	—
47	C₃×H(3,𝔽₃)	—	3	3	no	—
48	C₃²⋊C₉	—	37	9	no	—
49	C₃²⋊C₉	—	111	6	no	—
50	C₃₇	#10624769	1	1	yes	characteristic, complemented, Sylow
51	C₃³	—	3	3	no	—
52	C₃³	—	3	6	no	—
53	C₃³	—	3	3	no	—
54	C₃³	—	9	3	no	—
55	C₃³	—	9	9	no	—
56	C₃³	#11758608	1	9	yes	complemented
57	C₃³	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
58	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
59	3₋¹⁺²	—	333	9	no	—
60	H(3,𝔽₃)	—	3	27	no	—
61	C₃×C₉	—	111	9	no	—
62	C₃²	—	3	3	no	—
63	C₃²	—	3	9	no	—
64	C₃²	—	3	3	no	—
65	C₃²	—	3	3	no	—
66	C₃²	—	3	6	no	—
67	C₃²	—	9	9	no	—
68	C₃²	—	9	9	no	—
69	C₃²	—	9	18	no	—
70	C₃²	—	9	9	no	—
71	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
72	C₃²	C₃×#11758608	1	3	yes	—
73	C₉	—	333	9	no	—
74	C₃	—	3	3	no	—
75	C₃	—	9	3	no	—
76	C₃	—	9	9	no	—
77	C₃	#22582554	1	1	yes	characteristic
78	C₃	#22582638	1	3	yes	—

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting, prime radical	C₃²×H(3,𝔽₃)×C₃₇	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃²×H(3,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624769
LCS	G ▷ C₃³×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. #22582554	C₃	semidirect
2. #22582468	C₃	semidirect
17. C₃³×C₃₇	3₋¹⁺²	semidirect
50. C₃₇	#10624769	semidirect
56. C₃³	#11758608	semidirect

Maximal subgroups

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

subgroup	index
#22582554	3
#22582468	3
C₃²×H(3,𝔽₃)×C₃₇	3
#10624769	37