Subgroups — #41006672

Order 26973.

476 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582642	C₃	1	8	yes	maximal, complemented
2	C₃×#15688512	C₃	1	4	yes	maximal, complemented
3	C₃₇×H(5,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic, complemented
5	C₃×#11758608	C₃²	1	8	yes	complemented
6	#15688515	—	3	24	no	—
7	#15688512	—	3	12	no	—
8	C₃×H(3,𝔽₃)×C₃₇	—	3	12	no	—
9	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	4	yes	—
10	C₃³×C₃₇	—	3	9	no	—
11	C₃³×C₃₇	—	3	4	no	—
12	C₃³×C₃₇	C₃³	1	1	yes	characteristic
13	C₃²×C₃₇⋊C₃	—	3	4	no	—
14	C₃²×C₃₇⋊C₃	—	9	3	no	—
15	H(3,𝔽₃)×C₃₇	—	3	18	no	—
16	H(3,𝔽₃)×C₃₇	—	3	8	no	—
17	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	12	yes	—
18	#11758608	—	3	8	no	—
19	#11758608	—	3	16	no	—
20	C₃×#66524	—	3	8	no	—
21	#10624761	—	37	1	no	maximal, Sylow
22	C₃²×C₃₇	—	3	12	no	—
23	C₃²×C₃₇	—	9	1	no	—
24	C₃²×C₃₇	—	27	4	no	—
25	C₃²×C₃₇	—	27	9	no	—
26	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	4	yes	—
27	C₃×C₃₇⋊C₃	—	9	1	no	—
28	C₃×C₃₇⋊C₃	—	9	12	no	—
29	#66524	—	9	8	no	—
30	H(5,𝔽₃)	C₃₇⋊C₃	1	1	yes	characteristic, complemented
31	#6910	—	37	8	no	—
32	C₃×AΣL(1,𝔽₂₇)	—	37	4	no	—
33	C₃×C₃₇	—	3	4	no	—
34	C₃×C₃₇	—	9	12	no	—
35	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
36	C₃₇⋊C₃	—	9	3	no	—
37	C₃⁴	—	37	1	no	—
38	C₃×3₋¹⁺²	—	37	8	no	—
39	C₃×H(3,𝔽₃)	—	3	12	no	—
40	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	4	yes	—
41	#731	—	111	24	no	—
42	AΣL(1,𝔽₂₇)	—	111	12	no	—
43	C₃₇	#10624761	1	1	yes	characteristic, complemented, Sylow
44	C₃³	—	3	9	no	—
45	C₃³	—	3	4	no	—
46	C₃³	—	333	3	no	—
47	C₃³	—	111	4	no	—
48	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
49	3₋¹⁺²	—	111	8	no	—
50	3₋¹⁺²	—	111	16	no	—
51	H(3,𝔽₃)	—	3	18	no	—
52	H(3,𝔽₃)	—	3	8	no	—
53	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	12	yes	—
54	C₃×C₉	—	111	8	no	—
55	C₃²	—	3	12	no	—
56	C₃²	—	9	1	no	—
57	C₃²	—	27	4	no	—
58	C₃²	—	27	9	no	—
59	C₃²	—	333	12	no	—
60	C₃²	—	333	1	no	—
61	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	4	yes	—
62	C₉	—	333	8	no	—
63	C₃	—	3	4	no	—
64	C₃	—	9	12	no	—
65	C₃	—	333	3	no	—
66	C₃	#22582550	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582550
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×H(5,𝔽₃)	C₃
socle	C₃×C₃₇	C₃²⋊H(3,𝔽₃)
hypercenter	H(5,𝔽₃)	C₃₇⋊C₃
hypocenter	C₃₇	#10624761
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ H(5,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. #22582642	C₃	semidirect
2. C₃×#15688512	C₃	semidirect
3. C₃₇×H(5,𝔽₃)	C₃	semidirect
4. C₃³×C₃₇⋊C₃	C₃²	semidirect
5. C₃×#11758608	C₃²	semidirect
30. H(5,𝔽₃)	C₃₇⋊C₃	semidirect
43. C₃₇	#10624761	semidirect

Maximal subgroups

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

subgroup	index
#22582642	3
C₃×#15688512	3
C₃₇×H(5,𝔽₃)	3
#10624761	37