Subgroups — C₃×ΓL(1,𝔽₈)×#10596817 #49141456

Order 36540.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×ΓL(1,𝔽₈)×D₂₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×#10596817	C₃	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#10596817	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₅×ΓL(1,𝔽₈)×C₂₉	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×ΓL(1,𝔽₈)×#1181	—	5	1	no	maximal, Hall
6	C₅×ΓL(1,𝔽₈)×D₂₉	C₂×C₃	1	3	yes	—
7	C₃×C₅×C₇×D₂₉	C₂×C₃	1	1	yes	characteristic
8	C₃²×#10596817	—	7	1	no	maximal, Hall
9	C₇×#10596817	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₃×ΓL(1,𝔽₈)×D₂₉	D₅	1	1	yes	characteristic
11	C₅×ΓL(1,𝔽₈)×C₂₉	C₃×C₄	1	3	yes	complemented
12	C₃×C₅×C₇×C₂₉	C₃×C₄	1	1	yes	characteristic, complemented
13	C₃²×C₅×D₂₉	—	7	1	no	—
14	C₃×C₇×#1181	—	5	1	no	—
15	ΓL(1,𝔽₈)×#1181	—	5	3	no	—
16	C₅×C₇×D₂₉	C₂×C₃²	1	1	yes	characteristic
17	C₃×ΓL(1,𝔽₈)×C₂₉	Dic₅	1	1	yes	characteristic, complemented, Hall
18	C₃×#10596817	—	7	3	no	—
19	C₃×#10596817	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
20	C₃²×C₅×C₂₉	—	7	1	no	Hall
21	C₃×Dic₅×ΓL(1,𝔽₈)	—	29	1	no	maximal, Hall
22	C₃×C₇×D₂₉	C₃×D₅	1	1	yes	characteristic
23	ΓL(1,𝔽₈)×D₂₉	C₃×D₅	1	3	yes	—
24	C₃²×#1181	—	35	1	no	Hall
25	C₅×C₇×C₂₉	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
26	C₃×C₅×D₂₉	—	7	3	no	—
27	C₃×C₅×D₂₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
28	C₇×#1181	—	5	1	no	Hall
29	C₂×C₃×C₅×ΓL(1,𝔽₈)	—	29	1	no	—
30	C₃×C₇×C₂₉	C₃×Dic₅	1	1	yes	characteristic, complemented
31	ΓL(1,𝔽₈)×C₂₉	C₃×Dic₅	1	3	yes	complemented
32	#10596817	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
33	C₃²×D₂₉	—	7	1	no	—
34	C₃×C₅×C₂₉	—	7	3	no	—
35	C₃×C₅×C₂₉	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
36	C₃×C₇×Dic₅	—	29	1	no	—
37	Dic₅×ΓL(1,𝔽₈)	—	29	3	no	—
38	C₇×D₂₉	C₃²×D₅	1	1	yes	characteristic
39	C₃×#1181	—	5	1	no	—
40	C₃×#1181	—	35	3	no	—
41	C₃×C₅×ΓL(1,𝔽₈)	#1181	1	1	yes	characteristic, complemented, Hall
42	C₅×D₂₉	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
43	C₃²×C₂₉	—	7	1	no	Hall
44	C₃×C₄×ΓL(1,𝔽₈)	—	145	1	no	Hall
45	C₂×C₃×C₅×C₇	—	29	1	no	—
46	C₂×C₅×ΓL(1,𝔽₈)	—	29	3	no	—
47	C₇×C₂₉	C₃²×Dic₅	1	1	yes	characteristic, complemented, Hall
48	C₃²×Dic₅	—	203	1	no	Hall
49	C₃×D₂₉	—	7	3	no	—
50	C₃×D₂₉	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
51	C₅×C₂₉	C₃×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
52	C₇×Dic₅	—	29	1	no	Hall
53	C₂×C₃×ΓL(1,𝔽₈)	—	29	1	no	—
54	#1181	—	5	1	no	Hall
55	C₃×C₅×C₇	C₃×#1181	1	1	yes	characteristic, complemented
56	C₅×ΓL(1,𝔽₈)	C₃×#1181	1	3	yes	complemented
57	C₂×C₃²×C₅	—	203	1	no	—
58	C₃×C₂₉	—	7	3	no	—
59	C₃×C₂₉	Dic₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
60	C₃×C₄×C₇	—	145	1	no	—
61	C₄×ΓL(1,𝔽₈)	—	145	3	no	—
62	C₂×C₅×C₇	—	29	1	no	—
63	C₃×ΓL(1,𝔽₈)	#10596817	1	1	yes	characteristic, direct product factor, Hall
64	C₃×Dic₅	—	29	1	no	—
65	C₃×Dic₅	—	203	3	no	—
66	D₂₉	C₃×D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
67	C₃²×C₅	—	7	1	no	Hall
68	C₂×C₃×C₇	—	29	1	no	—
69	C₂×ΓL(1,𝔽₈)	—	29	3	no	—
70	C₃²×C₄	—	1015	1	no	Hall
71	C₅×C₇	C₃²×#1181	1	1	yes	characteristic, complemented, Hall
72	C₂×C₃×C₅	—	29	1	no	—
73	C₂×C₃×C₅	—	203	3	no	—
74	C₂₉	C₃×Dic₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
75	C₄×C₇	—	145	1	no	Hall
76	C₃×C₇	C₃×#10596817	1	1	yes	characteristic, complemented
77	ΓL(1,𝔽₈)	C₃×#10596817	1	3	yes	direct product factor
78	Dic₅	—	29	1	no	Hall
79	C₂×C₃²	—	203	1	no	—
80	C₃×C₅	—	7	3	no	—
81	C₃×C₅	ΓL(1,𝔽₈)×#1181	1	1	yes	characteristic, complemented
82	C₂×C₇	—	29	1	no	—
83	C₃×C₄	—	145	1	no	—
84	C₃×C₄	—	1015	3	no	—
85	C₂×C₅	—	29	1	no	—
86	C₃²	—	7	1	no	Sylow
87	C₇	C₃²×#10596817	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	29	1	no	—
89	C₂×C₃	—	203	3	no	—
90	C₅	C₃×ΓL(1,𝔽₈)×#1181	1	1	yes	characteristic, complemented, Sylow
91	C₄	—	145	1	no	Sylow
92	C₃	—	7	3	no	—
93	C₃	ΓL(1,𝔽₈)×#10596817	1	1	yes	characteristic, direct product factor
94	C₂	—	29	1	no	—

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)×#10596817
commutator, hypocenter	C₅×C₇×C₂₉	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₂₉	C₃×C₄
prime radical	C₃×C₅×ΓL(1,𝔽₈)×D₂₉	C₂
cosocle	C₅×C₇×D₂₉	C₂×C₃²
LCS	G ▷ C₅×C₇×C₂₉
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×#10596817	C₃	semidirect
3. ΓL(1,𝔽₈)×#10596817	C₃	direct (N × H)
4. C₃×C₅×ΓL(1,𝔽₈)×C₂₉	C₄	semidirect
9. C₇×#10596817	C₃²	semidirect
11. C₅×ΓL(1,𝔽₈)×C₂₉	C₃×C₄	semidirect
12. C₃×C₅×C₇×C₂₉	C₃×C₄	semidirect
17. C₃×ΓL(1,𝔽₈)×C₂₉	Dic₅	semidirect
19. C₃×#10596817	ΓL(1,𝔽₈)	direct (N × H)
25. C₅×C₇×C₂₉	C₃²×C₄	semidirect
30. C₃×C₇×C₂₉	C₃×Dic₅	semidirect
31. ΓL(1,𝔽₈)×C₂₉	C₃×Dic₅	semidirect
32. #10596817	C₃×ΓL(1,𝔽₈)	direct (N × H)
35. C₃×C₅×C₂₉	C₄×ΓL(1,𝔽₈)	semidirect
41. C₃×C₅×ΓL(1,𝔽₈)	#1181	semidirect
47. C₇×C₂₉	C₃²×Dic₅	semidirect
51. C₅×C₂₉	C₃×C₄×ΓL(1,𝔽₈)	semidirect
55. C₃×C₅×C₇	C₃×#1181	semidirect
56. C₅×ΓL(1,𝔽₈)	C₃×#1181	semidirect
59. C₃×C₂₉	Dic₅×ΓL(1,𝔽₈)	semidirect
63. C₃×ΓL(1,𝔽₈)	#10596817	direct (N × H)
71. C₅×C₇	C₃²×#1181	semidirect
74. C₂₉	C₃×Dic₅×ΓL(1,𝔽₈)	semidirect
76. C₃×C₇	C₃×#10596817	semidirect
77. ΓL(1,𝔽₈)	C₃×#10596817	direct (N × H)
81. C₃×C₅	ΓL(1,𝔽₈)×#1181	semidirect
87. C₇	C₃²×#10596817	semidirect
90. C₅	C₃×ΓL(1,𝔽₈)×#1181	semidirect
93. C₃	ΓL(1,𝔽₈)×#10596817	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×ΓL(1,𝔽₈)×D₂₉	2
C₃×C₇×#10596817	3
ΓL(1,𝔽₈)×#10596817	3
C₃×ΓL(1,𝔽₈)×#1181	5
C₃²×#10596817	7
C₃×Dic₅×ΓL(1,𝔽₈)	29

Subgroups — C3×ΓL(1,𝔽8)×#10596817 #49141456

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₈)×#10596817 #49141456