Subgroups — C₃×ΓL(1,𝔽₈)×#66772 #36339355

Order 21420.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×ΓL(1,𝔽₈)×D₁₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×#66772	C₃	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#66772	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₅×C₁₇×ΓL(1,𝔽₈)	C₄	1	1	yes	characteristic, complemented, Hall
5	C₃×ΓL(1,𝔽₈)×#595	—	5	1	no	maximal, Hall
6	C₅×ΓL(1,𝔽₈)×D₁₇	C₂×C₃	1	3	yes	—
7	C₃×C₅×C₇×D₁₇	C₂×C₃	1	1	yes	characteristic
8	C₃²×#66772	—	7	1	no	maximal, Hall
9	C₇×#66772	C₃²	1	1	yes	characteristic, complemented, Hall
10	C₃×ΓL(1,𝔽₈)×D₁₇	D₅	1	1	yes	characteristic
11	C₃×C₅×C₇×C₁₇	C₃×C₄	1	1	yes	characteristic, complemented
12	C₅×C₁₇×ΓL(1,𝔽₈)	C₃×C₄	1	3	yes	complemented
13	C₃²×C₅×D₁₇	—	7	1	no	—
14	C₃×C₇×#595	—	5	1	no	—
15	ΓL(1,𝔽₈)×#595	—	5	3	no	—
16	C₃×Dic₅×ΓL(1,𝔽₈)	—	17	1	no	maximal, Hall
17	C₅×C₇×D₁₇	C₂×C₃²	1	1	yes	characteristic
18	C₃×C₁₇×ΓL(1,𝔽₈)	Dic₅	1	1	yes	characteristic, complemented, Hall
19	C₃×#66772	—	7	3	no	—
20	C₃×#66772	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
21	C₃²×C₅×C₁₇	—	7	1	no	Hall
22	C₃×C₇×D₁₇	C₃×D₅	1	1	yes	characteristic
23	ΓL(1,𝔽₈)×D₁₇	C₃×D₅	1	3	yes	—
24	C₂×C₃×C₅×ΓL(1,𝔽₈)	—	17	1	no	—
25	C₃²×#595	—	35	1	no	Hall
26	C₅×C₇×C₁₇	C₃²×C₄	1	1	yes	characteristic, complemented, Hall
27	C₃×C₅×D₁₇	—	7	3	no	—
28	C₃×C₅×D₁₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
29	C₇×#595	—	5	1	no	Hall
30	C₃×C₇×Dic₅	—	17	1	no	—
31	Dic₅×ΓL(1,𝔽₈)	—	17	3	no	—
32	C₃×C₇×C₁₇	C₃×Dic₅	1	1	yes	characteristic, complemented
33	C₁₇×ΓL(1,𝔽₈)	C₃×Dic₅	1	3	yes	complemented
34	#66772	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
35	C₃×C₅×ΓL(1,𝔽₈)	#595	1	1	yes	characteristic, complemented, Hall
36	C₃²×D₁₇	—	7	1	no	—
37	C₃×C₅×C₁₇	—	7	3	no	—
38	C₃×C₅×C₁₇	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
39	C₃×C₄×ΓL(1,𝔽₈)	—	85	1	no	Hall
40	C₇×D₁₇	C₃²×D₅	1	1	yes	characteristic
41	C₂×C₃×C₅×C₇	—	17	1	no	—
42	C₂×C₅×ΓL(1,𝔽₈)	—	17	3	no	—
43	C₃×#595	—	5	1	no	—
44	C₃×#595	—	35	3	no	—
45	C₃²×Dic₅	—	119	1	no	Hall
46	C₅×D₁₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
47	C₃²×C₁₇	—	7	1	no	Hall
48	C₇×Dic₅	—	17	1	no	Hall
49	C₂×C₃×ΓL(1,𝔽₈)	—	17	1	no	—
50	C₇×C₁₇	C₃²×Dic₅	1	1	yes	characteristic, complemented, Hall
51	C₃×C₅×C₇	C₃×#595	1	1	yes	characteristic, complemented
52	C₅×ΓL(1,𝔽₈)	C₃×#595	1	3	yes	complemented
53	C₃×D₁₇	—	7	3	no	—
54	C₃×D₁₇	D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
55	C₂×C₃²×C₅	—	119	1	no	—
56	C₅×C₁₇	C₃×C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
57	C₃×C₄×C₇	—	85	1	no	—
58	C₄×ΓL(1,𝔽₈)	—	85	3	no	—
59	C₂×C₅×C₇	—	17	1	no	—
60	#595	—	5	1	no	Hall
61	C₃×ΓL(1,𝔽₈)	#66772	1	1	yes	characteristic, direct product factor, Hall
62	C₃×Dic₅	—	17	1	no	—
63	C₃×Dic₅	—	119	3	no	—
64	C₃×C₁₇	—	7	3	no	—
65	C₃×C₁₇	Dic₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
66	C₃²×C₅	—	7	1	no	Hall
67	C₂×C₃×C₇	—	17	1	no	—
68	C₂×ΓL(1,𝔽₈)	—	17	3	no	—
69	C₃²×C₄	—	595	1	no	Hall
70	C₅×C₇	C₃²×#595	1	1	yes	characteristic, complemented, Hall
71	D₁₇	C₃×D₅×ΓL(1,𝔽₈)	1	1	yes	characteristic
72	C₂×C₃×C₅	—	17	1	no	—
73	C₂×C₃×C₅	—	119	3	no	—
74	C₄×C₇	—	85	1	no	Hall
75	C₃×C₇	C₃×#66772	1	1	yes	characteristic, complemented
76	ΓL(1,𝔽₈)	C₃×#66772	1	3	yes	direct product factor
77	Dic₅	—	17	1	no	Hall
78	C₂×C₃²	—	119	1	no	—
79	C₁₇	C₃×Dic₅×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
80	C₃×C₅	—	7	3	no	—
81	C₃×C₅	ΓL(1,𝔽₈)×#595	1	1	yes	characteristic, complemented
82	C₂×C₇	—	17	1	no	—
83	C₃×C₄	—	595	3	no	—
84	C₃×C₄	—	85	1	no	—
85	C₂×C₅	—	17	1	no	—
86	C₃²	—	7	1	no	Sylow
87	C₇	C₃²×#66772	1	1	yes	characteristic, complemented, Sylow
88	C₂×C₃	—	17	1	no	—
89	C₂×C₃	—	119	3	no	—
90	C₅	C₃×ΓL(1,𝔽₈)×#595	1	1	yes	characteristic, complemented, Sylow
91	C₄	—	85	1	no	Sylow
92	C₃	—	7	3	no	—
93	C₃	ΓL(1,𝔽₈)×#66772	1	1	yes	characteristic, direct product factor
94	C₂	—	17	1	no	—

	subgroup	G/H
center, hypercenter	C₃	ΓL(1,𝔽₈)×#66772
commutator, hypocenter	C₅×C₇×C₁₇	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₇×C₁₇	C₃×C₄
prime radical	C₃×C₅×ΓL(1,𝔽₈)×D₁₇	C₂
cosocle	C₅×C₇×D₁₇	C₂×C₃²
LCS	G ▷ C₅×C₇×C₁₇
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×#66772	C₃	semidirect
3. ΓL(1,𝔽₈)×#66772	C₃	direct (N × H)
4. C₃×C₅×C₁₇×ΓL(1,𝔽₈)	C₄	semidirect
9. C₇×#66772	C₃²	semidirect
11. C₃×C₅×C₇×C₁₇	C₃×C₄	semidirect
12. C₅×C₁₇×ΓL(1,𝔽₈)	C₃×C₄	semidirect
18. C₃×C₁₇×ΓL(1,𝔽₈)	Dic₅	semidirect
20. C₃×#66772	ΓL(1,𝔽₈)	direct (N × H)
26. C₅×C₇×C₁₇	C₃²×C₄	semidirect
32. C₃×C₇×C₁₇	C₃×Dic₅	semidirect
33. C₁₇×ΓL(1,𝔽₈)	C₃×Dic₅	semidirect
34. #66772	C₃×ΓL(1,𝔽₈)	direct (N × H)
35. C₃×C₅×ΓL(1,𝔽₈)	#595	semidirect
38. C₃×C₅×C₁₇	C₄×ΓL(1,𝔽₈)	semidirect
50. C₇×C₁₇	C₃²×Dic₅	semidirect
51. C₃×C₅×C₇	C₃×#595	semidirect
52. C₅×ΓL(1,𝔽₈)	C₃×#595	semidirect
56. C₅×C₁₇	C₃×C₄×ΓL(1,𝔽₈)	semidirect
61. C₃×ΓL(1,𝔽₈)	#66772	direct (N × H)
65. C₃×C₁₇	Dic₅×ΓL(1,𝔽₈)	semidirect
70. C₅×C₇	C₃²×#595	semidirect
75. C₃×C₇	C₃×#66772	semidirect
76. ΓL(1,𝔽₈)	C₃×#66772	direct (N × H)
79. C₁₇	C₃×Dic₅×ΓL(1,𝔽₈)	semidirect
81. C₃×C₅	ΓL(1,𝔽₈)×#595	semidirect
87. C₇	C₃²×#66772	semidirect
90. C₅	C₃×ΓL(1,𝔽₈)×#595	semidirect
93. C₃	ΓL(1,𝔽₈)×#66772	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)×D₁₇	2
C₃×C₇×#66772	3
ΓL(1,𝔽₈)×#66772	3
C₃×ΓL(1,𝔽₈)×#595	5
C₃²×#66772	7
C₃×Dic₅×ΓL(1,𝔽₈)	17

Subgroups — C3×ΓL(1,𝔽8)×#66772 #36339355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×ΓL(1,𝔽₈)×#66772 #36339355