Subgroups — C₃×C₇×C₁₇×C₂²⋊ΓL(1,𝔽₈) #44473003

Order 29988.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₇×C₂²⋊ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₇²×C₁₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×C₁₇×ΓL(1,𝔽₈)	—	4	1	no	maximal, Hall
4	C₂×C₃×C₇²×C₁₇	—	3	1	no	—
5	C₃×C₇×A₄×C₁₇	—	7	1	no	maximal
6	C₃×C₁₇×C₂²⋊ΓL(1,𝔽₈)	C₇	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₇²×C₁₇	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₇×C₁₇×ΓL(1,𝔽₈)	—	4	3	no	—
9	C₃×C₇²×C₁₇	A₄	1	1	yes	characteristic, complemented
10	C₃×C₇×C₂²⋊ΓL(1,𝔽₈)	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₇²×C₁₇	—	3	1	no	—
12	C₂²×C₃×C₇×C₁₇	—	3	2	no	—
13	C₂²×C₃×C₇×C₁₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₇×C₁₇	C₃×C₇	1	1	yes	characteristic, complemented
15	C₁₇×C₂²⋊ΓL(1,𝔽₈)	C₃×C₇	1	3	yes	direct product factor
16	C₇×A₄×C₁₇	—	7	3	no	—
17	C₃²×C₇×C₁₇	—	28	1	no	—
18	C₃×C₁₇×ΓL(1,𝔽₈)	—	4	1	no	—
19	C₇²×C₁₇	C₃×A₄	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃×C₇×C₁₇	—	3	6	no	—
21	C₂×C₃×C₇×C₁₇	—	3	1	no	—
22	C₂×C₃×C₇×C₁₇	—	3	1	no	—
23	C₃×A₄×C₁₇	—	7	1	no	Hall
24	C₂²×C₃×C₇²	C₃×C₁₇	1	1	yes	characteristic, complemented
25	C₇×C₂²⋊ΓL(1,𝔽₈)	C₃×C₁₇	1	3	yes	direct product factor
26	C₂²×C₇×C₁₇	—	3	2	no	—
27	C₂²×C₇×C₁₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	C₂²×C₇×C₁₇	C₃²×C₇	1	1	yes	characteristic, complemented
29	C₃×C₇×ΓL(1,𝔽₈)	—	4	1	no	Hall
30	C₃×C₇×C₁₇	—	3	2	no	—
31	C₃×C₇×C₁₇	—	28	3	no	—
32	C₃×C₇×C₁₇	C₇×A₄	1	1	yes	characteristic, complemented
33	C₃×C₇×C₁₇	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
34	C₁₇×ΓL(1,𝔽₈)	—	4	3	no	—
35	C₂×C₃×C₇²	—	3	1	no	—
36	C₃×C₂²⋊ΓL(1,𝔽₈)	C₇×C₁₇	1	1	yes	characteristic, direct product factor
37	C₃×C₇×A₄	—	7	1	no	—
38	C₂×C₇×C₁₇	—	3	6	no	—
39	C₂×C₇×C₁₇	—	3	1	no	—
40	C₂×C₇×C₁₇	—	3	1	no	—
41	C₂²×C₃×C₁₇	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
42	A₄×C₁₇	—	7	3	no	—
43	C₂²×C₇²	C₃²×C₁₇	1	1	yes	characteristic, complemented, Hall
44	C₃²×C₁₇	—	28	1	no	Hall
45	C₃×C₇²	A₄×C₁₇	1	1	yes	characteristic, complemented
46	C₇×ΓL(1,𝔽₈)	—	4	3	no	—
47	C₇×C₁₇	—	3	2	no	—
48	C₇×C₁₇	C₃×C₇×A₄	1	1	yes	characteristic, complemented
49	C₇×C₁₇	C₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
50	C₂×C₃×C₁₇	—	3	1	no	—
51	C₂×C₇²	—	3	1	no	—
52	C₂²×C₃×C₇	—	3	2	no	—
53	C₂²×C₃×C₇	C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
54	C₂²×C₃×C₇	C₃×C₇×C₁₇	1	1	yes	characteristic, complemented
55	C₂²⋊ΓL(1,𝔽₈)	C₃×C₇×C₁₇	1	3	yes	direct product factor
56	C₇×A₄	—	7	3	no	—
57	C₂²×C₁₇	C₃×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
58	C₃²×C₇	—	28	1	no	—
59	C₃×ΓL(1,𝔽₈)	—	4	1	no	—
60	C₃×C₁₇	—	28	3	no	—
61	C₃×C₁₇	C₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
62	C₇²	C₃×A₄×C₁₇	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₃×C₇	—	3	6	no	—
64	C₂×C₃×C₇	—	3	1	no	—
65	C₂×C₃×C₇	—	3	1	no	—
66	C₃×A₄	—	7	1	no	Hall
67	C₂×C₁₇	—	3	1	no	—
68	C₂²×C₇	—	3	2	no	—
69	C₂²×C₇	C₃×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
70	C₂²×C₇	C₃²×C₇×C₁₇	1	1	yes	characteristic, complemented
71	C₃×C₇	—	3	2	no	—
72	C₃×C₇	—	28	3	no	—
73	C₃×C₇	C₇×A₄×C₁₇	1	1	yes	characteristic, complemented
74	C₃×C₇	C₁₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
75	ΓL(1,𝔽₈)	—	4	3	no	—
76	C₁₇	C₃×C₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×C₇	—	3	6	no	—
78	C₂×C₇	—	3	1	no	—
79	C₂×C₇	—	3	1	no	—
80	C₂²×C₃	C₇×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
81	A₄	—	7	3	no	—
82	C₃²	—	28	1	no	Sylow
83	C₇	—	3	2	no	—
84	C₇	C₃×C₁₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
85	C₇	C₃×C₇×A₄×C₁₇	1	1	yes	characteristic, complemented
86	C₂×C₃	—	3	1	no	—
87	C₂²	C₃×C₇×C₁₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
88	C₃	—	28	3	no	—
89	C₃	C₇×C₁₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
90	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₇×C₁₇	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃²×C₇×C₁₇
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇²×C₁₇	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₃×C₇×C₁₇

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₇×C₂²⋊ΓL(1,𝔽₈)	C₃	direct (N × H)
2. C₂²×C₃×C₇²×C₁₇	C₃	semidirect
6. C₃×C₁₇×C₂²⋊ΓL(1,𝔽₈)	C₇	direct (N × H)
7. C₂²×C₇²×C₁₇	C₃²	semidirect
9. C₃×C₇²×C₁₇	A₄	semidirect
10. C₃×C₇×C₂²⋊ΓL(1,𝔽₈)	C₁₇	direct (N × H)
13. C₂²×C₃×C₇×C₁₇	ΓL(1,𝔽₈)	semidirect
14. C₂²×C₃×C₇×C₁₇	C₃×C₇	semidirect
15. C₁₇×C₂²⋊ΓL(1,𝔽₈)	C₃×C₇	direct (N × H)
19. C₇²×C₁₇	C₃×A₄	semidirect
24. C₂²×C₃×C₇²	C₃×C₁₇	semidirect
25. C₇×C₂²⋊ΓL(1,𝔽₈)	C₃×C₁₇	direct (N × H)
27. C₂²×C₇×C₁₇	C₃×ΓL(1,𝔽₈)	semidirect
28. C₂²×C₇×C₁₇	C₃²×C₇	semidirect
32. C₃×C₇×C₁₇	C₇×A₄	semidirect
33. C₃×C₇×C₁₇	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
36. C₃×C₂²⋊ΓL(1,𝔽₈)	C₇×C₁₇	direct (N × H)
41. C₂²×C₃×C₁₇	C₇×ΓL(1,𝔽₈)	semidirect
43. C₂²×C₇²	C₃²×C₁₇	semidirect
45. C₃×C₇²	A₄×C₁₇	semidirect
48. C₇×C₁₇	C₃×C₇×A₄	semidirect
49. C₇×C₁₇	C₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
53. C₂²×C₃×C₇	C₁₇×ΓL(1,𝔽₈)	semidirect
54. C₂²×C₃×C₇	C₃×C₇×C₁₇	semidirect
55. C₂²⋊ΓL(1,𝔽₈)	C₃×C₇×C₁₇	direct (N × H)
57. C₂²×C₁₇	C₃×C₇×ΓL(1,𝔽₈)	semidirect
61. C₃×C₁₇	C₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
62. C₇²	C₃×A₄×C₁₇	semidirect
69. C₂²×C₇	C₃×C₁₇×ΓL(1,𝔽₈)	semidirect
70. C₂²×C₇	C₃²×C₇×C₁₇	semidirect
73. C₃×C₇	C₇×A₄×C₁₇	semidirect
74. C₃×C₇	C₁₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
76. C₁₇	C₃×C₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
80. C₂²×C₃	C₇×C₁₇×ΓL(1,𝔽₈)	semidirect
84. C₇	C₃×C₁₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
85. C₇	C₃×C₇×A₄×C₁₇	semidirect
87. C₂²	C₃×C₇×C₁₇×ΓL(1,𝔽₈)	semidirect
89. C₃	C₇×C₁₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×C₁₇×C₂²⋊ΓL(1,𝔽₈)	3
C₂²×C₃×C₇²×C₁₇	3
C₃×C₇×C₁₇×ΓL(1,𝔽₈)	4
C₃×C₇×A₄×C₁₇	7
C₃×C₁₇×C₂²⋊ΓL(1,𝔽₈)	7
C₃×C₇×C₂²⋊ΓL(1,𝔽₈)	17

Subgroups — C3×C7×C17×C22⋊ΓL(1,𝔽8) #44473003

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×C₁₇×C₂²⋊ΓL(1,𝔽₈) #44473003