Subgroups — C₃×C₇×C₁₃×C₂²⋊ΓL(1,𝔽₈) #37525652

Order 22932.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃	1	3	yes	maximal, direct product factor
2	C₂²×C₃×C₇²×C₁₃	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×C₁₃×ΓL(1,𝔽₈)	—	4	1	no	maximal, Hall
4	C₂×C₃×C₇²×C₁₃	—	3	1	no	—
5	C₃×C₇×A₄×C₁₃	—	7	1	no	maximal
6	C₃×C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₇	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₇²×C₁₃	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇²×C₁₃	A₄	1	1	yes	characteristic, complemented
9	C₇×C₁₃×ΓL(1,𝔽₈)	—	4	3	no	—
10	C₃×C₇×C₂²⋊ΓL(1,𝔽₈)	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₇²×C₁₃	—	3	1	no	—
12	C₂²×C₃×C₇×C₁₃	—	3	2	no	—
13	C₂²×C₃×C₇×C₁₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
14	C₂²×C₃×C₇×C₁₃	C₃×C₇	1	1	yes	characteristic, complemented
15	C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃×C₇	1	3	yes	direct product factor
16	C₇×A₄×C₁₃	—	7	3	no	—
17	C₃²×C₇×C₁₃	—	28	1	no	—
18	C₃×C₁₃×ΓL(1,𝔽₈)	—	4	1	no	—
19	C₇²×C₁₃	C₃×A₄	1	1	yes	characteristic, complemented, Hall
20	C₂²×C₃×C₇²	C₃×C₁₃	1	1	yes	characteristic, complemented
21	C₇×C₂²⋊ΓL(1,𝔽₈)	C₃×C₁₃	1	3	yes	direct product factor
22	C₂×C₃×C₇×C₁₃	—	3	6	no	—
23	C₂×C₃×C₇×C₁₃	—	3	1	no	—
24	C₂×C₃×C₇×C₁₃	—	3	1	no	—
25	C₃×A₄×C₁₃	—	7	1	no	Hall
26	C₃×C₇×ΓL(1,𝔽₈)	—	4	1	no	Hall
27	C₂²×C₇×C₁₃	—	3	2	no	—
28	C₂²×C₇×C₁₃	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
29	C₂²×C₇×C₁₃	C₃²×C₇	1	1	yes	characteristic, complemented
30	C₂×C₃×C₇²	—	3	1	no	—
31	C₃×C₇×C₁₃	—	3	2	no	—
32	C₃×C₇×C₁₃	—	28	3	no	—
33	C₃×C₇×C₁₃	C₇×A₄	1	1	yes	characteristic, complemented
34	C₃×C₇×C₁₃	C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
35	C₁₃×ΓL(1,𝔽₈)	—	4	3	no	—
36	C₃×C₂²⋊ΓL(1,𝔽₈)	C₇×C₁₃	1	1	yes	characteristic, direct product factor
37	C₃×C₇×A₄	—	7	1	no	—
38	C₂²×C₇²	C₃²×C₁₃	1	1	yes	characteristic, complemented, Hall
39	C₂×C₇×C₁₃	—	3	6	no	—
40	C₂×C₇×C₁₃	—	3	1	no	—
41	C₂×C₇×C₁₃	—	3	1	no	—
42	C₂²×C₃×C₁₃	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
43	A₄×C₁₃	—	7	3	no	—
44	C₃×C₇²	A₄×C₁₃	1	1	yes	characteristic, complemented
45	C₇×ΓL(1,𝔽₈)	—	4	3	no	—
46	C₃²×C₁₃	—	28	1	no	Hall
47	C₂×C₇²	—	3	1	no	—
48	C₇×C₁₃	—	3	2	no	—
49	C₇×C₁₃	C₃×C₇×A₄	1	1	yes	characteristic, complemented
50	C₇×C₁₃	C₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
51	C₂²×C₃×C₇	—	3	2	no	—
52	C₂²×C₃×C₇	C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
53	C₂²×C₃×C₇	C₃×C₇×C₁₃	1	1	yes	characteristic, complemented
54	C₂²⋊ΓL(1,𝔽₈)	C₃×C₇×C₁₃	1	3	yes	direct product factor
55	C₇×A₄	—	7	3	no	—
56	C₂×C₃×C₁₃	—	3	1	no	—
57	C₃²×C₇	—	28	1	no	—
58	C₃×ΓL(1,𝔽₈)	—	4	1	no	—
59	C₂²×C₁₃	C₃×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
60	C₇²	C₃×A₄×C₁₃	1	1	yes	characteristic, complemented, Sylow
61	C₂×C₃×C₇	—	3	6	no	—
62	C₂×C₃×C₇	—	3	1	no	—
63	C₂×C₃×C₇	—	3	1	no	—
64	C₃×C₁₃	—	28	3	no	—
65	C₃×C₁₃	C₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
66	C₃×A₄	—	7	1	no	Hall
67	C₂²×C₇	—	3	2	no	—
68	C₂²×C₇	C₃×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
69	C₂²×C₇	C₃²×C₇×C₁₃	1	1	yes	characteristic, complemented
70	C₂×C₁₃	—	3	1	no	—
71	C₃×C₇	—	3	2	no	—
72	C₃×C₇	—	28	3	no	—
73	C₃×C₇	C₇×A₄×C₁₃	1	1	yes	characteristic, complemented
74	C₃×C₇	C₁₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
75	ΓL(1,𝔽₈)	—	4	3	no	—
76	C₂×C₇	—	3	6	no	—
77	C₂×C₇	—	3	1	no	—
78	C₂×C₇	—	3	1	no	—
79	C₁₃	C₃×C₇×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Sylow
80	C₂²×C₃	C₇×C₁₃×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×C₁₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
85	C₇	C₃×C₇×A₄×C₁₃	1	1	yes	characteristic, complemented
86	C₂×C₃	—	3	1	no	—
87	C₂²	C₃×C₇×C₁₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
88	C₃	—	28	3	no	—
89	C₃	C₇×C₁₃×C₂²⋊ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
90	C₂	—	3	1	no	—

	subgroup	G/H
center, hypercenter	C₃×C₇×C₁₃	C₂²⋊ΓL(1,𝔽₈)
commutator, cosocle, hypocenter	C₂²×C₇	C₃²×C₇×C₁₃
Frattini	C₁	G
Fitting, socle	C₂²×C₃×C₇²×C₁₃	C₃
LCS	G ▷ C₂²×C₇
UCS	C₁ ◁ C₃×C₇×C₁₃

Complements

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

N	H ≅ G/N	type
1. C₇×C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃	direct (N × H)
2. C₂²×C₃×C₇²×C₁₃	C₃	semidirect
6. C₃×C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₇	direct (N × H)
7. C₂²×C₇²×C₁₃	C₃²	semidirect
8. C₃×C₇²×C₁₃	A₄	semidirect
10. C₃×C₇×C₂²⋊ΓL(1,𝔽₈)	C₁₃	direct (N × H)
13. C₂²×C₃×C₇×C₁₃	ΓL(1,𝔽₈)	semidirect
14. C₂²×C₃×C₇×C₁₃	C₃×C₇	semidirect
15. C₁₃×C₂²⋊ΓL(1,𝔽₈)	C₃×C₇	direct (N × H)
19. C₇²×C₁₃	C₃×A₄	semidirect
20. C₂²×C₃×C₇²	C₃×C₁₃	semidirect
21. C₇×C₂²⋊ΓL(1,𝔽₈)	C₃×C₁₃	direct (N × H)
28. C₂²×C₇×C₁₃	C₃×ΓL(1,𝔽₈)	semidirect
29. C₂²×C₇×C₁₃	C₃²×C₇	semidirect
33. C₃×C₇×C₁₃	C₇×A₄	semidirect
34. C₃×C₇×C₁₃	C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
36. C₃×C₂²⋊ΓL(1,𝔽₈)	C₇×C₁₃	direct (N × H)
38. C₂²×C₇²	C₃²×C₁₃	semidirect
42. C₂²×C₃×C₁₃	C₇×ΓL(1,𝔽₈)	semidirect
44. C₃×C₇²	A₄×C₁₃	semidirect
49. C₇×C₁₃	C₃×C₇×A₄	semidirect
50. C₇×C₁₃	C₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
52. C₂²×C₃×C₇	C₁₃×ΓL(1,𝔽₈)	semidirect
53. C₂²×C₃×C₇	C₃×C₇×C₁₃	semidirect
54. C₂²⋊ΓL(1,𝔽₈)	C₃×C₇×C₁₃	direct (N × H)
59. C₂²×C₁₃	C₃×C₇×ΓL(1,𝔽₈)	semidirect
60. C₇²	C₃×A₄×C₁₃	semidirect
65. C₃×C₁₃	C₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
68. C₂²×C₇	C₃×C₁₃×ΓL(1,𝔽₈)	semidirect
69. C₂²×C₇	C₃²×C₇×C₁₃	semidirect
73. C₃×C₇	C₇×A₄×C₁₃	semidirect
74. C₃×C₇	C₁₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
79. C₁₃	C₃×C₇×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
80. C₂²×C₃	C₇×C₁₃×ΓL(1,𝔽₈)	semidirect
84. C₇	C₃×C₁₃×C₂²⋊ΓL(1,𝔽₈)	direct (N × H)
85. C₇	C₃×C₇×A₄×C₁₃	semidirect
87. C₂²	C₃×C₇×C₁₃×ΓL(1,𝔽₈)	semidirect
89. C₃	C₇×C₁₃×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×C₁₃×C₂²⋊ΓL(1,𝔽₈)	3
C₂²×C₃×C₇²×C₁₃	3
C₃×C₇×C₁₃×ΓL(1,𝔽₈)	4
C₃×C₇×A₄×C₁₃	7
C₃×C₁₃×C₂²⋊ΓL(1,𝔽₈)	7
C₃×C₇×C₂²⋊ΓL(1,𝔽₈)	13

Subgroups — C3×C7×C13×C22⋊ΓL(1,𝔽8) #37525652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×C₁₃×C₂²⋊ΓL(1,𝔽₈) #37525652