Subgroups — C₂×C₃×#16861353 #37567926

Order 24444.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#14523271	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#16861353	C₂	1	1	yes	maximal, characteristic, direct product factor
3	C₃×#16861353	C₂	1	1	yes	maximal, characteristic, direct product factor
4	C₂×C₃×D₇×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×#16861353	C₃	1	3	yes	maximal, direct product factor
6	C₃×#14523271	C₂²	1	1	yes	characteristic, complemented, Hall
7	#16861353	C₂×C₃	1	3	yes	direct product factor
8	#16861353	C₂×C₃	1	3	yes	direct product factor
9	C₃×D₇×C₉₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₃×D₇×C₉₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×#14523271	C₂×C₃	1	3	yes	complemented
12	C₂×C₃×C₇×C₉₇	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂²×C₃×C₉₇⋊C₃	—	7	1	no	maximal, Hall
14	C₂×D₇×C₉₇	C₃²	1	1	yes	characteristic, complemented, Hall
15	C₃×C₇×C₉₇	C₂²×C₃	1	1	yes	characteristic, complemented
16	#14523271	C₂²×C₃	1	3	yes	complemented
17	C₂×C₃×C₉₇⋊C₃	—	7	1	no	—
18	C₂×C₃×C₉₇⋊C₃	—	7	1	no	—
19	C₂×C₃×C₉₇⋊C₃	—	7	1	no	—
20	C₂×C₇×C₉₇	C₂×C₃²	1	1	yes	characteristic, complemented
21	D₇×C₉₇	C₂×C₃²	1	1	yes	characteristic, complemented
22	D₇×C₉₇	C₂×C₃²	1	1	yes	characteristic, complemented
23	C₂²×C₃×C₉₇	—	21	1	no	—
24	C₂²×C₉₇⋊C₃	—	7	3	no	—
25	C₃×C₉₇⋊C₃	—	7	1	no	Hall
26	C₇×C₉₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
27	C₂×C₃×C₉₇	—	7	1	no	—
28	C₂×C₃×C₉₇	—	21	1	no	—
29	C₂×C₃×C₉₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
30	C₂×C₉₇⋊C₃	—	7	3	no	—
31	C₂×C₉₇⋊C₃	—	7	3	no	—
32	C₂×C₉₇⋊C₃	—	7	3	no	—
33	C₂²×C₉₇	—	21	1	no	Hall
34	C₃×C₉₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
35	C₉₇⋊C₃	—	7	3	no	—
36	C₂×C₃×AGL(1,𝔽₇)	—	97	1	no	maximal, Hall
37	C₂×C₉₇	—	7	1	no	—
38	C₂×C₉₇	—	21	1	no	—
39	C₂×C₉₇	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
40	C₂×C₃×ΓL(1,𝔽₈)	—	97	1	no	—
41	C₃×AGL(1,𝔽₇)	—	97	1	no	—
42	C₃×AGL(1,𝔽₇)	—	97	1	no	—
43	C₉₇	C₂×C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
44	C₂×C₃×D₇	C₉₇⋊C₃	1	1	yes	characteristic, complemented
45	C₂×AGL(1,𝔽₇)	—	97	3	no	—
46	C₃×ΓL(1,𝔽₈)	—	97	1	no	Hall
47	C₂×C₃×C₇	C₂×C₉₇⋊C₃	1	1	yes	characteristic, complemented
48	C₃×D₇	C₂×C₉₇⋊C₃	1	1	yes	characteristic, complemented
49	C₃×D₇	C₂×C₉₇⋊C₃	1	1	yes	characteristic, complemented
50	C₂×ΓL(1,𝔽₈)	—	97	3	no	—
51	AGL(1,𝔽₇)	—	97	3	no	—
52	AGL(1,𝔽₇)	—	97	3	no	—
53	C₂²×C₃²	—	679	1	no	Hall
54	C₂×D₇	C₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented, Hall
55	C₃×C₇	C₂²×C₉₇⋊C₃	1	1	yes	characteristic, complemented
56	ΓL(1,𝔽₈)	—	97	3	no	—
57	C₂×C₃²	—	679	1	no	—
58	C₂×C₃²	—	679	1	no	—
59	C₂×C₃²	—	679	1	no	—
60	C₂×C₇	C₂×C₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented
61	D₇	C₂×C₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented
62	D₇	C₂×C₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented
63	C₂²×C₃	—	21	1	no	—
64	C₂²×C₃	—	679	3	no	—
65	C₃²	—	679	1	no	Sylow
66	C₇	C₂²×C₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₃	—	7	1	no	—
68	C₂×C₃	—	21	1	no	—
69	C₂×C₃	—	679	3	no	—
70	C₂×C₃	—	679	3	no	—
71	C₂×C₃	—	679	3	no	—
72	C₂×C₃	#16861353	1	1	yes	characteristic, direct product factor
73	C₂²	—	21	1	no	Sylow
74	C₃	—	679	3	no	—
75	C₃	C₂×#16861353	1	1	yes	characteristic, direct product factor
76	C₂	—	7	1	no	—
77	C₂	—	21	1	no	—
78	C₂	C₃×#16861353	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	#16861353
commutator, cosocle, hypocenter	C₇×C₉₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₇×C₉₇	C₂×C₃
LCS	G ▷ C₇×C₉₇
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#14523271	C₂	semidirect
2. C₃×#16861353	C₂	direct (N × H)
3. C₃×#16861353	C₂	direct (N × H)
4. C₂×C₃×D₇×C₉₇	C₃	semidirect
5. C₂×#16861353	C₃	direct (N × H)
6. C₃×#14523271	C₂²	semidirect
7. #16861353	C₂×C₃	direct (N × H)
8. #16861353	C₂×C₃	direct (N × H)
9. C₃×D₇×C₉₇	C₂×C₃	semidirect
10. C₃×D₇×C₉₇	C₂×C₃	semidirect
11. C₂×#14523271	C₂×C₃	semidirect
12. C₂×C₃×C₇×C₉₇	C₂×C₃	semidirect
14. C₂×D₇×C₉₇	C₃²	semidirect
15. C₃×C₇×C₉₇	C₂²×C₃	semidirect
16. #14523271	C₂²×C₃	semidirect
20. C₂×C₇×C₉₇	C₂×C₃²	semidirect
21. D₇×C₉₇	C₂×C₃²	semidirect
22. D₇×C₉₇	C₂×C₃²	semidirect
26. C₇×C₉₇	C₂²×C₃²	semidirect
29. C₂×C₃×C₉₇	AGL(1,𝔽₇)	semidirect
34. C₃×C₉₇	C₂×AGL(1,𝔽₇)	semidirect
39. C₂×C₉₇	C₃×AGL(1,𝔽₇)	semidirect
43. C₉₇	C₂×C₃×AGL(1,𝔽₇)	semidirect
44. C₂×C₃×D₇	C₉₇⋊C₃	semidirect
47. C₂×C₃×C₇	C₂×C₉₇⋊C₃	semidirect
48. C₃×D₇	C₂×C₉₇⋊C₃	semidirect
49. C₃×D₇	C₂×C₉₇⋊C₃	semidirect
54. C₂×D₇	C₃×C₉₇⋊C₃	semidirect
55. C₃×C₇	C₂²×C₉₇⋊C₃	semidirect
60. C₂×C₇	C₂×C₃×C₉₇⋊C₃	semidirect
61. D₇	C₂×C₃×C₉₇⋊C₃	semidirect
62. D₇	C₂×C₃×C₉₇⋊C₃	semidirect
66. C₇	C₂²×C₃×C₉₇⋊C₃	semidirect
72. C₂×C₃	#16861353	direct (N × H)
75. C₃	C₂×#16861353	direct (N × H)
78. C₂	C₃×#16861353	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×#14523271	2
C₃×#16861353	2
C₂×C₃×D₇×C₉₇	3
C₂×#16861353	3
C₂²×C₃×C₉₇⋊C₃	7
C₂×C₃×AGL(1,𝔽₇)	97

Subgroups — C2×C3×#16861353 #37567926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#16861353 #37567926