Subgroups — C₂×C₃⋊#16861353 #37567939

Order 24444.

88 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	2	yes	maximal, direct product factor
3	C₂×D₂₁×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×#16861353	—	3	1	no	maximal
5	C₃×#14523271	C₂²	1	1	yes	characteristic, complemented, Hall
6	#16861353	—	3	2	no	—
7	D₂₁×C₉₇	C₂×C₃	1	2	yes	complemented
8	C₂×#14523271	—	2	1	no	—
9	C₂×#14523271	D₃	1	1	yes	characteristic, complemented
10	C₂×C₃×C₇×C₉₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×D₃×C₉₇⋊C₃	—	7	1	no	maximal, Hall
12	C₂×D₇×C₉₇	—	3	1	no	Hall
13	C₃×C₇×C₉₇	C₂²×C₃	1	1	yes	characteristic, complemented
14	#14523271	—	2	1	no	—
15	#14523271	C₂×D₃	1	1	yes	characteristic, complemented
16	C₂×C₃×C₉₇⋊C₃	—	14	1	no	—
17	D₃×C₉₇⋊C₃	—	7	2	no	—
18	C₂×C₇×C₉₇	C₃×D₃	1	1	yes	characteristic, complemented
19	D₇×C₉₇	—	3	2	no	—
20	C₂×D₃×C₉₇	—	7	1	no	—
21	C₂²×C₉₇⋊C₃	—	21	1	no	—
22	C₃×C₉₇⋊C₃	—	14	1	no	Hall
23	C₇×C₉₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×C₉₇	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
25	D₃×C₉₇	—	7	2	no	—
26	C₂×C₉₇⋊C₃	—	14	1	no	—
27	C₂×C₉₇⋊C₃	—	14	1	no	—
28	C₂×C₉₇⋊C₃	—	21	2	no	—
29	C₂²×C₉₇	—	21	1	no	Hall
30	C₃×C₉₇	C₂×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
31	C₉₇⋊C₃	—	14	1	no	—
32	C₉₇⋊C₃	—	14	1	no	—
33	C₂×C₃⋊AGL(1,𝔽₇)	—	97	1	no	maximal, Hall
34	C₂×C₉₇	—	21	2	no	—
35	C₂×C₉₇	C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
36	C₂×C₃×ΓL(1,𝔽₈)	—	97	1	no	—
37	C₃⋊AGL(1,𝔽₇)	—	97	2	no	—
38	C₉₇	C₂×C₃⋊AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
39	C₂×D₂₁	C₉₇⋊C₃	1	1	yes	characteristic, complemented
40	C₂×AGL(1,𝔽₇)	—	291	1	no	—
41	C₃×ΓL(1,𝔽₈)	—	97	1	no	Hall
42	C₂×C₃×C₇	C₂×C₉₇⋊C₃	1	1	yes	characteristic, complemented
43	D₂₁	C₂×C₉₇⋊C₃	1	2	yes	complemented
44	C₂×ΓL(1,𝔽₈)	—	97	1	no	—
45	C₂×ΓL(1,𝔽₈)	—	194	1	no	—
46	AGL(1,𝔽₇)	—	291	2	no	—
47	C₂×C₃×D₃	—	679	1	no	Hall
48	C₂×D₇	—	3	1	no	Hall
49	C₃×C₇	C₂²×C₉₇⋊C₃	1	1	yes	characteristic, complemented
50	ΓL(1,𝔽₈)	—	97	1	no	—
51	ΓL(1,𝔽₈)	—	194	1	no	—
52	C₂×C₃²	—	1358	1	no	—
53	C₃×D₃	—	679	2	no	—
54	C₂×C₇	D₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented
55	D₇	—	3	2	no	—
56	C₂²×C₃	—	2037	1	no	—
57	C₂×D₃	—	7	1	no	—
58	C₃²	—	1358	1	no	Sylow
59	C₇	C₂×D₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	1358	1	no	—
61	C₂×C₃	—	1358	1	no	—
62	C₂×C₃	—	2037	2	no	—
63	C₂×C₃	#16861353	1	1	yes	characteristic, complemented
64	D₃	—	7	2	no	—
65	C₂²	—	21	1	no	Sylow
66	C₃	—	1358	1	no	—
67	C₃	—	1358	1	no	—
68	C₃	C₂×#16861353	1	1	yes	characteristic, complemented
69	C₂	—	21	2	no	—
70	C₂	C₃⋊#16861353	1	1	yes	characteristic, direct product factor

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

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. C₂×D₂₁×C₉₇	C₃	semidirect
5. C₃×#14523271	C₂²	semidirect
7. D₂₁×C₉₇	C₂×C₃	semidirect
9. C₂×#14523271	D₃	semidirect
10. C₂×C₃×C₇×C₉₇	C₂×C₃	semidirect
13. C₃×C₇×C₉₇	C₂²×C₃	semidirect
15. #14523271	C₂×D₃	semidirect
18. C₂×C₇×C₉₇	C₃×D₃	semidirect
23. C₇×C₉₇	C₂×C₃×D₃	semidirect
24. C₂×C₃×C₉₇	AGL(1,𝔽₇)	semidirect
30. C₃×C₉₇	C₂×AGL(1,𝔽₇)	semidirect
35. C₂×C₉₇	C₃⋊AGL(1,𝔽₇)	semidirect
38. C₉₇	C₂×C₃⋊AGL(1,𝔽₇)	semidirect
39. C₂×D₂₁	C₉₇⋊C₃	semidirect
42. C₂×C₃×C₇	C₂×C₉₇⋊C₃	semidirect
43. D₂₁	C₂×C₉₇⋊C₃	semidirect
49. C₃×C₇	C₂²×C₉₇⋊C₃	semidirect
54. C₂×C₇	D₃×C₉₇⋊C₃	semidirect
59. C₇	C₂×D₃×C₉₇⋊C₃	semidirect
63. C₂×C₃	#16861353	semidirect
68. C₃	C₂×#16861353	semidirect
70. 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×D₂₁×C₉₇	3
C₂×#16861353	3
C₂×D₃×C₉₇⋊C₃	7
C₂×C₃⋊AGL(1,𝔽₇)	97

Subgroups — C2×C3⋊#16861353 #37567939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊#16861353 #37567939