Subgroups — C₂×C₃×#16861350 #37567930

Order 24444.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#14523272	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#16861350	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃×C₇×D₉₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×#16861350	C₃	1	3	yes	maximal, direct product factor
5	C₃×#14523272	C₂²	1	1	yes	characteristic, complemented, Hall
6	#16861350	C₂×C₃	1	6	yes	direct product factor
7	C₃×C₇×D₉₇	C₂×C₃	1	2	yes	complemented
8	C₂×#14523272	C₂×C₃	1	3	yes	complemented
9	C₂×C₃×C₇×C₉₇	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂×C₃×D₉₇⋊C₃	—	7	1	no	maximal, Hall
11	C₂×C₇×D₉₇	C₃²	1	1	yes	characteristic, complemented, Hall
12	C₃×C₇×C₉₇	C₂²×C₃	1	1	yes	characteristic, complemented
13	#14523272	C₂²×C₃	1	3	yes	complemented
14	C₂×C₃×C₉₇⋊C₃	—	7	1	no	—
15	C₃×D₉₇⋊C₃	—	7	2	no	—
16	C₂×C₇×C₉₇	C₂×C₃²	1	1	yes	characteristic, complemented
17	C₇×D₉₇	C₂×C₃²	1	2	yes	complemented
18	C₂×C₃×D₉₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
19	C₂×D₉₇⋊C₃	—	7	3	no	—
20	C₃×C₉₇⋊C₃	—	7	1	no	Hall
21	C₇×C₉₇	C₂²×C₃²	1	1	yes	characteristic, complemented, Hall
22	C₂×C₃×C₉₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
23	C₃×D₉₇	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
24	C₂×C₉₇⋊C₃	—	7	3	no	—
25	D₉₇⋊C₃	—	7	6	no	—
26	C₂×D₉₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
27	C₃×C₉₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	C₉₇⋊C₃	—	7	3	no	—
29	C₂²×C₃×ΓL(1,𝔽₈)	—	97	1	no	maximal, Hall
30	C₂×C₉₇	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	D₉₇	C₂×C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
32	C₂×C₃×ΓL(1,𝔽₈)	—	97	2	no	—
33	C₂×C₃×ΓL(1,𝔽₈)	—	97	1	no	—
34	C₉₇	C₂²×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
35	C₂²×C₃×C₇	—	291	1	no	—
36	C₂²×ΓL(1,𝔽₈)	—	97	3	no	—
37	C₃×ΓL(1,𝔽₈)	—	97	1	no	Hall
38	C₂×C₃×C₇	—	291	2	no	—
39	C₂×C₃×C₇	D₉₇⋊C₃	1	1	yes	characteristic, complemented
40	C₂×ΓL(1,𝔽₈)	—	97	6	no	—
41	C₂×ΓL(1,𝔽₈)	—	97	3	no	—
42	C₂²×C₃²	—	679	1	no	Hall
43	C₂²×C₇	—	291	1	no	Hall
44	C₃×C₇	C₂×D₉₇⋊C₃	1	1	yes	characteristic, complemented
45	ΓL(1,𝔽₈)	—	97	3	no	—
46	C₂×C₃²	—	679	2	no	—
47	C₂×C₃²	—	679	1	no	—
48	C₂×C₇	—	291	2	no	—
49	C₂×C₇	C₃×D₉₇⋊C₃	1	1	yes	characteristic, complemented
50	C₂²×C₃	—	291	1	no	—
51	C₂²×C₃	—	679	3	no	—
52	C₃²	—	679	1	no	Sylow
53	C₇	C₂×C₃×D₉₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₃	—	291	2	no	—
55	C₂×C₃	—	679	6	no	—
56	C₂×C₃	—	679	3	no	—
57	C₂×C₃	#16861350	1	1	yes	characteristic, direct product factor
58	C₂²	—	291	1	no	Sylow
59	C₃	—	679	3	no	—
60	C₃	C₂×#16861350	1	1	yes	characteristic, direct product factor
61	C₂	—	291	2	no	—
62	C₂	C₃×#16861350	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂×C₃	#16861350
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×#14523272	C₂	semidirect
2. C₃×#16861350	C₂	direct (N × H)
3. C₂×C₃×C₇×D₉₇	C₃	semidirect
4. C₂×#16861350	C₃	direct (N × H)
5. C₃×#14523272	C₂²	semidirect
6. #16861350	C₂×C₃	direct (N × H)
7. C₃×C₇×D₉₇	C₂×C₃	semidirect
8. C₂×#14523272	C₂×C₃	semidirect
9. C₂×C₃×C₇×C₉₇	C₂×C₃	semidirect
11. C₂×C₇×D₉₇	C₃²	semidirect
12. C₃×C₇×C₉₇	C₂²×C₃	semidirect
13. #14523272	C₂²×C₃	semidirect
16. C₂×C₇×C₉₇	C₂×C₃²	semidirect
17. C₇×D₉₇	C₂×C₃²	semidirect
18. C₂×C₃×D₉₇	ΓL(1,𝔽₈)	semidirect
21. C₇×C₉₇	C₂²×C₃²	semidirect
22. C₂×C₃×C₉₇	C₂×ΓL(1,𝔽₈)	semidirect
23. C₃×D₉₇	C₂×ΓL(1,𝔽₈)	semidirect
26. C₂×D₉₇	C₃×ΓL(1,𝔽₈)	semidirect
27. C₃×C₉₇	C₂²×ΓL(1,𝔽₈)	semidirect
30. C₂×C₉₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
31. D₉₇	C₂×C₃×ΓL(1,𝔽₈)	semidirect
34. C₉₇	C₂²×C₃×ΓL(1,𝔽₈)	semidirect
39. C₂×C₃×C₇	D₉₇⋊C₃	semidirect
44. C₃×C₇	C₂×D₉₇⋊C₃	semidirect
49. C₂×C₇	C₃×D₉₇⋊C₃	semidirect
53. C₇	C₂×C₃×D₉₇⋊C₃	semidirect
57. C₂×C₃	#16861350	direct (N × H)
60. C₃	C₂×#16861350	direct (N × H)
62. C₂	C₃×#16861350	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×#14523272	2
C₃×#16861350	2
C₂×C₃×C₇×D₉₇	3
C₂×#16861350	3
C₂×C₃×D₉₇⋊C₃	7
C₂²×C₃×ΓL(1,𝔽₈)	97

Subgroups — C2×C3×#16861350 #37567930

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃×#16861350 #37567930