Subgroups — C₂×D₃×#14523272 #37567946

Order 24444.

88 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	D₃×#14523272	C₂	1	2	yes	maximal, direct product factor
3	C₂×D₃×C₇×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×#14523272	—	3	1	no	maximal
5	C₃×#14523272	C₂²	1	1	yes	characteristic, complemented, Hall
6	D₃×C₇×C₉₇	C₂×C₃	1	2	yes	complemented
7	C₂×#14523272	—	2	1	no	—
8	C₂×#14523272	—	3	2	no	—
9	C₂×#14523272	D₃	1	1	yes	characteristic, direct product factor
10	C₂×C₃×C₇×C₉₇	C₂×C₃	1	1	yes	characteristic, complemented
11	C₂×D₃×C₉₇⋊C₃	—	7	1	no	maximal, Hall
12	C₂²×C₇×C₉₇	—	3	1	no	Hall
13	C₃×C₇×C₉₇	C₂²×C₃	1	1	yes	characteristic, complemented
14	#14523272	—	2	1	no	—
15	#14523272	C₂×D₃	1	1	yes	characteristic, direct product factor
16	C₂×C₃×C₉₇⋊C₃	—	7	1	no	—
17	D₃×C₉₇⋊C₃	—	7	2	no	—
18	C₂×C₇×C₉₇	—	3	2	no	—
19	C₂×C₇×C₉₇	C₃×D₃	1	1	yes	characteristic, complemented
20	C₂×D₃×C₉₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
21	C₂²×C₉₇⋊C₃	—	21	1	no	—
22	C₃×C₉₇⋊C₃	—	7	1	no	Hall
23	C₇×C₉₇	C₂×C₃×D₃	1	1	yes	characteristic, complemented, Hall
24	C₂×C₃×C₉₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
25	D₃×C₉₇	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
26	C₂×C₉₇⋊C₃	—	7	1	no	—
27	C₂×C₉₇⋊C₃	—	14	1	no	—
28	C₂×C₉₇⋊C₃	—	21	2	no	—
29	C₂²×C₉₇	—	3	1	no	Hall
30	C₃×C₉₇	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
31	C₉₇⋊C₃	—	7	1	no	—
32	C₉₇⋊C₃	—	14	1	no	—
33	C₂×D₃×ΓL(1,𝔽₈)	—	97	1	no	maximal, Hall
34	C₂×C₉₇	—	3	2	no	—
35	C₂×C₉₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
36	C₂×C₃×ΓL(1,𝔽₈)	—	97	1	no	—
37	D₃×ΓL(1,𝔽₈)	—	97	2	no	—
38	C₉₇	C₂×D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
39	C₂×D₃×C₇	C₉₇⋊C₃	1	1	yes	characteristic, complemented
40	C₂²×ΓL(1,𝔽₈)	—	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	D₃×C₇	C₂×C₉₇⋊C₃	1	2	yes	complemented
44	C₂×ΓL(1,𝔽₈)	—	291	2	no	—
45	C₂×ΓL(1,𝔽₈)	—	97	1	no	—
46	C₂×ΓL(1,𝔽₈)	—	194	1	no	—
47	C₂×C₃×D₃	—	679	1	no	Hall
48	C₂²×C₇	—	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	—	679	1	no	—
53	C₃×D₃	—	679	2	no	—
54	C₂×C₇	—	3	2	no	—
55	C₂×C₇	D₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented
56	C₂²×C₃	—	2037	1	no	—
57	C₂×D₃	#14523272	1	1	yes	characteristic, direct product factor
58	C₃²	—	679	1	no	Sylow
59	C₇	C₂×D₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
60	C₂×C₃	—	1358	1	no	—
61	C₂×C₃	—	679	1	no	—
62	C₂×C₃	—	2037	2	no	—
63	C₂×C₃	C₂×#14523272	1	1	yes	characteristic, complemented
64	D₃	C₂×#14523272	1	2	yes	direct product factor
65	C₂²	—	3	1	no	Sylow
66	C₃	—	1358	1	no	—
67	C₃	—	679	1	no	—
68	C₃	C₂²×#14523272	1	1	yes	characteristic, complemented
69	C₂	—	3	2	no	—
70	C₂	D₃×#14523272	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	D₃×#14523272
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×#14523272	C₂	semidirect
2. D₃×#14523272	C₂	direct (N × H)
3. C₂×D₃×C₇×C₉₇	C₃	semidirect
5. C₃×#14523272	C₂²	semidirect
6. D₃×C₇×C₉₇	C₂×C₃	semidirect
9. C₂×#14523272	D₃	direct (N × H)
10. C₂×C₃×C₇×C₉₇	C₂×C₃	semidirect
13. C₃×C₇×C₉₇	C₂²×C₃	semidirect
15. #14523272	C₂×D₃	direct (N × H)
19. C₂×C₇×C₉₇	C₃×D₃	semidirect
20. C₂×D₃×C₉₇	ΓL(1,𝔽₈)	semidirect
23. C₇×C₉₇	C₂×C₃×D₃	semidirect
24. C₂×C₃×C₉₇	C₂×ΓL(1,𝔽₈)	semidirect
25. D₃×C₉₇	C₂×ΓL(1,𝔽₈)	semidirect
30. C₃×C₉₇	C₂²×ΓL(1,𝔽₈)	semidirect
35. C₂×C₉₇	D₃×ΓL(1,𝔽₈)	semidirect
38. C₉₇	C₂×D₃×ΓL(1,𝔽₈)	semidirect
39. C₂×D₃×C₇	C₉₇⋊C₃	semidirect
42. C₂×C₃×C₇	C₂×C₉₇⋊C₃	semidirect
43. D₃×C₇	C₂×C₉₇⋊C₃	semidirect
49. C₃×C₇	C₂²×C₉₇⋊C₃	semidirect
55. C₂×C₇	D₃×C₉₇⋊C₃	semidirect
57. C₂×D₃	#14523272	direct (N × H)
59. C₇	C₂×D₃×C₉₇⋊C₃	semidirect
63. C₂×C₃	C₂×#14523272	semidirect
64. D₃	C₂×#14523272	direct (N × H)
68. C₃	C₂²×#14523272	semidirect
70. C₂	D₃×#14523272	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
D₃×#14523272	2
C₂×D₃×C₇×C₉₇	3
C₂²×#14523272	3
C₂×D₃×C₉₇⋊C₃	7
C₂×D₃×ΓL(1,𝔽₈)	97

Subgroups — C2×D3×#14523272 #37567946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×#14523272 #37567946