Subgroups — A₄×#14523272 #37567918

Order 24444.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523272	C₃	1	1	yes	maximal, characteristic, complemented
2	C₇×A₄×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#14523272	C₃	1	2	yes	maximal, complemented
4	C₃×#14523272	—	4	1	no	maximal, Hall
5	C₂×#14523272	—	3	1	no	—
6	A₄×C₉₇⋊C₃	—	7	1	no	maximal, Hall
7	C₂²×C₇×C₉₇	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₃×C₇×C₉₇	—	4	1	no	—
9	#14523272	—	4	2	no	—
10	#14523272	A₄	1	1	yes	characteristic, direct product factor
11	C₂×C₇×C₉₇	—	3	1	no	—
12	C₂²⋊(C₉₇⋊C₃)	—	7	2	no	—
13	A₄×C₉₇	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
14	C₂²×C₉₇⋊C₃	—	7	1	no	—
15	C₃×C₉₇⋊C₃	—	28	1	no	Hall
16	C₇×C₉₇	C₃×A₄	1	1	yes	characteristic, complemented, Hall
17	C₂×C₉₇⋊C₃	—	21	1	no	—
18	C₂²×C₉₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
19	C₃×C₉₇	—	4	1	no	—
20	C₉₇⋊C₃	—	7	1	no	—
21	C₉₇⋊C₃	—	28	2	no	—
22	A₄×ΓL(1,𝔽₈)	—	97	1	no	maximal, Hall
23	C₂×C₉₇	—	3	1	no	—
24	C₉₇	A₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
25	C₂²⋊ΓL(1,𝔽₈)	—	97	2	no	—
26	C₇×A₄	C₉₇⋊C₃	1	1	yes	characteristic, complemented
27	C₂²×ΓL(1,𝔽₈)	—	97	1	no	—
28	C₃×ΓL(1,𝔽₈)	—	388	1	no	Hall
29	C₂×ΓL(1,𝔽₈)	—	291	1	no	—
30	C₃×A₄	—	679	1	no	Hall
31	C₂²×C₇	C₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented, Hall
32	C₃×C₇	—	4	1	no	—
33	ΓL(1,𝔽₈)	—	97	1	no	—
34	ΓL(1,𝔽₈)	—	388	2	no	—
35	C₂×C₇	—	3	1	no	—
36	C₂²×C₃	—	679	1	no	—
37	A₄	—	679	2	no	—
38	A₄	#14523272	1	1	yes	characteristic, direct product factor
39	C₃²	—	2716	1	no	Sylow
40	C₇	A₄×C₉₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	2037	1	no	—
42	C₂²	C₃×#14523272	1	1	yes	characteristic, complemented, Sylow
43	C₃	—	4	1	no	—
44	C₃	—	2716	2	no	—
45	C₃	—	679	1	no	—
46	C₂	—	3	1	no	—

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

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523272	C₃	semidirect
2. C₇×A₄×C₉₇	C₃	semidirect
3. C₂²⋊#14523272	C₃	semidirect
7. C₂²×C₇×C₉₇	C₃²	semidirect
10. #14523272	A₄	direct (N × H)
13. A₄×C₉₇	ΓL(1,𝔽₈)	semidirect
16. C₇×C₉₇	C₃×A₄	semidirect
18. C₂²×C₉₇	C₃×ΓL(1,𝔽₈)	semidirect
24. C₉₇	A₄×ΓL(1,𝔽₈)	semidirect
26. C₇×A₄	C₉₇⋊C₃	semidirect
31. C₂²×C₇	C₃×C₉₇⋊C₃	semidirect
38. A₄	#14523272	direct (N × H)
40. C₇	A₄×C₉₇⋊C₃	semidirect
42. C₂²	C₃×#14523272	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523272	3
C₇×A₄×C₉₇	3
C₂²⋊#14523272	3
C₃×#14523272	4
A₄×C₉₇⋊C₃	7
A₄×ΓL(1,𝔽₈)	97

Subgroups — A4×#14523272 #37567918

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — A₄×#14523272 #37567918