Subgroups — ΓL(1,𝔽₈)×C₂²⋊(C₉₇⋊C₃) #37567916

Order 24444.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×ΓL(1,𝔽₈)×C₉₇	C₃	1	1	yes	maximal, characteristic, complemented
2	C₇×C₂²⋊(C₉₇⋊C₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#14523272	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊#14523271	C₃	1	1	yes	maximal, characteristic, complemented
5	ΓL(1,𝔽₈)×C₉₇⋊C₃	—	4	1	no	maximal, Hall
6	C₂×ΓL(1,𝔽₈)×C₉₇	—	3	1	no	—
7	C₃×C₂²⋊(C₉₇⋊C₃)	—	7	1	no	maximal, Hall
8	C₂²×C₇×C₉₇	C₃²	1	1	yes	characteristic, complemented, Hall
9	#14523272	—	4	1	no	—
10	#14523271	—	4	1	no	—
11	C₇×C₉₇⋊C₃	—	4	1	no	—
12	ΓL(1,𝔽₈)×C₉₇	A₄	1	1	yes	characteristic, complemented
13	C₂×C₇×C₉₇	—	3	1	no	—
14	C₂²×C₃×C₉₇	—	7	1	no	—
15	C₂²⋊(C₉₇⋊C₃)	—	7	1	no	—
16	C₂²⋊(C₉₇⋊C₃)	—	7	1	no	—
17	C₂²⋊(C₉₇⋊C₃)	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
18	C₃×C₉₇⋊C₃	—	28	1	no	Hall
19	C₇×C₉₇	C₃×A₄	1	1	yes	characteristic, complemented, Hall
20	C₂×C₃×C₉₇	—	21	1	no	—
21	C₂²×C₉₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
22	C₃×C₉₇	—	7	1	no	—
23	C₉₇⋊C₃	—	4	1	no	—
24	C₉₇⋊C₃	—	28	1	no	—
25	C₉₇⋊C₃	—	28	1	no	—
26	A₄×ΓL(1,𝔽₈)	—	97	1	no	maximal, Hall
27	C₂×C₉₇	—	3	1	no	—
28	C₉₇	A₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
29	C₂²⋊ΓL(1,𝔽₈)	—	97	1	no	—
30	C₂²⋊ΓL(1,𝔽₈)	—	97	1	no	—
31	C₇×A₄	—	97	1	no	—
32	C₂²×ΓL(1,𝔽₈)	C₉₇⋊C₃	1	1	yes	characteristic, complemented
33	C₃×ΓL(1,𝔽₈)	—	388	1	no	Hall
34	C₂×ΓL(1,𝔽₈)	—	3	1	no	—
35	C₃×A₄	—	679	1	no	Hall
36	C₂²×C₇	C₃×C₉₇⋊C₃	1	1	yes	characteristic, complemented, Hall
37	C₃×C₇	—	388	1	no	—
38	ΓL(1,𝔽₈)	—	388	1	no	—
39	ΓL(1,𝔽₈)	—	388	1	no	—
40	ΓL(1,𝔽₈)	C₂²⋊(C₉₇⋊C₃)	1	1	yes	characteristic, direct product factor
41	C₂×C₇	—	3	1	no	—
42	C₂²×C₃	—	7	1	no	—
43	A₄	—	97	1	no	—
44	A₄	—	679	1	no	—
45	A₄	—	679	1	no	—
46	C₃²	—	2716	1	no	Sylow
47	C₇	C₃×C₂²⋊(C₉₇⋊C₃)	1	1	yes	characteristic, complemented, Sylow
48	C₂×C₃	—	21	1	no	—
49	C₂²	ΓL(1,𝔽₈)×C₉₇⋊C₃	1	1	yes	characteristic, complemented, Sylow
50	C₃	—	7	1	no	—
51	C₃	—	388	1	no	—
52	C₃	—	2716	1	no	—
53	C₃	—	2716	1	no	—
54	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×ΓL(1,𝔽₈)×C₉₇	C₃	semidirect
2. C₇×C₂²⋊(C₉₇⋊C₃)	C₃	semidirect
3. C₂²⋊#14523272	C₃	semidirect
4. C₂²⋊#14523271	C₃	semidirect
8. C₂²×C₇×C₉₇	C₃²	semidirect
12. ΓL(1,𝔽₈)×C₉₇	A₄	semidirect
17. C₂²⋊(C₉₇⋊C₃)	ΓL(1,𝔽₈)	direct (N × H)
19. C₇×C₉₇	C₃×A₄	semidirect
21. C₂²×C₉₇	C₃×ΓL(1,𝔽₈)	semidirect
28. C₉₇	A₄×ΓL(1,𝔽₈)	semidirect
32. C₂²×ΓL(1,𝔽₈)	C₉₇⋊C₃	semidirect
36. C₂²×C₇	C₃×C₉₇⋊C₃	semidirect
40. ΓL(1,𝔽₈)	C₂²⋊(C₉₇⋊C₃)	direct (N × H)
47. C₇	C₃×C₂²⋊(C₉₇⋊C₃)	semidirect
49. C₂²	ΓL(1,𝔽₈)×C₉₇⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×ΓL(1,𝔽₈)×C₉₇	3
C₇×C₂²⋊(C₉₇⋊C₃)	3
C₂²⋊#14523272	3
C₂²⋊#14523271	3
ΓL(1,𝔽₈)×C₉₇⋊C₃	4
C₃×C₂²⋊(C₉₇⋊C₃)	7
A₄×ΓL(1,𝔽₈)	97

Subgroups — ΓL(1,𝔽8)×C22⋊(C97⋊C3) #37567916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₂²⋊(C₉₇⋊C₃) #37567916