Subgroups — ΓL(1,𝔽₈)×C₃⋊#87604 #37567905

Order 24444.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×ΓL(1,𝔽₈)×D₉₇	C₂	1	1	yes	maximal, characteristic
2	C₇×C₃⋊#87604	C₃	1	1	yes	maximal, characteristic, complemented
3	ΓL(1,𝔽₈)×#87604	—	3	1	no	maximal
4	C₃×ΓL(1,𝔽₈)×C₉₇	C₄	1	1	yes	characteristic, complemented, Hall
5	ΓL(1,𝔽₈)×D₉₇	—	2	1	no	—
6	ΓL(1,𝔽₈)×D₉₇	D₃	1	1	yes	characteristic
7	C₃×C₇×D₉₇	C₂×C₃	1	1	yes	characteristic
8	C₃×C₃⋊#87604	—	7	1	no	maximal, Hall
9	C₇×#87604	—	3	1	no	Hall
10	C₃×C₇×C₉₇	C₃×C₄	1	1	yes	characteristic, complemented
11	ΓL(1,𝔽₈)×C₉₇	—	2	1	no	—
12	ΓL(1,𝔽₈)×C₉₇	Dic₃	1	1	yes	characteristic, complemented
13	C₃²×D₉₇	—	7	1	no	—
14	C₇×D₉₇	C₃×D₃	1	1	yes	characteristic
15	C₃⋊#87604	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
16	C₃×#87604	—	21	1	no	—
17	C₃²×C₉₇	—	7	1	no	Hall
18	C₇×C₉₇	C₃×Dic₃	1	1	yes	characteristic, complemented, Hall
19	C₃×D₉₇	—	7	1	no	—
20	C₃×D₉₇	—	14	1	no	—
21	C₃×D₉₇	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic
22	#87604	—	3	1	no	Hall
23	C₃×C₉₇	—	7	1	no	—
24	C₃×C₉₇	—	14	1	no	—
25	C₃×C₉₇	C₄×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
26	Dic₃×ΓL(1,𝔽₈)	—	97	1	no	maximal, Hall
27	D₉₇	D₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
28	C₂×C₃×ΓL(1,𝔽₈)	—	97	1	no	—
29	C₉₇	Dic₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
30	C₇×Dic₃	—	97	1	no	—
31	C₄×ΓL(1,𝔽₈)	—	291	1	no	—
32	C₃×ΓL(1,𝔽₈)	#87604	1	1	yes	characteristic, complemented, Hall
33	C₂×C₃×C₇	—	97	1	no	—
34	C₂×ΓL(1,𝔽₈)	—	97	1	no	—
35	C₂×ΓL(1,𝔽₈)	—	194	1	no	—
36	C₃×Dic₃	—	679	1	no	Hall
37	C₄×C₇	—	291	1	no	Hall
38	C₃×C₇	C₃×#87604	1	1	yes	characteristic, complemented
39	ΓL(1,𝔽₈)	—	2	1	no	—
40	ΓL(1,𝔽₈)	C₃⋊#87604	1	1	yes	characteristic, direct product factor
41	C₂×C₃²	—	679	1	no	—
42	C₂×C₇	—	97	1	no	—
43	C₃×C₄	—	2037	1	no	—
44	Dic₃	—	97	1	no	—
45	C₃²	—	7	1	no	Sylow
46	C₇	C₃×C₃⋊#87604	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃	—	1358	1	no	—
48	C₂×C₃	—	97	1	no	—
49	C₂×C₃	—	679	1	no	—
50	C₄	—	291	1	no	Sylow
51	C₃	—	7	1	no	—
52	C₃	—	14	1	no	—
53	C₃	ΓL(1,𝔽₈)×#87604	1	1	yes	characteristic, complemented
54	C₂	—	97	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₇×C₉₇	C₃×C₄
prime radical	C₃×ΓL(1,𝔽₈)×D₉₇	C₂
cosocle	C₃×C₇×D₉₇	C₂×C₃
LCS	G ▷ C₃×C₇×C₉₇
UCS	C₁

Complements

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

N	H ≅ G/N	type
2. C₇×C₃⋊#87604	C₃	semidirect
4. C₃×ΓL(1,𝔽₈)×C₉₇	C₄	semidirect
10. C₃×C₇×C₉₇	C₃×C₄	semidirect
12. ΓL(1,𝔽₈)×C₉₇	Dic₃	semidirect
15. C₃⋊#87604	ΓL(1,𝔽₈)	direct (N × H)
18. C₇×C₉₇	C₃×Dic₃	semidirect
25. C₃×C₉₇	C₄×ΓL(1,𝔽₈)	semidirect
29. C₉₇	Dic₃×ΓL(1,𝔽₈)	semidirect
32. C₃×ΓL(1,𝔽₈)	#87604	semidirect
38. C₃×C₇	C₃×#87604	semidirect
40. ΓL(1,𝔽₈)	C₃⋊#87604	direct (N × H)
46. C₇	C₃×C₃⋊#87604	semidirect
53. C₃	ΓL(1,𝔽₈)×#87604	semidirect

Maximal subgroups

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

subgroup	index
C₃×ΓL(1,𝔽₈)×D₉₇	2
C₇×C₃⋊#87604	3
ΓL(1,𝔽₈)×#87604	3
C₃×C₃⋊#87604	7
Dic₃×ΓL(1,𝔽₈)	97

Subgroups — ΓL(1,𝔽8)×C3⋊#87604 #37567905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — ΓL(1,𝔽₈)×C₃⋊#87604 #37567905