Subgroups — C₇₉×ΓL(1,𝔽₈)⋊D₇ #37551119

Order 23226.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×ΓL(1,𝔽₈)×C₇₉	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇₉×C₇⋊D₇	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇²×C₇₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	AGL(1,𝔽₇)×C₇₉	—	7	1	no	maximal
5	C₃×D₇×C₇₉	—	7	1	no	maximal
6	C₃×C₇×C₇₉	—	7	1	no	—
7	ΓL(1,𝔽₈)×C₇₉	D₇	1	1	yes	characteristic, complemented
8	D₇×C₇₉	—	7	1	no	—
9	D₇×C₇₉	—	7	1	no	—
10	D₇×C₇₉	—	21	2	no	—
11	C₇×C₇₉	—	3	2	no	—
12	C₇×C₇₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
13	C₇×C₇₉	C₃×D₇	1	1	yes	characteristic, complemented
14	C₂×C₃×C₇₉	—	49	1	no	Hall
15	ΓL(1,𝔽₈)⋊D₇	C₇₉	1	1	yes	maximal, characteristic, direct product factor, Hall
16	C₃×C₇₉	—	7	1	no	Hall
17	C₂×C₇₉	—	49	1	no	Hall
18	C₇×ΓL(1,𝔽₈)	C₂×C₇₉	1	1	yes	characteristic, complemented, Hall
19	C₇⋊D₇	C₃×C₇₉	1	1	yes	characteristic, complemented, Hall
20	C₇₉	ΓL(1,𝔽₈)⋊D₇	1	1	yes	characteristic, direct product factor, Sylow
21	C₇²	C₂×C₃×C₇₉	1	1	yes	characteristic, complemented, Sylow
22	C₃×D₇	—	7	1	no	—
23	AGL(1,𝔽₇)	—	7	1	no	—
24	C₃×C₇	—	7	1	no	—
25	ΓL(1,𝔽₈)	D₇×C₇₉	1	1	yes	characteristic, complemented
26	D₇	—	7	1	no	—
27	D₇	—	7	1	no	—
28	D₇	—	21	2	no	—
29	C₇	—	3	2	no	—
30	C₇	C₃×D₇×C₇₉	1	1	yes	characteristic, complemented
31	C₇	AGL(1,𝔽₇)×C₇₉	1	1	yes	characteristic, complemented
32	C₂×C₃	—	49	1	no	Hall
33	C₃	—	7	1	no	Sylow
34	C₂	—	49	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇₉	ΓL(1,𝔽₈)⋊D₇
commutator, cosocle, hypocenter	C₇²	C₂×C₃×C₇₉
Frattini	C₁	G
Fitting, socle	C₇²×C₇₉	C₂×C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₇₉

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

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