Subgroups — ΓL(1,𝔽₈)×D₅₅₃ #37551130

Order 23226.

34 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	C₇×D₅₅₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇²×C₇₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	ΓL(1,𝔽₈)×D₇₉	—	7	1	no	maximal
5	C₃×D₅₅₃	—	7	1	no	maximal
6	C₃×C₇×C₇₉	—	7	1	no	—
7	ΓL(1,𝔽₈)×C₇₉	D₇	1	1	yes	characteristic, complemented
8	D₅₅₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor
9	C₇×D₇₉	—	7	1	no	—
10	C₇×C₇₉	—	6	1	no	—
11	C₇×C₇₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
12	C₇×C₇₉	C₃×D₇	1	1	yes	characteristic, complemented
13	C₃×D₇₉	—	49	1	no	Hall
14	D₇×ΓL(1,𝔽₈)	—	79	1	no	maximal, Hall
15	C₃×C₇₉	—	7	1	no	Hall
16	D₇₉	—	7	1	no	Hall
17	C₇×ΓL(1,𝔽₈)	D₇₉	1	1	yes	characteristic, complemented, Hall
18	C₇×D₇	—	79	1	no	Hall
19	C₇₉	D₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
20	C₇²	C₃×D₇₉	1	1	yes	characteristic, complemented, Sylow
21	C₃×D₇	—	553	1	no	—
22	C₂×ΓL(1,𝔽₈)	—	553	1	no	—
23	C₃×C₇	—	7	1	no	—
24	ΓL(1,𝔽₈)	D₅₅₃	1	1	yes	characteristic, direct product factor
25	C₂×C₇	—	553	1	no	—
26	D₇	—	79	1	no	—
27	C₇	—	6	1	no	—
28	C₇	C₃×D₅₅₃	1	1	yes	characteristic, complemented
29	C₇	ΓL(1,𝔽₈)×D₇₉	1	1	yes	characteristic, complemented
30	C₂×C₃	—	3871	1	no	Hall
31	C₃	—	7	1	no	Sylow
32	C₂	—	553	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₇₉	C₂×C₃
LCS	G ▷ C₇²×C₇₉
UCS	C₁

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.