Subgroups — #37551100

Order 23226.

24 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042470	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	C₇₉×D₄₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄₉×C₇₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	#15693302	—	7	1	no	maximal
5	#13110305	—	7	1	no	—
6	D₇×C₇₉	—	7	1	no	—
7	C₇×C₇₉	AGL(1,𝔽₇)	1	1	yes	characteristic
8	C₂×C₇₉⋊C₃	—	49	1	no	Hall
9	D₄₉⋊C₃	—	79	1	no	maximal, Hall
10	C₇₉⋊C₃	—	49	1	no	Hall
11	C₂×C₇₉	—	147	1	no	Hall
12	C₄₉⋊C₃	—	79	1	no	Hall
13	D₄₉	C₇₉⋊C₃	1	1	yes	characteristic, complemented, Hall
14	C₇₉	D₄₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
15	C₄₉	C₂×C₇₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
16	AGL(1,𝔽₇)	—	553	1	no	—
17	ΓL(1,𝔽₈)	—	553	1	no	—
18	D₇	—	21	1	no	—
19	C₇	#15693302	1	1	yes	characteristic
20	C₂×C₃	—	3871	1	no	Hall
21	C₃	—	3871	1	no	Sylow
22	C₂	—	147	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₄₉×C₇₉	C₂×C₃
Frattini	C₇	#15693302
socle	C₇×C₇₉	AGL(1,𝔽₇)
LCS	G ▷ C₄₉×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.