Subgroups — C₂×#26042478 #37551167

Order 23226.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26042478	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₇²×C₇₉	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₇²×C₇₉	C₂×C₃	1	1	yes	characteristic, complemented, Hall
4	C₂×#13110305	—	7	8	no	maximal
5	#13110305	—	7	8	no	—
6	C₂×C₇×C₇₉	ΓL(1,𝔽₈)	1	8	yes	complemented
7	C₇×C₇₉	C₂×ΓL(1,𝔽₈)	1	8	yes	complemented
8	C₂×C₇₉⋊C₃	—	49	1	no	Hall
9	C₂×#3861	—	79	1	no	maximal, Hall
10	C₇₉⋊C₃	—	49	1	no	Hall
11	C₂×C₇₉	#3861	1	1	yes	characteristic, complemented, Hall
12	#3861	—	79	1	no	Hall
13	C₂×C₇²	C₇₉⋊C₃	1	1	yes	characteristic, complemented, Hall
14	C₇₉	C₂×#3861	1	1	yes	characteristic, complemented, Sylow
15	C₇²	C₂×C₇₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
16	C₂×ΓL(1,𝔽₈)	—	553	8	no	—
17	ΓL(1,𝔽₈)	—	553	8	no	—
18	C₂×C₇	#13110305	1	8	yes	complemented
19	C₇	C₂×#13110305	1	8	yes	complemented
20	C₂×C₃	—	3871	1	no	Hall
21	C₃	—	3871	1	no	Sylow
22	C₂	#26042478	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂	#26042478
commutator, cosocle, hypocenter	C₇²×C₇₉	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₇₉	C₃
LCS	G ▷ C₇²×C₇₉
UCS	C₁ ◁ C₂

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.