Subgroups — C₇×#15693306 #37551114

Order 23226.

34 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#13110305	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	#15693306	C₇	1	1	yes	maximal, characteristic, direct product factor
5	C₇×D₇₉⋊C₃	—	7	1	no	maximal
6	#13110305	C₂×C₇	1	1	yes	characteristic, complemented
7	C₇×C₇₉⋊C₃	—	7	1	no	—
8	D₅₅₃	C₃×C₇	1	1	yes	characteristic, complemented
9	C₇×D₇₉	—	7	1	no	—
10	C₇×C₇₉	—	6	1	no	—
11	C₇×C₇₉	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
12	C₇×C₇₉	C₂×C₃×C₇	1	1	yes	characteristic, complemented
13	D₇₉⋊C₃	—	7	1	no	Hall
14	C₇×AGL(1,𝔽₇)	—	79	1	no	maximal, Hall
15	C₇₉⋊C₃	—	7	1	no	Hall
16	D₇₉	—	21	1	no	Hall
17	C₇×ΓL(1,𝔽₈)	—	79	1	no	Hall
18	C₇×D₇	—	237	1	no	Hall
19	C₇₉	C₇×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Sylow
20	C₇²	D₇₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
21	C₂×C₃×C₇	—	553	1	no	—
22	AGL(1,𝔽₇)	—	79	1	no	—
23	C₃×C₇	—	553	1	no	—
24	ΓL(1,𝔽₈)	—	79	1	no	—
25	C₂×C₇	—	1659	1	no	—
26	D₇	—	237	1	no	—
27	C₇	—	6	1	no	—
28	C₇	C₇×D₇₉⋊C₃	1	1	yes	characteristic, complemented
29	C₇	#15693306	1	1	yes	characteristic, direct product factor
30	C₂×C₃	—	553	1	no	Hall
31	C₃	—	553	1	no	Sylow
32	C₂	—	1659	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₇	#15693306
commutator, cosocle, hypocenter	C₇×C₇₉	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₇²×C₇₉	C₂×C₃
LCS	G ▷ C₇×C₇₉
UCS	C₁ ◁ C₇

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.