Subgroups — C₂×C₇×#13110306 #37551165

Order 23226.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#13110306	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×#13110306	C₇	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₇×C₇₉⋊C₃	—	7	1	no	maximal
6	#13110306	C₂×C₇	1	1	yes	characteristic, direct product factor
7	C₇×C₇₉⋊C₃	—	7	1	no	—
8	C₂×C₇×C₇₉	—	3	2	no	—
9	C₂×C₇×C₇₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
10	C₂×C₇×C₇₉	C₃×C₇	1	1	yes	characteristic, complemented
11	C₇×C₇₉	—	3	2	no	—
12	C₇×C₇₉	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
13	C₇×C₇₉	C₂×C₃×C₇	1	1	yes	characteristic, complemented
14	C₂×C₇₉⋊C₃	—	7	1	no	Hall
15	C₂×C₇×ΓL(1,𝔽₈)	—	79	1	no	maximal, Hall
16	C₇₉⋊C₃	—	7	1	no	Hall
17	C₂×C₇₉	C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
18	C₇×ΓL(1,𝔽₈)	—	79	1	no	Hall
19	C₂×C₇²	C₇₉⋊C₃	1	1	yes	characteristic, complemented, Hall
20	C₇₉	C₂×C₇×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
21	C₇²	C₂×C₇₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
22	C₂×C₃×C₇	—	553	1	no	—
23	C₂×ΓL(1,𝔽₈)	—	79	1	no	—
24	C₃×C₇	—	553	1	no	—
25	ΓL(1,𝔽₈)	—	79	1	no	—
26	C₂×C₇	—	3	2	no	—
27	C₂×C₇	C₇×C₇₉⋊C₃	1	1	yes	characteristic, complemented
28	C₂×C₇	#13110306	1	1	yes	characteristic, direct product factor
29	C₇	—	3	2	no	—
30	C₇	C₂×C₇×C₇₉⋊C₃	1	1	yes	characteristic, complemented
31	C₇	C₂×#13110306	1	1	yes	characteristic, direct product factor
32	C₂×C₃	—	553	1	no	Hall
33	C₃	—	553	1	no	Sylow
34	C₂	C₇×#13110306	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center, hypercenter	C₂×C₇	#13110306
commutator, cosocle, hypocenter	C₇×C₇₉	C₂×C₃×C₇
Frattini	C₁	G
Fitting, socle	C₂×C₇²×C₇₉	C₃
LCS	G ▷ C₇×C₇₉
UCS	C₁ ◁ C₂×C₇

Complements

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

N	H ≅ G/N	type
1. C₇×#13110306	C₂	direct (N × H)
2. C₂×C₇²×C₇₉	C₃	semidirect
3. C₇²×C₇₉	C₂×C₃	semidirect
4. C₂×#13110306	C₇	direct (N × H)
6. #13110306	C₂×C₇	direct (N × H)
9. C₂×C₇×C₇₉	ΓL(1,𝔽₈)	semidirect
10. C₂×C₇×C₇₉	C₃×C₇	semidirect
12. C₇×C₇₉	C₂×ΓL(1,𝔽₈)	semidirect
13. C₇×C₇₉	C₂×C₃×C₇	semidirect
17. C₂×C₇₉	C₇×ΓL(1,𝔽₈)	semidirect
19. C₂×C₇²	C₇₉⋊C₃	semidirect
20. C₇₉	C₂×C₇×ΓL(1,𝔽₈)	semidirect
21. C₇²	C₂×C₇₉⋊C₃	semidirect
27. C₂×C₇	C₇×C₇₉⋊C₃	semidirect
28. C₂×C₇	#13110306	direct (N × H)
30. C₇	C₂×C₇×C₇₉⋊C₃	semidirect
31. C₇	C₂×#13110306	direct (N × H)
34. C₂	C₇×#13110306	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#13110306	2
C₂×C₇²×C₇₉	3
C₂×#13110306	7
C₂×C₇×C₇₉⋊C₃	7
C₂×C₇×ΓL(1,𝔽₈)	79

Subgroups — C2×C7×#13110306 #37551165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#13110306 #37551165