Subgroups — C₂²×C₄₁×#3861 #37563166

Order 24108.

200 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#3861	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₇²×C₄₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄₁×#3861	C₂²	1	1	yes	characteristic, direct product factor, Hall
4	C₂×C₇²×C₄₁	C₂×C₃	1	3	yes	complemented
5	C₂²×ΓL(1,𝔽₈)×C₄₁	—	7	8	no	maximal
6	C₇²×C₄₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
7	C₂×ΓL(1,𝔽₈)×C₄₁	—	7	24	no	—
8	C₂²×C₇×C₄₁	ΓL(1,𝔽₈)	1	8	yes	complemented
9	ΓL(1,𝔽₈)×C₄₁	—	7	8	no	—
10	C₂²×#3861	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂×C₇×C₄₁	C₂×ΓL(1,𝔽₈)	1	24	yes	complemented
12	C₂²×C₃×C₄₁	—	49	1	no	Hall
13	C₂×#3861	C₂×C₄₁	1	3	yes	direct product factor
14	C₇×C₄₁	C₂²×ΓL(1,𝔽₈)	1	8	yes	complemented
15	C₂×C₃×C₄₁	—	49	3	no	—
16	C₂²×C₇²	C₃×C₄₁	1	1	yes	characteristic, complemented, Hall
17	C₂²×C₄₁	#3861	1	1	yes	characteristic, direct product factor, Hall
18	#3861	C₂²×C₄₁	1	1	yes	characteristic, direct product factor, Hall
19	C₃×C₄₁	—	49	1	no	Hall
20	C₂×C₇²	C₂×C₃×C₄₁	1	3	yes	complemented
21	C₂²×ΓL(1,𝔽₈)	—	7	8	no	—
22	C₂×C₄₁	C₂×#3861	1	3	yes	direct product factor
23	C₇²	C₂²×C₃×C₄₁	1	1	yes	characteristic, complemented, Sylow
24	C₂×ΓL(1,𝔽₈)	—	7	24	no	—
25	C₄₁	C₂²×#3861	1	1	yes	characteristic, direct product factor, Sylow
26	C₂²×C₇	ΓL(1,𝔽₈)×C₄₁	1	8	yes	complemented
27	ΓL(1,𝔽₈)	—	7	8	no	—
28	C₂×C₇	C₂×ΓL(1,𝔽₈)×C₄₁	1	24	yes	complemented
29	C₂²×C₃	—	49	1	no	Hall
30	C₇	C₂²×ΓL(1,𝔽₈)×C₄₁	1	8	yes	complemented
31	C₂×C₃	—	49	3	no	—
32	C₂²	C₄₁×#3861	1	1	yes	characteristic, direct product factor, Sylow
33	C₃	—	49	1	no	Sylow
34	C₂	C₂×C₄₁×#3861	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₄₁	#3861
commutator, cosocle, hypocenter	C₇²	C₂²×C₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₂²×C₇²×C₄₁	C₃
LCS	G ▷ C₇²
UCS	C₁ ◁ C₂²×C₄₁

Complements

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

N	H ≅ G/N	type
1. C₂×C₄₁×#3861	C₂	direct (N × H)
2. C₂²×C₇²×C₄₁	C₃	semidirect
3. C₄₁×#3861	C₂²	direct (N × H)
4. C₂×C₇²×C₄₁	C₂×C₃	semidirect
6. C₇²×C₄₁	C₂²×C₃	semidirect
8. C₂²×C₇×C₄₁	ΓL(1,𝔽₈)	semidirect
10. C₂²×#3861	C₄₁	direct (N × H)
11. C₂×C₇×C₄₁	C₂×ΓL(1,𝔽₈)	semidirect
13. C₂×#3861	C₂×C₄₁	direct (N × H)
14. C₇×C₄₁	C₂²×ΓL(1,𝔽₈)	semidirect
16. C₂²×C₇²	C₃×C₄₁	semidirect
17. C₂²×C₄₁	#3861	direct (N × H)
18. #3861	C₂²×C₄₁	direct (N × H)
20. C₂×C₇²	C₂×C₃×C₄₁	semidirect
22. C₂×C₄₁	C₂×#3861	direct (N × H)
23. C₇²	C₂²×C₃×C₄₁	semidirect
25. C₄₁	C₂²×#3861	direct (N × H)
26. C₂²×C₇	ΓL(1,𝔽₈)×C₄₁	semidirect
28. C₂×C₇	C₂×ΓL(1,𝔽₈)×C₄₁	semidirect
30. C₇	C₂²×ΓL(1,𝔽₈)×C₄₁	semidirect
32. C₂²	C₄₁×#3861	direct (N × H)
34. C₂	C₂×C₄₁×#3861	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄₁×#3861	2
C₂²×C₇²×C₄₁	3
C₂²×ΓL(1,𝔽₈)×C₄₁	7
C₂²×#3861	41

Subgroups — C22×C41×#3861 #37563166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄₁×#3861 #37563166