Subgroups — C₂²×C₄₁×#3862 #37563167

Order 24108.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄₁×#3862	C₂	1	3	yes	maximal, direct product factor
2	C₂²×C₇²×C₄₁	C₃	1	1	yes	maximal, characteristic, complemented, Hall
3	C₄₁×#3862	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	2	no	maximal
6	C₇²×C₄₁	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
7	C₂×ΓL(1,𝔽₈)×C₄₁	—	7	6	no	—
8	C₂²×C₇×C₄₁	—	3	2	no	—
9	C₂²×C₇×C₄₁	ΓL(1,𝔽₈)	1	2	yes	complemented
10	ΓL(1,𝔽₈)×C₄₁	—	7	2	no	—
11	C₂²×#3862	C₄₁	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₇×C₄₁	—	3	6	no	—
13	C₂×C₇×C₄₁	C₂×ΓL(1,𝔽₈)	1	6	yes	complemented
14	C₂²×C₃×C₄₁	—	49	1	no	Hall
15	C₂×#3862	C₂×C₄₁	1	3	yes	direct product factor
16	C₇×C₄₁	—	3	2	no	—
17	C₇×C₄₁	C₂²×ΓL(1,𝔽₈)	1	2	yes	complemented
18	C₂×C₃×C₄₁	—	49	3	no	—
19	C₂²×C₇²	C₃×C₄₁	1	1	yes	characteristic, complemented, Hall
20	C₂²×C₄₁	#3862	1	1	yes	characteristic, direct product factor, Hall
21	#3862	C₂²×C₄₁	1	1	yes	characteristic, direct product factor, Hall
22	C₃×C₄₁	—	49	1	no	Hall
23	C₂×C₇²	C₂×C₃×C₄₁	1	3	yes	complemented
24	C₂²×ΓL(1,𝔽₈)	—	7	2	no	—
25	C₂×C₄₁	C₂×#3862	1	3	yes	direct product factor
26	C₇²	C₂²×C₃×C₄₁	1	1	yes	characteristic, complemented, Sylow
27	C₂×ΓL(1,𝔽₈)	—	7	6	no	—
28	C₄₁	C₂²×#3862	1	1	yes	characteristic, direct product factor, Sylow
29	C₂²×C₇	—	3	2	no	—
30	C₂²×C₇	ΓL(1,𝔽₈)×C₄₁	1	2	yes	complemented
31	ΓL(1,𝔽₈)	—	7	2	no	—
32	C₂×C₇	—	3	6	no	—
33	C₂×C₇	C₂×ΓL(1,𝔽₈)×C₄₁	1	6	yes	complemented
34	C₂²×C₃	—	49	1	no	Hall
35	C₇	—	3	2	no	—
36	C₇	C₂²×ΓL(1,𝔽₈)×C₄₁	1	2	yes	complemented
37	C₂×C₃	—	49	3	no	—
38	C₂²	C₄₁×#3862	1	1	yes	characteristic, direct product factor, Sylow
39	C₃	—	49	1	no	Sylow
40	C₂	C₂×C₄₁×#3862	1	3	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂²×C₄₁	#3862
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×#3862	C₂	direct (N × H)
2. C₂²×C₇²×C₄₁	C₃	semidirect
3. C₄₁×#3862	C₂²	direct (N × H)
4. C₂×C₇²×C₄₁	C₂×C₃	semidirect
6. C₇²×C₄₁	C₂²×C₃	semidirect
9. C₂²×C₇×C₄₁	ΓL(1,𝔽₈)	semidirect
11. C₂²×#3862	C₄₁	direct (N × H)
13. C₂×C₇×C₄₁	C₂×ΓL(1,𝔽₈)	semidirect
15. C₂×#3862	C₂×C₄₁	direct (N × H)
17. C₇×C₄₁	C₂²×ΓL(1,𝔽₈)	semidirect
19. C₂²×C₇²	C₃×C₄₁	semidirect
20. C₂²×C₄₁	#3862	direct (N × H)
21. #3862	C₂²×C₄₁	direct (N × H)
23. C₂×C₇²	C₂×C₃×C₄₁	semidirect
25. C₂×C₄₁	C₂×#3862	direct (N × H)
26. C₇²	C₂²×C₃×C₄₁	semidirect
28. C₄₁	C₂²×#3862	direct (N × H)
30. C₂²×C₇	ΓL(1,𝔽₈)×C₄₁	semidirect
33. C₂×C₇	C₂×ΓL(1,𝔽₈)×C₄₁	semidirect
36. C₇	C₂²×ΓL(1,𝔽₈)×C₄₁	semidirect
38. C₂²	C₄₁×#3862	direct (N × H)
40. C₂	C₂×C₄₁×#3862	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×#3862	2
C₂²×C₇²×C₄₁	3
C₂²×ΓL(1,𝔽₈)×C₄₁	7
C₂²×#3862	41

Subgroups — C22×C41×#3862 #37563167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄₁×#3862 #37563167