Subgroups — C₂×ΓL(1,𝔽₈)×D₃₇₃ #44531286

Order 31332.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	ΓL(1,𝔽₈)×D₃₇₃	C₂	1	2	yes	maximal, direct product factor
2	C₂×ΓL(1,𝔽₈)×C₃₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×D₃₇₃	C₃	1	1	yes	maximal, characteristic, complemented, Hall
4	ΓL(1,𝔽₈)×C₃₇₃	C₂²	1	1	yes	characteristic, complemented, Hall
5	C₂×C₇×C₃₇₃	C₂×C₃	1	1	yes	characteristic, complemented
6	C₇×D₃₇₃	C₂×C₃	1	2	yes	complemented
7	C₂×C₃×D₃₇₃	—	7	1	no	maximal, Hall
8	C₇×C₃₇₃	C₂²×C₃	1	1	yes	characteristic, complemented, Hall
9	C₂×C₃×C₃₇₃	—	7	1	no	—
10	C₃×D₃₇₃	—	7	2	no	—
11	C₂×D₃₇₃	ΓL(1,𝔽₈)	1	1	yes	characteristic, direct product factor, Hall
12	C₃×C₃₇₃	—	7	1	no	Hall
13	C₂×C₃₇₃	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
14	D₃₇₃	C₂×ΓL(1,𝔽₈)	1	2	yes	direct product factor
15	C₃₇₃	C₂²×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Sylow
16	C₂²×ΓL(1,𝔽₈)	—	373	1	no	maximal, Hall
17	C₂×ΓL(1,𝔽₈)	—	373	2	no	—
18	C₂×ΓL(1,𝔽₈)	D₃₇₃	1	1	yes	characteristic, direct product factor
19	C₂²×C₇	—	373	1	no	Hall
20	ΓL(1,𝔽₈)	C₂×D₃₇₃	1	1	yes	characteristic, direct product factor, Hall
21	C₂×C₇	—	373	2	no	—
22	C₂×C₇	C₃×D₃₇₃	1	1	yes	characteristic, complemented
23	C₂²×C₃	—	2611	1	no	Hall
24	C₇	C₂×C₃×D₃₇₃	1	1	yes	characteristic, complemented, Sylow
25	C₂×C₃	—	7	1	no	—
26	C₂×C₃	—	2611	2	no	—
27	C₂²	—	373	1	no	Sylow
28	C₃	—	7	1	no	Sylow
29	C₂	—	373	2	no	—
30	C₂	ΓL(1,𝔽₈)×D₃₇₃	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	ΓL(1,𝔽₈)×D₃₇₃
commutator, cosocle, hypocenter	C₇×C₃₇₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₇×C₃₇₃	C₂×C₃
LCS	G ▷ C₇×C₃₇₃
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. ΓL(1,𝔽₈)×D₃₇₃	C₂	direct (N × H)
2. C₂×ΓL(1,𝔽₈)×C₃₇₃	C₂	semidirect
3. C₂×C₇×D₃₇₃	C₃	semidirect
4. ΓL(1,𝔽₈)×C₃₇₃	C₂²	semidirect
5. C₂×C₇×C₃₇₃	C₂×C₃	semidirect
6. C₇×D₃₇₃	C₂×C₃	semidirect
8. C₇×C₃₇₃	C₂²×C₃	semidirect
11. C₂×D₃₇₃	ΓL(1,𝔽₈)	direct (N × H)
13. C₂×C₃₇₃	C₂×ΓL(1,𝔽₈)	semidirect
14. D₃₇₃	C₂×ΓL(1,𝔽₈)	direct (N × H)
15. C₃₇₃	C₂²×ΓL(1,𝔽₈)	semidirect
18. C₂×ΓL(1,𝔽₈)	D₃₇₃	direct (N × H)
20. ΓL(1,𝔽₈)	C₂×D₃₇₃	direct (N × H)
22. C₂×C₇	C₃×D₃₇₃	semidirect
24. C₇	C₂×C₃×D₃₇₃	semidirect
30. C₂	ΓL(1,𝔽₈)×D₃₇₃	direct (N × H)

Maximal subgroups

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

subgroup	index
ΓL(1,𝔽₈)×D₃₇₃	2
C₂×ΓL(1,𝔽₈)×C₃₇₃	2
C₂×C₇×D₃₇₃	3
C₂×C₃×D₃₇₃	7
C₂²×ΓL(1,𝔽₈)	373

Subgroups — C2×ΓL(1,𝔽8)×D373 #44531286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×ΓL(1,𝔽₈)×D₃₇₃ #44531286