Subgroups — C₇×ΓL(1,𝔽₂₅) #66606

Order 336.

40 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₃×C₇	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇×M₁₆	—	3	1	no	maximal, Hall
5	C₂×D₃×C₇	C₄	1	1	yes	characteristic
6	C₃×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₇×Dic₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₇	—	3	1	no	—
9	C₇×C₈	—	3	1	no	—
10	C₇×C₈	D₃	1	1	yes	characteristic, complemented
11	ΓL(1,𝔽₂₅)	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
12	C₂×C₃×C₇	C₂×C₄	1	1	yes	characteristic
13	D₃×C₇	—	2	1	no	—
14	C₂²×C₇	—	3	1	no	—
15	C₄×C₇	—	3	1	no	—
16	C₄×C₇	C₂×D₃	1	1	yes	characteristic
17	C₄×D₃	C₂×C₇	1	1	yes	characteristic
18	C₃×C₈	C₂×C₇	1	1	yes	characteristic, complemented
19	C₃⋊C₈	C₂×C₇	1	1	yes	characteristic, complemented
20	C₃×C₇	M₁₆	1	1	yes	characteristic, complemented, Hall
21	M₁₆	—	3	1	no	Sylow
22	C₂×C₇	—	6	1	no	—
23	C₂×C₇	C₄×D₃	1	1	yes	characteristic
24	C₂×D₃	C₄×C₇	1	1	yes	characteristic
25	C₃×C₄	C₂²×C₇	1	1	yes	characteristic
26	Dic₃	C₄×C₇	1	1	yes	characteristic
27	C₂×C₄	—	3	1	no	—
28	C₈	—	3	1	no	—
29	C₈	D₃×C₇	1	1	yes	characteristic, complemented
30	C₇	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃	C₂×C₄×C₇	1	1	yes	characteristic
32	D₃	—	2	1	no	—
33	C₂²	—	3	1	no	—
34	C₄	—	3	1	no	—
35	C₄	C₂×D₃×C₇	1	1	yes	characteristic
36	C₃	C₇×M₁₆	1	1	yes	characteristic, complemented, Sylow
37	C₂	—	6	1	no	—
38	C₂	C₄×D₃×C₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₇
Frattini	C₄	C₂×D₃×C₇
Fitting	C₃×C₇×C₈	C₂
socle	C₂×C₃×C₇	C₂×C₄
prime radical	C₂×D₃×C₇	C₄
cosocle	C₃×C₄	C₂²×C₇
hypercenter	C₇×C₈	D₃
hypocenter	C₃	C₇×M₁₆
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₇ ◁ C₇×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×C₈	C₂	semidirect
3. C₇×C₃⋊C₈	C₂	semidirect
10. C₇×C₈	D₃	semidirect
11. ΓL(1,𝔽₂₅)	C₇	direct (N × H)
18. C₃×C₈	C₂×C₇	semidirect
19. C₃⋊C₈	C₂×C₇	semidirect
20. C₃×C₇	M₁₆	semidirect
29. C₈	D₃×C₇	semidirect
30. C₇	ΓL(1,𝔽₂₅)	direct (N × H)
36. C₃	C₇×M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₃×C₇	2
C₃×C₇×C₈	2
C₇×C₃⋊C₈	2
C₇×M₁₆	3
ΓL(1,𝔽₂₅)	7

Subgroups — C7×ΓL(1,𝔽25) #66606

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×ΓL(1,𝔽₂₅) #66606