Subgroups — C₄×C₃⋊(C₇⋊C₈) #10620641

Order 672.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊(C₇⋊C₈)	C₂	1	2	yes	maximal
3	C₄×C₇⋊C₈	—	3	1	no	maximal, Hall
4	C₂×C₃×C₄×C₇	C₄	1	2	yes	—
5	C₂×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
6	C₃⋊(C₇⋊C₈)	C₄	1	4	yes	direct product factor
7	C₄²×C₇	D₃	1	1	yes	characteristic
8	C₂×C₇⋊C₈	—	3	2	no	—
9	C₄×C₃⋊C₈	—	7	1	no	maximal, Hall
10	C₂²×C₃×C₇	C₂×C₄	1	1	yes	characteristic
11	C₃×C₄×C₇	C₈	1	4	yes	complemented
12	C₃×C₄×C₇	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₇	Dic₃	1	2	yes	—
14	C₂×C₄×C₇	C₂×D₃	1	1	yes	characteristic
15	C₇⋊C₈	—	3	4	no	—
16	C₃×C₄²	D₇	1	1	yes	characteristic
17	C₂×C₃⋊C₈	—	7	2	no	—
18	C₂×C₃×C₇	C₄²	1	1	yes	characteristic
19	C₂×C₃×C₇	C₂×C₈	1	2	yes	—
20	C₄×C₈	—	21	1	no	Sylow
21	C₂²×C₇	C₂×Dic₃	1	1	yes	characteristic
22	C₄×C₇	C₃⋊C₈	1	4	yes	complemented
23	C₄×C₇	C₄×D₃	1	2	yes	—
24	C₂×C₃×C₄	Dic₇	1	2	yes	—
25	C₂×C₃×C₄	C₂×D₇	1	1	yes	characteristic
26	C₃⋊C₈	—	7	4	no	—
27	C₃×C₇	C₄×C₈	1	1	yes	characteristic, complemented, Hall
28	C₂×C₈	—	21	2	no	—
29	C₄²	D₂₁	1	1	yes	characteristic
30	C₂×C₇	C₂×C₃⋊C₈	1	2	yes	—
31	C₂×C₇	C₄×Dic₃	1	1	yes	characteristic
32	C₂²×C₃	C₂×Dic₇	1	1	yes	characteristic
33	C₃×C₄	C₇⋊C₈	1	4	yes	complemented
34	C₃×C₄	C₄×D₇	1	2	yes	—
35	C₂×C₄	Dic₂₁	1	2	yes	—
36	C₂×C₄	C₂×D₂₁	1	1	yes	characteristic
37	C₈	—	21	4	no	—
38	C₇	C₄×C₃⋊C₈	1	1	yes	characteristic, complemented, Sylow
39	C₂×C₃	C₂×C₇⋊C₈	1	2	yes	—
40	C₂×C₃	C₄×Dic₇	1	1	yes	characteristic
41	C₂²	C₂×Dic₂₁	1	1	yes	characteristic
42	C₄	C₃⋊(C₇⋊C₈)	1	4	yes	direct product factor
43	C₄	C₄×D₂₁	1	2	yes	—
44	C₃	C₄×C₇⋊C₈	1	1	yes	characteristic, complemented, Sylow
45	C₂	C₂×C₃⋊(C₇⋊C₈)	1	2	yes	—
46	C₂	C₄×Dic₂₁	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄²	D₂₁
commutator, hypocenter	C₃×C₇	C₄×C₈
Frattini	C₂×C₄	C₂×D₂₁
Fitting	C₃×C₄²×C₇	C₂
socle, prime radical	C₂²×C₃×C₇	C₂×C₄
cosocle	C₂×C₃×C₄×C₇	C₂²
LCS	G ▷ C₃×C₇
UCS	C₁ ◁ C₄²

Complements

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

N	H ≅ G/N	type
6. C₃⋊(C₇⋊C₈)	C₄	direct (N × H)
11. C₃×C₄×C₇	C₈	semidirect
22. C₄×C₇	C₃⋊C₈	semidirect
27. C₃×C₇	C₄×C₈	semidirect
33. C₃×C₄	C₇⋊C₈	semidirect
38. C₇	C₄×C₃⋊C₈	semidirect
42. C₄	C₃⋊(C₇⋊C₈)	direct (N × H)
44. C₃	C₄×C₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄²×C₇	2
C₂×C₃⋊(C₇⋊C₈)	2
C₄×C₇⋊C₈	3
C₄×C₃⋊C₈	7

Subgroups — C4×C3⋊(C7⋊C8) #10620641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₃⋊(C₇⋊C₈) #10620641