Subgroups — C₇×AGL(1,ℤ/8) #6539

Order 224.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇×SD₁₆	C₂	1	2	yes	maximal, complemented
4	C₇×D₈	C₂	1	2	yes	maximal, complemented
5	C₇×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₇	—	2	1	no	—
7	C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
8	C₇×D₄	—	2	1	no	—
9	C₇×D₄	—	2	1	no	—
10	C₇×D₄	C₂²	1	2	yes	—
11	C₇×D₄	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₄×C₇	—	2	1	no	—
13	C₂×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₇×C₈	C₂²	1	2	yes	complemented
15	AGL(1,ℤ/8)	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂²×C₇	—	2	1	no	—
17	C₂²×C₇	—	2	2	no	—
18	C₂²×C₇	—	4	2	no	—
19	C₂²×C₇	D₄	1	1	yes	characteristic
20	C₄×C₇	—	2	1	no	—
21	C₄×C₇	D₄	1	1	yes	characteristic
22	C₄×C₇	C₂³	1	1	yes	characteristic
23	#41	C₂×C₇	1	1	yes	characteristic, complemented
24	C₂×D₄	C₂×C₇	1	1	yes	characteristic, complemented
25	SD₁₆	C₂×C₇	1	2	yes	complemented
26	D₈	C₂×C₇	1	2	yes	complemented
27	M₁₆	C₂×C₇	1	1	yes	characteristic, complemented
28	C₂×C₇	—	2	1	no	—
29	C₂×C₇	—	4	1	no	—
30	C₂×C₇	—	4	2	no	—
31	C₂×C₇	C₂×D₄	1	1	yes	characteristic
32	C₂³	—	2	1	no	—
33	Q₈	C₂²×C₇	1	1	yes	characteristic, complemented
34	D₄	—	2	1	no	—
35	D₄	—	2	1	no	—
36	D₄	C₂²×C₇	1	2	yes	—
37	D₄	C₂²×C₇	1	1	yes	characteristic, complemented
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	C₂²×C₇	1	1	yes	characteristic
40	C₈	C₂²×C₇	1	2	yes	complemented
41	C₇	AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
42	C₂²	—	2	1	no	—
43	C₂²	—	2	2	no	—
44	C₂²	—	4	2	no	—
45	C₂²	C₇×D₄	1	1	yes	characteristic
46	C₄	—	2	1	no	—
47	C₄	C₇×D₄	1	1	yes	characteristic
48	C₄	C₂³×C₇	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	4	2	no	—
51	C₂	—	4	1	no	—
52	C₂	C₂×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₇	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂³×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₇ ◁ C₂×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇×#41	C₂	semidirect
2. C₂×C₇×D₄	C₂	semidirect
3. C₇×SD₁₆	C₂	semidirect
4. C₇×D₈	C₂	semidirect
5. C₇×M₁₆	C₂	semidirect
7. C₇×Q₈	C₂²	semidirect
11. C₇×D₄	C₂²	semidirect
14. C₇×C₈	C₂²	semidirect
15. AGL(1,ℤ/8)	C₇	direct (N × H)
23. #41	C₂×C₇	semidirect
24. C₂×D₄	C₂×C₇	semidirect
25. SD₁₆	C₂×C₇	semidirect
26. D₈	C₂×C₇	semidirect
27. M₁₆	C₂×C₇	semidirect
33. Q₈	C₂²×C₇	semidirect
37. D₄	C₂²×C₇	semidirect
40. C₈	C₂²×C₇	semidirect
41. C₇	AGL(1,ℤ/8)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#41	2
C₂×C₇×D₄	2
C₇×SD₁₆	2
C₇×D₈	2
C₇×M₁₆	2
AGL(1,ℤ/8)	7

Subgroups — C7×AGL(1,ℤ/8) #6539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×AGL(1,ℤ/8) #6539