Subgroups — AGL(1,ℤ/8) #136

Order 32.

34 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₁₆	C₂	1	2	yes	maximal, complemented
4	D₈	C₂	1	2	yes	maximal, complemented
5	M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³	—	2	1	no	—
7	Q₈	C₂²	1	1	yes	characteristic, complemented
8	D₄	—	2	1	no	—
9	D₄	—	2	1	no	—
10	D₄	C₂²	1	2	yes	—
11	D₄	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₄	—	2	1	no	—
13	C₂×C₄	C₂²	1	1	yes	characteristic
14	C₈	C₂²	1	2	yes	complemented
15	C₂²	—	2	1	no	—
16	C₂²	—	2	2	no	—
17	C₂²	—	4	2	no	—
18	C₂²	D₄	1	1	yes	characteristic
19	C₄	—	2	1	no	—
20	C₄	D₄	1	1	yes	characteristic
21	C₄	C₂³	1	1	yes	characteristic
22	C₂	—	2	1	no	—
23	C₂	—	4	1	no	—
24	C₂	—	4	2	no	—
25	C₂	C₂×D₄	1	1	yes	characteristic

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

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

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