Subgroups — C₂×H(5,𝔽₂) #583

Order 64.

449 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)	C₂	1	16	yes	maximal, direct product factor
2	C₂×#41	C₂	1	6	yes	maximal, complemented
3	C₂²×D₄	C₂	1	9	yes	maximal, complemented
4	C₂⁴	C₂²	1	6	yes	complemented
5	#41	C₂²	1	48	yes	complemented
6	C₂×Q₈	C₂²	1	2	yes	complemented
7	C₂×D₄	C₂²	1	18	yes	complemented
8	C₂×D₄	C₂²	1	72	yes	complemented
9	C₂²×C₄	C₂²	1	9	yes	complemented
10	C₂³	—	4	12	no	—
11	C₂³	C₂³	1	9	yes	—
12	C₂³	C₂³	1	24	yes	complemented
13	Q₈	C₂³	1	8	yes	complemented
14	D₄	C₂³	1	72	yes	complemented
15	C₂×C₄	C₂³	1	6	yes	—
16	C₂×C₄	C₂³	1	36	yes	complemented
17	C₂²	—	2	18	no	—
18	C₂²	—	4	24	no	—
19	C₂²	C₂⁴	1	18	yes	—
20	C₂²	C₂⁴	1	1	yes	characteristic
21	C₄	C₂⁴	1	12	yes	—
22	C₂	—	2	18	no	—
23	C₂	H(5,𝔽₂)	1	2	yes	direct product factor
24	C₂	C₂⁵	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂⁴
commutator, Frattini, cosocle	C₂	C₂⁵
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ 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.