Subgroups — C₂×Q₆₄ #2261

Order 128.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂	C₂	1	2	yes	maximal
2	Q₆₄	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₁₆	—	2	2	no	—
5	Q₃₂	—	2	2	no	—
6	Q₃₂	C₂²	1	4	yes	—
7	C₂×C₁₆	C₂²	1	1	yes	characteristic
8	C₃₂	C₂²	1	2	yes	—
9	C₂×Q₈	—	4	2	no	—
10	Q₁₆	—	2	4	no	—
11	Q₁₆	—	4	2	no	—
12	C₂×C₈	D₄	1	1	yes	characteristic
13	C₁₆	D₄	1	1	yes	characteristic
14	C₁₆	C₂³	1	1	yes	characteristic
15	Q₈	—	4	4	no	—
16	Q₈	—	8	2	no	—
17	C₂×C₄	—	8	2	no	—
18	C₂×C₄	D₈	1	1	yes	characteristic
19	C₈	D₈	1	1	yes	characteristic
20	C₈	C₂×D₄	1	1	yes	characteristic
21	C₂²	D₁₆	1	1	yes	characteristic
22	C₄	—	8	4	no	—
23	C₄	D₁₆	1	1	yes	characteristic
24	C₄	C₂×D₈	1	1	yes	characteristic
25	C₂	Q₆₄	1	2	yes	direct product factor
26	C₂	C₂×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

N	H ≅ G/N	type
2. Q₆₄	C₂	direct (N × H)
25. C₂	Q₆₄	direct (N × H)

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