Subgroups — C₁₁×Q₆₄ #10622258

Order 704.

30 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₁×Q₃₂	C₂	1	2	yes	maximal
2	C₁₁×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₁₁×Q₁₆	—	2	2	no	—
4	C₁₁×C₁₆	C₂²	1	1	yes	characteristic
5	Q₈×C₁₁	—	4	2	no	—
6	C₈×C₁₁	D₄	1	1	yes	characteristic
7	Q₆₄	C₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
8	C₄×C₁₁	—	8	2	no	—
9	C₄×C₁₁	D₈	1	1	yes	characteristic
10	Q₃₂	C₂×C₁₁	1	2	yes	—
11	C₃₂	C₂×C₁₁	1	1	yes	characteristic
12	C₂×C₁₁	D₁₆	1	1	yes	characteristic
13	Q₁₆	—	2	2	no	—
14	C₁₆	C₂²×C₁₁	1	1	yes	characteristic
15	C₁₁	Q₆₄	1	1	yes	characteristic, direct product factor, Sylow
16	Q₈	—	4	2	no	—
17	C₈	D₄×C₁₁	1	1	yes	characteristic
18	C₄	—	8	2	no	—
19	C₄	C₁₁×D₈	1	1	yes	characteristic
20	C₂	C₁₁×D₁₆	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₁₁	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂²×C₁₁
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₁ ◁ C₄×C₁₁ ◁ C₈×C₁₁ ◁ C₁₁×C₁₆ ◁ G

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

N	H ≅ G/N	type
7. Q₆₄	C₁₁	direct (N × H)
15. C₁₁	Q₆₄	direct (N × H)

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