Subgroups — SD₁₂₈×C₂₁₁ #41009050

Order 27008.

38 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₆₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃₂×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₆₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₃₂×C₂₁₁	—	2	1	no	—
5	D₁₆×C₂₁₁	—	2	1	no	—
6	C₃₂×C₂₁₁	C₂²	1	1	yes	characteristic
7	Q₁₆×C₂₁₁	—	4	1	no	—
8	D₈×C₂₁₁	—	4	1	no	—
9	C₁₆×C₂₁₁	D₄	1	1	yes	characteristic
10	Q₈×C₂₁₁	—	8	1	no	—
11	D₄×C₂₁₁	—	8	1	no	—
12	C₈×C₂₁₁	D₈	1	1	yes	characteristic
13	C₂²×C₂₁₁	—	16	1	no	—
14	C₄×C₂₁₁	—	16	1	no	—
15	C₄×C₂₁₁	D₁₆	1	1	yes	characteristic
16	C₂×C₂₁₁	—	32	1	no	—
17	C₂×C₂₁₁	D₃₂	1	1	yes	characteristic
18	C₂₁₁	SD₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
19	SD₁₂₈	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	Q₆₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
21	D₃₂	C₂×C₂₁₁	1	1	yes	characteristic
22	C₆₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
23	Q₃₂	—	2	1	no	—
24	D₁₆	—	2	1	no	—
25	C₃₂	C₂²×C₂₁₁	1	1	yes	characteristic
26	Q₁₆	—	4	1	no	—
27	D₈	—	4	1	no	—
28	C₁₆	D₄×C₂₁₁	1	1	yes	characteristic
29	Q₈	—	8	1	no	—
30	D₄	—	8	1	no	—
31	C₈	D₈×C₂₁₁	1	1	yes	characteristic
32	C₂²	—	16	1	no	—
33	C₄	—	16	1	no	—
34	C₄	D₁₆×C₂₁₁	1	1	yes	characteristic
35	C₂	—	32	1	no	—
36	C₂	D₃₂×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₁₁	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂²×C₂₁₁
prime radical	D₃₂×C₂₁₁	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₁₁ ◁ C₄×C₂₁₁ ◁ C₈×C₂₁₁ ◁ C₁₆×C₂₁₁ ◁ C₃₂×C₂₁₁ ◁ G

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.