Subgroups — Q₁₂₈×C₂₂₃ #43250446

Order 28544.

36 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₆₄×C₂₂₃	C₂	1	2	yes	maximal
2	C₆₄×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	Q₃₂×C₂₂₃	—	2	2	no	—
4	C₃₂×C₂₂₃	C₂²	1	1	yes	characteristic
5	Q₁₆×C₂₂₃	—	4	2	no	—
6	C₁₆×C₂₂₃	D₄	1	1	yes	characteristic
7	Q₈×C₂₂₃	—	8	2	no	—
8	C₈×C₂₂₃	D₈	1	1	yes	characteristic
9	C₄×C₂₂₃	—	16	2	no	—
10	C₄×C₂₂₃	D₁₆	1	1	yes	characteristic
11	C₂×C₂₂₃	D₃₂	1	1	yes	characteristic
12	C₂₂₃	Q₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
13	Q₁₂₈	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
14	Q₆₄	C₂×C₂₂₃	1	2	yes	—
15	C₆₄	C₂×C₂₂₃	1	1	yes	characteristic
16	Q₃₂	—	2	2	no	—
17	C₃₂	C₂²×C₂₂₃	1	1	yes	characteristic
18	Q₁₆	—	4	2	no	—
19	C₁₆	D₄×C₂₂₃	1	1	yes	characteristic
20	Q₈	—	8	2	no	—
21	C₈	D₈×C₂₂₃	1	1	yes	characteristic
22	C₄	—	16	2	no	—
23	C₄	D₁₆×C₂₂₃	1	1	yes	characteristic
24	C₂	D₃₂×C₂₂₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₂₂₃	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂²×C₂₂₃
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₂₃ ◁ C₄×C₂₂₃ ◁ C₈×C₂₂₃ ◁ C₁₆×C₂₂₃ ◁ C₃₂×C₂₂₃ ◁ G

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

N	H ≅ G/N	type
12. C₂₂₃	Q₁₂₈	direct (N × H)
13. Q₁₂₈	C₂₂₃	direct (N × H)

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