Subgroups — M₁₂₈×C₂₂₃ #43250443

Order 28544.

38 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₂×C₂₂₃	C₂	1	1	yes	maximal, characteristic
2	C₆₄×C₂₂₃	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₆×C₂₂₃	C₄	1	1	yes	characteristic
4	C₃₂×C₂₂₃	C₄	1	1	yes	characteristic
5	C₃₂×C₂₂₃	C₂²	1	1	yes	characteristic
6	C₂×C₈×C₂₂₃	C₈	1	1	yes	characteristic
7	C₁₆×C₂₂₃	C₈	1	1	yes	characteristic
8	C₁₆×C₂₂₃	C₂×C₄	1	1	yes	characteristic
9	C₂×C₄×C₂₂₃	C₁₆	1	1	yes	characteristic
10	C₈×C₂₂₃	C₁₆	1	1	yes	characteristic
11	C₈×C₂₂₃	C₂×C₈	1	1	yes	characteristic
12	C₂²×C₂₂₃	C₃₂	1	1	yes	characteristic
13	C₄×C₂₂₃	C₃₂	1	1	yes	characteristic
14	C₄×C₂₂₃	C₂×C₁₆	1	1	yes	characteristic
15	C₂×C₂₂₃	—	2	1	no	—
16	C₂×C₂₂₃	C₂×C₃₂	1	1	yes	characteristic
17	C₂₂₃	M₁₂₈	1	1	yes	characteristic, direct product factor, Sylow
18	M₁₂₈	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₃₂	C₂×C₂₂₃	1	1	yes	characteristic
20	C₆₄	C₂×C₂₂₃	1	2	yes	complemented
21	C₂×C₁₆	C₄×C₂₂₃	1	1	yes	characteristic
22	C₃₂	C₄×C₂₂₃	1	1	yes	characteristic
23	C₃₂	C₂²×C₂₂₃	1	1	yes	characteristic
24	C₂×C₈	C₈×C₂₂₃	1	1	yes	characteristic
25	C₁₆	C₈×C₂₂₃	1	1	yes	characteristic
26	C₁₆	C₂×C₄×C₂₂₃	1	1	yes	characteristic
27	C₂×C₄	C₁₆×C₂₂₃	1	1	yes	characteristic
28	C₈	C₁₆×C₂₂₃	1	1	yes	characteristic
29	C₈	C₂×C₈×C₂₂₃	1	1	yes	characteristic
30	C₂²	C₃₂×C₂₂₃	1	1	yes	characteristic
31	C₄	C₃₂×C₂₂₃	1	1	yes	characteristic
32	C₄	C₂×C₁₆×C₂₂₃	1	1	yes	characteristic
33	C₂	—	2	1	no	—
34	C₂	C₂×C₃₂×C₂₂₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₆₄×C₂₂₃	C₂	semidirect
17. C₂₂₃	M₁₂₈	direct (N × H)
18. M₁₂₈	C₂₂₃	direct (N × H)
20. C₆₄	C₂×C₂₂₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₂×C₂₂₃	2
C₆₄×C₂₂₃	2
M₁₂₈	223

Subgroups — M128×C223 #43250443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₂₈×C₂₂₃ #43250443