Subgroups — C₂×M₁₂₈ #13736

Order 256.

67 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₂	C₂	1	1	yes	maximal, characteristic
2	M₁₂₈	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₆₄	C₂	1	2	yes	maximal, complemented
4	C₂²×C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₃₂	C₄	1	1	yes	characteristic
6	C₂×C₃₂	C₄	1	2	yes	—
7	C₂×C₃₂	C₂²	1	2	yes	—
8	C₂×C₃₂	C₂²	1	1	yes	characteristic
9	C₆₄	C₂²	1	4	yes	complemented
10	C₂²×C₈	C₈	1	1	yes	characteristic
11	C₂×C₁₆	C₈	1	1	yes	characteristic
12	C₂×C₁₆	C₈	1	2	yes	—
13	C₂×C₁₆	C₂×C₄	1	2	yes	—
14	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
15	C₃₂	C₂×C₄	1	2	yes	—
16	C₃₂	C₂×C₄	1	1	yes	characteristic
17	C₃₂	C₂³	1	1	yes	characteristic
18	C₂²×C₄	C₁₆	1	1	yes	characteristic
19	C₂×C₈	C₁₆	1	2	yes	—
20	C₂×C₈	C₁₆	1	1	yes	characteristic
21	C₂×C₈	C₂×C₈	1	2	yes	—
22	C₂×C₈	C₂×C₈	1	1	yes	characteristic
23	C₁₆	C₂×C₈	1	2	yes	—
24	C₁₆	C₂×C₈	1	1	yes	characteristic
25	C₁₆	C₂²×C₄	1	1	yes	characteristic
26	C₂³	C₃₂	1	1	yes	characteristic
27	C₂×C₄	C₃₂	1	2	yes	—
28	C₂×C₄	C₃₂	1	1	yes	characteristic
29	C₂×C₄	C₂×C₁₆	1	2	yes	—
30	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
31	C₈	C₂×C₁₆	1	2	yes	—
32	C₈	C₂×C₁₆	1	1	yes	characteristic
33	C₈	C₂²×C₈	1	1	yes	characteristic
34	C₂²	—	2	2	no	—
35	C₂²	C₂×C₃₂	1	2	yes	—
36	C₂²	C₂×C₃₂	1	1	yes	characteristic
37	C₄	C₂×C₃₂	1	1	yes	characteristic
38	C₄	C₂×C₃₂	1	2	yes	—
39	C₄	C₂²×C₁₆	1	1	yes	characteristic
40	C₂	—	2	2	no	—
41	C₂	M₁₂₈	1	2	yes	direct product factor
42	C₂	C₂²×C₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃₂	C₂²
commutator	C₂	C₂²×C₃₂
Frattini, cosocle	C₃₂	C₂³
socle	C₂²	C₂×C₃₂
prime radical	C₂³	C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. M₁₂₈	C₂	direct (N × H)
3. C₂×C₆₄	C₂	semidirect
9. C₆₄	C₂²	semidirect
41. C₂	M₁₂₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₂	2
M₁₂₈	2
C₂×C₆₄	2

Subgroups — C2×M128 #13736

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₂₈ #13736