Subgroups — C₂×SD₁₂₈ #13739

Order 256.

67 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃₂	C₂	1	1	yes	maximal, characteristic
3	SD₁₂₈	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₃₂	—	2	1	no	—
6	C₂×D₁₆	—	2	1	no	—
7	Q₆₄	—	2	1	no	—
8	Q₆₄	C₂²	1	2	yes	complemented
9	D₃₂	—	2	1	no	—
10	D₃₂	C₂²	1	2	yes	—
11	C₂×C₃₂	C₂²	1	1	yes	characteristic
12	C₆₄	C₂²	1	2	yes	complemented
13	C₂×Q₁₆	—	4	1	no	—
14	C₂×D₈	—	4	1	no	—
15	Q₃₂	—	2	2	no	—
16	Q₃₂	—	4	1	no	—
17	D₁₆	—	2	2	no	—
18	D₁₆	—	4	1	no	—
19	C₂×C₁₆	D₄	1	1	yes	characteristic
20	C₃₂	D₄	1	1	yes	characteristic
21	C₃₂	C₂³	1	1	yes	characteristic
22	C₂×Q₈	—	8	1	no	—
23	C₂×D₄	—	8	1	no	—
24	Q₁₆	—	4	2	no	—
25	Q₁₆	—	8	1	no	—
26	D₈	—	4	2	no	—
27	D₈	—	8	1	no	—
28	C₂×C₈	D₈	1	1	yes	characteristic
29	C₁₆	D₈	1	1	yes	characteristic
30	C₁₆	C₂×D₄	1	1	yes	characteristic
31	C₂³	—	16	1	no	—
32	Q₈	—	8	2	no	—
33	Q₈	—	16	1	no	—
34	D₄	—	8	2	no	—
35	D₄	—	16	1	no	—
36	C₂×C₄	—	16	1	no	—
37	C₂×C₄	D₁₆	1	1	yes	characteristic
38	C₈	D₁₆	1	1	yes	characteristic
39	C₈	C₂×D₈	1	1	yes	characteristic
40	C₂²	—	16	2	no	—
41	C₂²	—	32	2	no	—
42	C₂²	D₃₂	1	1	yes	characteristic
43	C₄	—	16	2	no	—
44	C₄	D₃₂	1	1	yes	characteristic
45	C₄	C₂×D₁₆	1	1	yes	characteristic
46	C₂	—	32	2	no	—
47	C₂	SD₁₂₈	1	2	yes	direct product factor
48	C₂	C₂×D₃₂	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	D₃₂
commutator, Frattini, cosocle	C₃₂	C₂³
prime radical	C₂×D₃₂	C₂
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ G
LECS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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.