Subgroups — #94817

Order 512.

53 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₂₈	C₂	1	1	yes	maximal, characteristic
2	M₂₅₆	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₂	C₄	1	1	yes	characteristic
4	M₁₂₈	—	2	2	no	—
5	C₂×C₆₄	C₄	1	1	yes	characteristic
6	C₂×C₆₄	C₂²	1	1	yes	characteristic
7	C₁₂₈	—	2	2	no	—
8	C₂²×C₁₆	C₈	1	1	yes	characteristic
9	C₂×C₃₂	—	2	1	no	—
10	C₂×C₃₂	—	2	1	no	—
11	C₂×C₃₂	C₈	1	1	yes	characteristic
12	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
13	C₆₄	—	2	1	no	—
14	C₆₄	D₄	1	2	yes	—
15	C₂²×C₈	C₁₆	1	1	yes	characteristic
16	C₂×C₁₆	—	2	1	no	—
17	C₂×C₁₆	—	2	1	no	—
18	C₂×C₁₆	C₁₆	1	1	yes	characteristic
19	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
20	C₃₂	—	2	1	no	—
21	C₃₂	C₂²⋊C₄	1	1	yes	characteristic
22	C₃₂	M₁₆	1	1	yes	characteristic
23	C₂²×C₄	C₃₂	1	1	yes	characteristic
24	C₂×C₈	—	2	1	no	—
25	C₂×C₈	—	2	1	no	—
26	C₂×C₈	C₃₂	1	1	yes	characteristic
27	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
28	C₁₆	—	2	1	no	—
29	C₁₆	C₂²⋊C₈	1	1	yes	characteristic
30	C₁₆	M₃₂	1	1	yes	characteristic
31	C₂³	C₆₄	1	1	yes	characteristic
32	C₂×C₄	—	2	1	no	—
33	C₂×C₄	—	2	1	no	—
34	C₂×C₄	C₆₄	1	1	yes	characteristic
35	C₂×C₄	C₂×C₃₂	1	1	yes	characteristic
36	C₈	—	2	1	no	—
37	C₈	C₂²⋊C₁₆	1	1	yes	characteristic
38	C₈	M₆₄	1	1	yes	characteristic
39	C₂²	—	2	1	no	—
40	C₂²	—	4	1	no	—
41	C₂²	C₂×C₆₄	1	1	yes	characteristic
42	C₄	—	2	1	no	—
43	C₄	C₂²⋊C₃₂	1	1	yes	characteristic
44	C₄	M₁₂₈	1	1	yes	characteristic
45	C₂	—	2	1	no	—
46	C₂	—	4	1	no	—
47	C₂	C₂²⋊C₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₂²⋊C₄
commutator	C₂²	C₂×C₆₄
Frattini, cosocle	C₂×C₆₄	C₂²
socle	C₂	C₂²⋊C₆₄
prime radical	C₂³	C₆₄
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ C₂×C₆₄ ◁ G
LECS	G ▷ C₂×C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. M₂₅₆	C₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₂₈	2
M₂₅₆	2