Subgroups — #94818

Order 512.

52 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13737	C₂	1	1	yes	maximal, characteristic
2	M₂₅₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#2258	C₄	1	1	yes	characteristic
5	M₁₂₈	—	2	1	no	—
6	M₁₂₈	C₄	1	1	yes	characteristic
7	C₂×C₆₄	—	2	1	no	—
8	C₂×C₆₄	C₂²	1	1	yes	characteristic
9	C₁₂₈	—	2	1	no	—
10	C₁₂₈	—	2	1	no	—
11	#504	C₈	1	1	yes	characteristic
12	M₆₄	—	2	1	no	—
13	M₆₄	C₈	1	1	yes	characteristic
14	C₂×C₃₂	—	2	1	no	—
15	C₂×C₃₂	C₂×C₄	1	1	yes	characteristic
16	C₆₄	—	2	1	no	—
17	C₆₄	D₄	1	1	yes	characteristic
18	C₆₄	D₄	1	1	yes	characteristic
19	#131	C₁₆	1	1	yes	characteristic
20	M₃₂	—	2	1	no	—
21	M₃₂	C₁₆	1	1	yes	characteristic
22	C₂×C₁₆	—	2	1	no	—
23	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
24	C₃₂	—	2	1	no	—
25	C₃₂	C₂²⋊C₄	1	1	yes	characteristic
26	C₃₂	M₁₆	1	1	yes	characteristic
27	#41	C₃₂	1	1	yes	characteristic
28	M₁₆	—	2	1	no	—
29	M₁₆	C₃₂	1	1	yes	characteristic
30	C₂×C₈	—	2	1	no	—
31	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
32	C₁₆	—	2	1	no	—
33	C₁₆	C₂²⋊C₈	1	1	yes	characteristic
34	C₁₆	M₃₂	1	1	yes	characteristic
35	Q₈	C₆₄	1	1	yes	characteristic
36	D₄	—	2	1	no	—
37	D₄	C₆₄	1	1	yes	characteristic
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	C₂×C₃₂	1	1	yes	characteristic
40	C₈	—	2	1	no	—
41	C₈	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₈	M₆₄	1	1	yes	characteristic
43	C₂²	—	2	1	no	—
44	C₂²	M₁₂₈	1	1	yes	characteristic
45	C₄	—	2	1	no	—
46	C₄	C₂²⋊C₃₂	1	1	yes	characteristic
47	C₄	C₂×C₆₄	1	1	yes	characteristic
48	C₂	—	2	1	no	—
49	C₂	—	4	1	no	—
50	C₂	C₂²⋊C₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₆₄	D₄
commutator	C₄	C₂×C₆₄
Frattini, cosocle	C₂×C₆₄	C₂²
socle	C₂	C₂²⋊C₆₄
prime radical	#41	C₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₆₄ ◁ 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
3. C₂×C₁₂₈	C₂	semidirect

Maximal subgroups

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

subgroup	index
#13737	2
M₂₅₆	2
C₂×C₁₂₈	2