Subgroups — C₄×M₆₄ #13548

Order 256.

125 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#1397	C₂	1	2	yes	maximal, complemented
4	C₄×C₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈	C₄	1	1	yes	characteristic
8	M₆₄	C₄	1	8	yes	direct product factor
9	C₂×C₃₂	C₂²	1	4	yes	—
10	C₄×C₁₆	C₄	1	1	yes	characteristic
11	C₄×C₁₆	C₄	1	1	yes	characteristic
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈	C₈	1	1	yes	characteristic
15	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²	C₈	1	1	yes	characteristic
17	C₂×C₁₆	C₈	1	4	yes	—
18	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆	C₂×C₄	1	4	yes	—
20	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂³	1	1	yes	characteristic
23	C₄×C₈	C₈	1	1	yes	characteristic
24	C₄×C₈	C₈	1	1	yes	characteristic
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₃₂	C₂×C₄	1	8	yes	complemented
28	C₂²×C₄	C₁₆	1	2	yes	—
29	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈	C₁₆	1	4	yes	—
31	C₂×C₈	C₄²	1	2	yes	—
32	C₂×C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈	C₂×C₈	1	2	yes	—
34	C₂×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₄²	C₁₆	1	2	yes	—
38	C₄²	C₂×C₈	1	1	yes	characteristic
39	C₄²	C₂×C₈	1	1	yes	characteristic
40	C₁₆	C₄²	1	2	yes	—
41	C₁₆	C₂×C₈	1	2	yes	—
42	C₁₆	C₂×C₈	1	2	yes	—
43	C₁₆	C₂²×C₄	1	2	yes	—
44	C₂³	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄	—	2	4	no	—
46	C₂×C₄	C₄×C₈	1	2	yes	—
47	C₂×C₄	C₂×C₁₆	1	2	yes	—
48	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄	C₂×C₁₆	1	2	yes	—
50	C₂×C₄	C₂×C₁₆	1	2	yes	—
51	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
52	C₈	C₄×C₈	1	2	yes	—
53	C₈	C₂×C₁₆	1	2	yes	—
54	C₈	C₂×C₁₆	1	2	yes	—
55	C₈	C₂×C₄²	1	1	yes	characteristic
56	C₈	C₂²×C₈	1	1	yes	characteristic
57	C₂²	—	2	2	no	—
58	C₂²	C₄×C₁₆	1	2	yes	—
59	C₂²	C₂²×C₁₆	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	C₄×C₁₆	1	2	yes	—
62	C₄	M₆₄	1	4	yes	direct product factor
63	C₄	C₂×C₄×C₈	1	1	yes	characteristic
64	C₄	C₂²×C₁₆	1	1	yes	characteristic
65	C₂	—	2	2	no	—
66	C₂	C₂×C₄×C₁₆	1	1	yes	characteristic
67	C₂	C₂×M₆₄	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #1397	C₂	semidirect
4. C₄×C₃₂	C₂	semidirect
8. M₆₄	C₄	direct (N × H)
27. C₃₂	C₂×C₄	semidirect
62. C₄	M₆₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×M₆₄	2
C₂×C₄×C₁₆	2
#1397	2
C₄×C₃₂	2

Subgroups — C4×M64 #13548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×M₆₄ #13548