Subgroups — C₂×#1397 #13547

Order 256.

127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²×C₁₆	C₂²
commutator	C₂	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³
socle, prime radical	C₂³	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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	direct (N × H)
7. C₂×C₃₂	C₄	semidirect
21. C₃₂	C₂×C₄	semidirect
53. C₂	#1397	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
C₂×C₄×C₁₆	2
#1397	2

Subgroups — C2×#1397 #13547

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1397 #13547