Subgroups — C₂×#1398 #13550

Order 256.

111 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×#346	C₂	1	1	yes	maximal, characteristic
3	#1398	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	M₆₄	C₄	1	4	yes	complemented
8	M₆₄	C₂²	1	4	yes	—
9	C₂×C₃₂	C₄	1	4	yes	complemented
10	#346	C₄	1	2	yes	—
11	#346	C₂²	1	2	yes	—
12	C₂²×C₈	C₈	1	1	yes	characteristic
13	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²	C₈	1	1	yes	characteristic
15	C₂×C₁₆	C₈	1	2	yes	—
16	C₂×C₁₆	C₈	1	2	yes	—
17	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₆	C₂³	1	1	yes	characteristic
22	C₄×C₈	C₈	1	2	yes	—
23	C₄×C₈	C₂×C₄	1	2	yes	—
24	C₃₂	C₂×C₄	1	8	yes	complemented
25	C₂²×C₄	—	2	1	no	—
26	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈	—	2	1	no	—
28	C₂×C₈	—	2	1	no	—
29	C₂×C₈	C₄²	1	1	yes	characteristic
30	C₂×C₈	C₄²	1	1	yes	characteristic
31	C₂×C₈	C₂×C₈	1	2	yes	—
32	C₂×C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
36	C₄²	—	2	1	no	—
37	C₄²	C₂×C₈	1	2	yes	—
38	C₁₆	C₄²	1	2	yes	—
39	C₁₆	C₂×C₈	1	4	yes	—
40	C₁₆	C₂²×C₄	1	2	yes	—
41	C₂³	M₃₂	1	1	yes	characteristic
42	C₂×C₄	—	2	2	no	—
43	C₂×C₄	—	4	2	no	—
44	C₂×C₄	C₄×C₈	1	1	yes	characteristic
45	C₂×C₄	C₄×C₈	1	1	yes	characteristic
46	C₂×C₄	M₃₂	1	1	yes	characteristic
47	C₂×C₄	M₃₂	1	1	yes	characteristic
48	C₂×C₄	M₃₂	1	1	yes	characteristic
49	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	C₄×C₈	1	1	yes	characteristic
52	C₈	C₄×C₈	1	1	yes	characteristic
53	C₈	C₂×C₄²	1	1	yes	characteristic
54	C₈	C₂²×C₈	1	1	yes	characteristic
55	C₂²	—	2	2	no	—
56	C₂²	#346	1	1	yes	characteristic
57	C₂²	#346	1	1	yes	characteristic
58	C₂²	C₂×M₃₂	1	1	yes	characteristic
59	C₄	—	4	2	no	—
60	C₄	#346	1	1	yes	characteristic
61	C₄	#346	1	1	yes	characteristic
62	C₄	C₂×C₄×C₈	1	1	yes	characteristic
63	C₄	C₂×M₃₂	1	1	yes	characteristic
64	C₂	—	2	1	no	—
65	C₂	—	2	1	no	—
66	C₂	#1398	1	2	yes	direct product factor
67	C₂	C₂×#346	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄²
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂²	#346
prime radical	C₂³	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ 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. #1398	C₂	direct (N × H)
7. M₆₄	C₄	semidirect
9. C₂×C₃₂	C₄	semidirect
24. C₃₂	C₂×C₄	semidirect
66. C₂	#1398	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×#346	2
#1398	2

Subgroups — C2×#1398 #13550

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1398 #13550