Subgroups — C₂×#568 #3572

Order 128.

730 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#41	C₂	1	1	yes	maximal, characteristic
2	#568	C₂	1	16	yes	maximal, direct product factor
3	C₂×#131	C₂	1	8	yes	maximal, complemented
4	C₂²×M₁₆	C₂	1	6	yes	maximal, complemented
5	C₂×#41	C₄	1	4	yes	—
6	C₂×#41	C₄	1	8	yes	—
7	C₂×#41	C₂²	1	4	yes	—
8	C₂×#41	C₂²	1	8	yes	—
9	C₂²×Q₈	C₄	1	1	yes	characteristic
10	C₂²×D₄	C₄	1	3	yes	—
11	C₂³×C₄	C₂²	1	3	yes	—
12	#131	C₂²	1	64	yes	complemented
13	C₂×M₁₆	C₂²	1	4	yes	—
14	C₂×M₁₆	C₂²	1	12	yes	complemented
15	C₂×M₁₆	C₂²	1	48	yes	complemented
16	C₂²×C₈	C₂²	1	12	yes	complemented
17	C₂⁴	C₂×C₄	1	3	yes	—
18	#41	C₂×C₄	1	16	yes	—
19	#41	C₂×C₄	1	32	yes	—
20	#41	C₂³	1	16	yes	—
21	C₂×Q₈	C₂×C₄	1	4	yes	—
22	C₂×Q₈	C₂×C₄	1	8	yes	—
23	C₂×D₄	C₂×C₄	1	12	yes	—
24	C₂×D₄	C₂×C₄	1	24	yes	—
25	C₂²×C₄	C₂×C₄	1	3	yes	—
26	C₂²×C₄	C₂×C₄	1	6	yes	—
27	C₂²×C₄	C₂×C₄	1	12	yes	—
28	C₂²×C₄	C₂³	1	12	yes	—
29	C₂²×C₄	C₂³	1	6	yes	—
30	C₂²×C₄	C₂³	1	1	yes	characteristic
31	M₁₆	C₂³	1	16	yes	—
32	M₁₆	C₂³	1	48	yes	complemented
33	C₂×C₈	C₂³	1	8	yes	—
34	C₂×C₈	C₂³	1	48	yes	complemented
35	C₂³	—	4	6	no	—
36	C₂³	C₂²×C₄	1	1	yes	characteristic
37	C₂³	C₂²×C₄	1	6	yes	—
38	C₂³	C₂²×C₄	1	12	yes	—
39	Q₈	C₂²×C₄	1	16	yes	—
40	D₄	C₂²×C₄	1	48	yes	—
41	C₂×C₄	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄	C₂²×C₄	1	2	yes	—
43	C₂×C₄	C₂²×C₄	1	12	yes	—
44	C₂×C₄	C₂²×C₄	1	6	yes	—
45	C₂×C₄	C₂²×C₄	1	24	yes	—
46	C₂×C₄	C₂²×C₄	1	12	yes	—
47	C₂×C₄	C₂⁴	1	12	yes	—
48	C₂×C₄	C₂⁴	1	2	yes	—
49	C₂×C₄	C₂⁴	1	1	yes	characteristic
50	C₈	C₂⁴	1	16	yes	—
51	C₂²	—	2	2	no	—
52	C₂²	—	2	12	no	—
53	C₂²	—	4	12	no	—
54	C₂²	C₂³×C₄	1	2	yes	—
55	C₂²	C₂³×C₄	1	12	yes	—
56	C₂²	C₂³×C₄	1	1	yes	characteristic
57	C₄	C₂³×C₄	1	2	yes	—
58	C₄	C₂³×C₄	1	1	yes	characteristic
59	C₄	C₂³×C₄	1	12	yes	—
60	C₄	C₂⁵	1	1	yes	characteristic
61	C₂	—	2	2	no	—
62	C₂	—	2	12	no	—
63	C₂	#568	1	2	yes	direct product factor
64	C₂	C₂⁴×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂⁴
commutator	C₂	C₂⁴×C₄
Frattini, cosocle	C₄	C₂⁵
socle	C₂²	C₂³×C₄
prime radical	C₂²×#41	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #568	C₂	direct (N × H)
3. C₂×#131	C₂	semidirect
4. C₂²×M₁₆	C₂	semidirect
12. #131	C₂²	semidirect
14. C₂×M₁₆	C₂²	semidirect
15. C₂×M₁₆	C₂²	semidirect
16. C₂²×C₈	C₂²	semidirect
32. M₁₆	C₂³	semidirect
34. C₂×C₈	C₂³	semidirect
63. C₂	#568	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×#41	2
#568	2
C₂×#131	2
C₂²×M₁₆	2

Subgroups — C2×#568 #3572

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#568 #3572