Subgroups — C₂×#560 #3525

Order 128.

672 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#560	C₂	1	16	yes	maximal, direct product factor
2	C₂×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#126	C₂	1	2	yes	maximal, complemented
4	C₂×#123	C₂	1	3	yes	maximal, complemented
5	C₂×#121	C₂	1	6	yes	maximal, complemented
6	C₂×#120	C₂	1	3	yes	maximal, complemented
7	C₂⁵	C₂²	1	1	yes	characteristic
8	C₂²×D₄	C₂²	1	6	yes	—
9	C₂²×D₄	C₂²	1	3	yes	—
10	C₂²×D₄	C₂²	1	3	yes	complemented
11	C₂³×C₄	C₂²	1	3	yes	complemented
12	#127	C₂²	1	8	yes	complemented
13	#126	C₂²	1	16	yes	complemented
14	#123	C₂²	1	24	yes	complemented
15	#121	C₂²	1	48	yes	complemented
16	#120	C₂²	1	24	yes	complemented
17	C₂×C₄⋊C₄	C₂²	1	6	yes	complemented
18	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
19	C₂×C₂²⋊C₄	C₂²	1	3	yes	—
20	C₂×C₂²⋊C₄	C₂²	1	3	yes	complemented
21	C₂×C₄²	C₂²	1	1	yes	characteristic, complemented
22	C₂⁴	—	4	6	no	—
23	C₂⁴	C₂³	1	2	yes	—
24	C₂⁴	C₂³	1	1	yes	characteristic
25	C₂⁴	C₂³	1	3	yes	—
26	C₂⁴	C₂³	1	4	yes	—
27	C₂×D₄	—	4	12	no	—
28	C₂×D₄	—	4	6	no	—
29	C₂×D₄	—	4	6	no	—
30	C₂×D₄	C₂³	1	24	yes	—
31	C₂×D₄	C₂³	1	12	yes	—
32	C₂×D₄	C₂³	1	12	yes	complemented
33	C₂²×C₄	—	4	6	no	—
34	C₂²×C₄	C₂³	1	3	yes	—
35	C₂²×C₄	C₂³	1	6	yes	—
36	C₂²×C₄	C₂³	1	12	yes	complemented
37	C₄⋊C₄	C₂³	1	24	yes	complemented
38	C₂²⋊C₄	C₂³	1	24	yes	—
39	C₂²⋊C₄	C₂³	1	12	yes	—
40	C₂²⋊C₄	C₂³	1	12	yes	complemented
41	C₄²	C₂³	1	4	yes	complemented
42	C₂³	—	2	6	no	—
43	C₂³	—	2	12	no	—
44	C₂³	—	2	12	no	—
45	C₂³	—	2	6	no	—
46	C₂³	—	4	12	no	—
47	C₂³	—	4	16	no	—
48	C₂³	C₂⁴	1	4	yes	—
49	C₂³	C₂⁴	1	2	yes	—
50	C₂³	C₂⁴	1	6	yes	—
51	C₂³	C₂⁴	1	1	yes	characteristic
52	D₄	—	4	24	no	—
53	D₄	—	4	12	no	—
54	D₄	—	4	12	no	—
55	C₂×C₄	—	2	12	no	—
56	C₂×C₄	—	2	6	no	—
57	C₂×C₄	—	4	12	no	—
58	C₂×C₄	C₂⁴	1	6	yes	—
59	C₂×C₄	C₂⁴	1	12	yes	—
60	C₂²	—	2	6	no	—
61	C₂²	—	2	12	no	—
62	C₂²	—	2	12	no	—
63	C₂²	—	2	6	no	—
64	C₂²	—	4	16	no	—
65	C₂²	—	4	12	no	—
66	C₂²	—	4	8	no	—
67	C₂²	—	4	4	no	—
68	C₂²	H(5,𝔽₂)	1	6	yes	—
69	C₂²	C₂⁵	1	1	yes	characteristic
70	C₄	—	2	12	no	—
71	C₄	—	2	6	no	—
72	C₂	—	4	6	no	—
73	C₂	—	4	4	no	—
74	C₂	—	4	2	no	—
75	C₂	#560	1	4	yes	direct product factor
76	C₂	C₂×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #560	C₂	direct (N × H)
2. C₂×#127	C₂	semidirect
3. C₂×#126	C₂	semidirect
4. C₂×#123	C₂	semidirect
5. C₂×#121	C₂	semidirect
6. C₂×#120	C₂	semidirect
10. C₂²×D₄	C₂²	semidirect
11. C₂³×C₄	C₂²	semidirect
12. #127	C₂²	semidirect
13. #126	C₂²	semidirect
14. #123	C₂²	semidirect
15. #121	C₂²	semidirect
16. #120	C₂²	semidirect
17. C₂×C₄⋊C₄	C₂²	semidirect
20. C₂×C₂²⋊C₄	C₂²	semidirect
21. C₂×C₄²	C₂²	semidirect
32. C₂×D₄	C₂³	semidirect
36. C₂²×C₄	C₂³	semidirect
37. C₄⋊C₄	C₂³	semidirect
40. C₂²⋊C₄	C₂³	semidirect
41. C₄²	C₂³	semidirect
75. C₂	#560	direct (N × H)

Maximal subgroups

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

subgroup	index
#560	2
C₂×#127	2
C₂×#126	2
C₂×#123	2
C₂×#121	2
C₂×#120	2

Subgroups — C2×#560 #3525

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#560 #3525