Subgroups — C₂×#518 #3425

Order 128.

960 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄	C₂	1	1	yes	maximal, characteristic
2	#518	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×D₄	C₂	1	8	yes	maximal, complemented
4	C₂×#117	C₂	1	2	yes	maximal, complemented
5	C₂²×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
6	C₂⁵	C₂²	1	2	yes	—
7	C₂²×D₄	C₄	1	16	yes	complemented
8	C₂²×D₄	C₂²	1	4	yes	—
9	C₂²×D₄	C₂²	1	8	yes	—
10	C₂³×C₄	C₂²	1	1	yes	characteristic
11	C₂³×C₄	C₂²	1	8	yes	complemented
12	C₄×D₄	C₂²	1	64	yes	complemented
13	#117	C₂²	1	16	yes	complemented
14	C₂×C₄⋊C₄	C₂²	1	4	yes	complemented
15	C₂×C₂²⋊C₄	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₄	C₂²	1	32	yes	complemented
17	C₂×C₂²⋊C₄	C₂²	1	8	yes	complemented
18	C₂×C₄²	C₂²	1	4	yes	complemented
19	C₂⁴	—	4	8	no	—
20	C₂⁴	C₂×C₄	1	16	yes	complemented
21	C₂⁴	C₂³	1	1	yes	characteristic
22	C₂⁴	C₂³	1	4	yes	—
23	C₂⁴	C₂³	1	8	yes	—
24	C₂×D₄	C₂×C₄	1	32	yes	complemented
25	C₂×D₄	C₂×C₄	1	64	yes	complemented
26	C₂×D₄	C₂³	1	16	yes	—
27	C₂²×C₄	—	4	16	no	—
28	C₂²×C₄	C₂×C₄	1	8	yes	complemented
29	C₂²×C₄	C₂³	1	8	yes	—
30	C₂²×C₄	C₂³	1	2	yes	—
31	C₂²×C₄	C₂³	1	4	yes	—
32	C₂²×C₄	C₂³	1	32	yes	complemented
33	C₄⋊C₄	C₂³	1	16	yes	complemented
34	C₂²⋊C₄	C₂³	1	16	yes	—
35	C₂²⋊C₄	C₂³	1	32	yes	complemented
36	C₄²	C₂³	1	16	yes	complemented
37	C₂³	—	2	16	no	—
38	C₂³	—	2	4	no	—
39	C₂³	—	4	32	no	—
40	C₂³	—	4	16	no	—
41	C₂³	C₂²×C₄	1	4	yes	—
42	C₂³	C₂²×C₄	1	16	yes	—
43	C₂³	C₂²×C₄	1	32	yes	complemented
44	C₂³	C₂⁴	1	2	yes	—
45	C₂³	C₂⁴	1	8	yes	—
46	C₂³	C₂⁴	1	1	yes	characteristic
47	D₄	C₂²×C₄	1	64	yes	complemented
48	C₂×C₄	—	2	16	no	—
49	C₂×C₄	—	4	32	no	—
50	C₂×C₄	C₂²×C₄	1	8	yes	—
51	C₂×C₄	C₂²×C₄	1	16	yes	complemented
52	C₂×C₄	C₂⁴	1	16	yes	—
53	C₂×C₄	C₂⁴	1	4	yes	—
54	C₂²	—	2	8	no	—
55	C₂²	—	2	32	no	—
56	C₂²	—	2	16	no	—
57	C₂²	—	2	4	no	—
58	C₂²	—	4	32	no	—
59	C₂²	C₂³×C₄	1	4	yes	—
60	C₂²	C₂³×C₄	1	16	yes	—
61	C₂²	C₂³×C₄	1	2	yes	—
62	C₂²	H(5,𝔽₂)	1	4	yes	—
63	C₂²	C₂⁵	1	1	yes	characteristic
64	C₄	—	2	16	no	—
65	C₄	C₂³×C₄	1	8	yes	—
66	C₂	—	2	4	no	—
67	C₂	—	2	16	no	—
68	C₂	#518	1	4	yes	direct product factor
69	C₂	C₂⁴×C₄	1	1	yes	characteristic
70	C₂	C₂×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator	C₂	C₂⁴×C₄
Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂³×D₄	C₂
LCS	G ▷ C₂ ▷ 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
2. #518	C₂	direct (N × H)
3. C₂×C₄×D₄	C₂	semidirect
4. C₂×#117	C₂	semidirect
5. C₂²×C₂²⋊C₄	C₂	semidirect
7. C₂²×D₄	C₄	semidirect
11. C₂³×C₄	C₂²	semidirect
12. C₄×D₄	C₂²	semidirect
13. #117	C₂²	semidirect
14. C₂×C₄⋊C₄	C₂²	semidirect
16. C₂×C₂²⋊C₄	C₂²	semidirect
17. C₂×C₂²⋊C₄	C₂²	semidirect
18. C₂×C₄²	C₂²	semidirect
20. C₂⁴	C₂×C₄	semidirect
24. C₂×D₄	C₂×C₄	semidirect
25. C₂×D₄	C₂×C₄	semidirect
28. C₂²×C₄	C₂×C₄	semidirect
32. C₂²×C₄	C₂³	semidirect
33. C₄⋊C₄	C₂³	semidirect
35. C₂²⋊C₄	C₂³	semidirect
36. C₄²	C₂³	semidirect
43. C₂³	C₂²×C₄	semidirect
47. D₄	C₂²×C₄	semidirect
51. C₂×C₄	C₂²×C₄	semidirect
68. C₂	#518	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄	2
#518	2
C₂×C₄×D₄	2
C₂×#117	2
C₂²×C₂²⋊C₄	2

Subgroups — C2×#518 #3425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#518 #3425