Subgroups — C₂×#520 #3427

Order 128.

820 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	#520	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄×Q₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₄	C₂	1	6	yes	maximal, complemented
5	C₂×#117	C₂	1	3	yes	maximal, complemented
6	C₂²×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
7	C₂×#41	C₄	1	16	yes	complemented
8	C₂×#41	C₂²	1	8	yes	—
9	C₂²×Q₈	C₂²	1	1	yes	characteristic
10	C₂²×D₄	C₂²	1	3	yes	—
11	C₂³×C₄	C₂²	1	6	yes	—
12	C₂³×C₄	C₂²	1	3	yes	—
13	C₄×Q₈	C₂²	1	16	yes	complemented
14	C₄×D₄	C₂²	1	48	yes	complemented
15	#117	C₂²	1	24	yes	complemented
16	C₂×C₄⋊C₄	C₂²	1	3	yes	—
17	C₂×C₄⋊C₄	C₂²	1	6	yes	—
18	C₂×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄	C₂²	1	24	yes	complemented
20	C₂×C₂²⋊C₄	C₂²	1	6	yes	—
21	C₂×C₄²	C₂²	1	6	yes	—
22	C₂⁴	C₂³	1	3	yes	—
23	#41	C₂×C₄	1	64	yes	complemented
24	C₂×Q₈	C₂×C₄	1	8	yes	complemented
25	C₂×Q₈	C₂³	1	4	yes	—
26	C₂×D₄	C₂×C₄	1	24	yes	complemented
27	C₂×D₄	C₂³	1	12	yes	—
28	C₂²×C₄	—	4	12	no	—
29	C₂²×C₄	C₂×C₄	1	24	yes	complemented
30	C₂²×C₄	C₂³	1	2	yes	—
31	C₂²×C₄	C₂³	1	6	yes	—
32	C₂²×C₄	C₂³	1	24	yes	—
33	C₂²×C₄	C₂³	1	3	yes	—
34	C₂²×C₄	C₂³	1	1	yes	characteristic
35	C₂²×C₄	C₂³	1	12	yes	—
36	C₄⋊C₄	C₂³	1	12	yes	—
37	C₄⋊C₄	C₂³	1	24	yes	—
38	C₄⋊C₄	C₂³	1	4	yes	—
39	C₂²⋊C₄	C₂³	1	24	yes	—
40	C₄²	C₂³	1	24	yes	—
41	C₂³	—	2	6	no	—
42	C₂³	—	2	6	no	—
43	C₂³	C₂²×C₄	1	12	yes	—
44	C₂³	C₂⁴	1	6	yes	—
45	C₂³	C₂⁴	1	1	yes	characteristic
46	Q₈	C₂²×C₄	1	16	yes	complemented
47	D₄	C₂²×C₄	1	48	yes	complemented
48	C₂×C₄	—	2	12	no	—
49	C₂×C₄	—	2	4	no	—
50	C₂×C₄	—	4	24	no	—
51	C₂×C₄	C₂²×C₄	1	4	yes	—
52	C₂×C₄	C₂²×C₄	1	12	yes	—
53	C₂×C₄	C₂²×C₄	1	48	yes	complemented
54	C₂×C₄	C₂⁴	1	4	yes	—
55	C₂×C₄	C₂⁴	1	12	yes	—
56	C₂×C₄	C₂⁴	1	6	yes	—
57	C₂×C₄	C₂⁴	1	2	yes	—
58	C₂²	—	2	24	no	—
59	C₂²	—	2	6	no	—
60	C₂²	—	2	6	no	—
61	C₂²	C₂³×C₄	1	12	yes	—
62	C₂²	C₂³×C₄	1	2	yes	—
63	C₂²	H(5,𝔽₂)	1	2	yes	—
64	C₂²	2₋¹⁺⁴	1	2	yes	—
65	C₂²	C₂⁵	1	1	yes	characteristic
66	C₄	—	2	12	no	—
67	C₄	—	2	4	no	—
68	C₄	C₂³×C₄	1	4	yes	—
69	C₄	C₂³×C₄	1	12	yes	—
70	C₂	—	2	12	no	—
71	C₂	#520	1	4	yes	direct product factor
72	C₂	C₂⁴×C₄	1	1	yes	characteristic
73	C₂	C₂×H(5,𝔽₂)	1	1	yes	characteristic
74	C₂	C₂×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³	C₂⁴
commutator	C₂	C₂⁴×C₄
Frattini, cosocle	C₂²	C₂⁵
prime radical	C₂²×#41	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. #520	C₂	direct (N × H)
3. C₂×C₄×Q₈	C₂	semidirect
4. C₂×C₄×D₄	C₂	semidirect
5. C₂×#117	C₂	semidirect
6. C₂²×C₄⋊C₄	C₂	semidirect
7. C₂×#41	C₄	semidirect
13. C₄×Q₈	C₂²	semidirect
14. C₄×D₄	C₂²	semidirect
15. #117	C₂²	semidirect
19. C₂×C₄⋊C₄	C₂²	semidirect
23. #41	C₂×C₄	semidirect
24. C₂×Q₈	C₂×C₄	semidirect
26. C₂×D₄	C₂×C₄	semidirect
29. C₂²×C₄	C₂×C₄	semidirect
46. Q₈	C₂²×C₄	semidirect
47. D₄	C₂²×C₄	semidirect
53. C₂×C₄	C₂²×C₄	semidirect
71. C₂	#520	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
#520	2
C₂×C₄×Q₈	2
C₂×C₄×D₄	2
C₂×#117	2
C₂²×C₄⋊C₄	2

Subgroups — C2×#520 #3427

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#520 #3427